converges, this is a subject of Lemma 6.2. Next, to prove that the series
(6.1) converge for any $(x,y)\in\Theta$, we show that for any
$(x,y)\in\Theta$, there are two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfying
(6.6) and such that

(6.10) $x^{k_{1}}y^{k_{2}}\leqslant
x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}},\quad\forall
k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}.$

####  6.2.2. Preliminary results for the proof of Proposition 6.1.

We begin our analysis with the following statements.

######  Lemma 6.1.

Suppose that the conditions (A1)-(A3) are satisfied and let two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfy (6.6).
Then for some finite set $E\subset\mathbb{Z}^{2}_{+}$, the function
$f:\mathbb{Z}^{2}_{+}\to]0,+\infty[$ defined by (6.7) satisfies (6.8) with
some $0<\theta<1$.

###### Proof.

Consider two functions $f_{1},f_{2}:\mathbb{Z}_{+}^{2}\to\mathbb{R}$ defined
by

$f_{1}(j_{1},j_{2})=x_{1}^{j_{1}}y_{1}^{j_{2}},\quad\text{and}\quad
f_{2}(j_{1},j_{2})=x_{2}^{j_{1}}y_{2}^{j_{2}},\quad\forall(j_{1},j_{2})\in\mathbb{Z}_{+}^{2},$

and let

$\tilde{\theta}=\max\\{\phi_{1}(x_{1},y_{1}),\;P(x_{1},y_{1}),\;\phi_{2}(x_{2},y_{2}),\;P(x_{2},y_{2})\\}.$

Then $0<\tilde{\theta}<1$ because
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$, and moreover,
for any $(j=(j_{1},j_{2})\in\mathbb{Z}_{+}^{2}{\setminus}\\{0\\}$,

${\mathbb{E}}_{(j_{1},j_{2})}(f_{1}(Z(1))\leqslant\begin{cases}\tilde{\theta}f_{1}(j_{1},j_{2}),&\text{if}\;j_{1}\not=0,\\\
\phi_{2}(x_{1},y_{1})y_{1}^{j_{2}},&\text{if}\;j_{1}=0,\end{cases}$

and

${\mathbb{E}}_{(j_{1},j_{2})}(f_{2}(Z(1))\leqslant\begin{cases}\tilde{\theta}f_{2}(j_{1},j_{2}),&\text{if}\;j_{2}\not=0,\\\
\phi_{1}(x_{2},y_{2})x_{2}^{j_{1}}&\text{if}\;j_{2}=0.\end{cases}$

For the function $f=f_{1}+f_{2}$, we obtain therefore that for any
$j=(j_{1},j_{2})\in\mathbb{Z}_{+}^{2}{\setminus}\\{0\\}$,

(6.11)
${\mathbb{E}}_{(j_{1},j_{2})}(f(Z(1))\leqslant\begin{cases}\tilde{\theta}f(j_{1},j_{2})&\text{if
$j_{1}>0$ and $j_{2}>0$}\\\
\tilde{\theta}y_{2}^{j_{2}}+\phi_{2}(x_{1},y_{1})y_{1}^{j_{2}}&\text{if
$j_{1}=0$ }\\\
\tilde{\theta}x_{1}^{j_{1}}+\phi_{1}(x_{2},y_{2})x_{2}^{j_{1}}&\text{if
$j_{2}=0$}.\end{cases}$

Since under hypotheses of our lemma, $x_{1}>x_{2}>0$ and $0<y_{1}<y_{2}$, we
have moreover

$\lim_{j_{1}}(x_{2}/x_{1})^{j_{1}}=\lim_{j_{2}\to\infty}(y_{1}/y_{2})^{j_{2}}=0,$

and consequently, for any ${\varepsilon}>0$ there is $N_{\varepsilon}>0$ such
that for any $(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with $j_{2}>N_{\varepsilon}$
and $j_{1}=0$,

$\tilde{\theta}y_{2}^{j_{2}}+\phi_{2}(x_{1},y_{1})y_{1}^{j_{2}}\leqslant(\tilde{\theta}+{\varepsilon})y_{2}^{j_{2}}=(\tilde{\theta}+{\varepsilon})f_{2}(j_{1},j_{2})\leqslant(\tilde{\theta}+{\varepsilon})f(j_{1},j_{2})$

and similarly, for any $(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with
$j_{1}>N_{\varepsilon}$ and $j_{2}=0$,

$\tilde{\theta}x_{1}^{j_{1}}+\phi_{1}(x_{2},y_{2})x_{2}^{j_{1}}\leqslant(\tilde{\theta}+{\varepsilon})f(j_{1},j_{2}).$

Hence, for ${\varepsilon}>0$ such that $\tilde{\theta}+{\varepsilon}<1$,
letting

$E=\\{(j_{1},0)\in\mathbb{Z}_{+}^{2}\leavevmode\nobreak\ :\leavevmode\nobreak\
j_{1}\leqslant
N_{\varepsilon}\\}\cup\\{(0,j_{2})\in\mathbb{Z}^{2}_{+}\leavevmode\nobreak\
:\;j_{2}\leqslant N_{\varepsilon}\\}$

one gets (6.8) with $\theta=\tilde{\theta}+{\varepsilon}<1$. ∎

######  Lemma 6.2.

Under the hypotheses of Lemma 6.1, for any $j\in\mathbb{Z}_{+}^{2}$, the
series (6.9) converges.

###### Proof.

Indeed, consider two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfying
(6.6) and let the function $f:\mathbb{Z}^{2}_{+}\to]0,+\infty[$ be defined by
(6.7). Then by Lemma 6.1, for some finite set $E\subset\mathbb{Z}^{2}_{+}$ and
some positive number $\theta<1$, (6.8) holds. Without any restriction of
generality, we will suppose that $E$ contains the origin $(0,0)$. Denote by
$\tau_{E}$ the first time when the process $(Z(n))$ hits the set $E$ :

$\tau_{E}=\inf\\{n\geqslant 1:\leavevmode\nobreak\ Z(n)\in E\\},$

and let

$g_{E}(j,k)=\sum_{n=0}^{\infty}{\mathbb{P}}_{j}(Z(n)=k,\;\tau_{E}\geqslant
n),\quad j\in\mathbb{Z}_{+}^{2}{\setminus}E,\;k\in\mathbb{Z}^{2}_{+}.$

Then by (6.8), for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E$,

${\mathbb{E}}_{j}(f(Z(n));\;\tau_{E}\geqslant
n)\leqslant\theta^{n}f(j),\quad\forall n\in{\mathbb{N}},$

and consequently,

(6.12)
$\sum_{k\in\mathbb{Z}^{2}_{+}}g_{E}(j,k)\\!\\!\left(x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}}\right)=\sum_{k\in\mathbb{Z}^{2}_{+}}g_{E}(j,k)f(k)=\sum_{n=1}^{\infty}{\mathbb{E}}_{j}\bigl{(}f(Z(n));\;\tau_{E}\geqslant
n\bigr{)}\leqslant f(j)(1-\theta)^{-1}.$

Using the identity

$\displaystyle g(j,k)=g_{E}(j,k)+\sum_{\ell\in
E{\setminus}\\{0\\}}\sum_{m\in\mathbb{Z}_{+}^{2}{\setminus}E}g(j,\ell){\mathbb{P}}_{\ell}(Z(1)=m)g_{E}(m,k)$

from the above relation it follows that for any
$j\in\mathbb{Z}^{2}_{+}{\setminus}E$,

$\displaystyle\sum_{k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}}g(j,k)\bigl{(}x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}}\bigr{)}$
$\displaystyle\leqslant\frac{1}{1-\theta}\Bigl{(}f(j)+\sum_{\ell\in
E{\setminus}\\{0\\}}\sum_{m\in\mathbb{Z}_{+}^{2}{\setminus}E}g(j,\ell){\mathbb{P}}_{\ell}(Z(1)=m)f(m)\Bigr{)}$
$\displaystyle\leqslant\frac{1}{1-\theta}\Bigl{(}f(j)+\sum_{\ell\in
E{\setminus}\\{0\\}}g(j,\ell){\mathbb{E}}_{\ell}(f(Z(1)))\Bigr{)}.$

Since the set $E$ is finite and under the hypotheses (A1) and (A3),
${\mathbb{E}}_{\ell}(f(Z(1)))<+\infty$ for any $\ell\in E$, this proves that
the series (6.9) converges for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E$. To
prove that this series converges for $j\in E$, it is sufficient now to notice
that for $j\in E$,

$g(j,k)=\sum_{\ell\in
E{\setminus}\\{0\\}}g(j,\ell)\sum_{m\in\mathbb{Z}_{+}^{2}{\setminus}E}{\mathbb{P}}_{\ell}(Z(1)=m)g_{E}(m,k),$

and consequently,

$\displaystyle\sum_{k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}}g(j,k)\left(x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}}\right)$
$\displaystyle\quad\leqslant\sum_{\ell\in
E{\setminus}\\{0\\}}g(j,\ell)\sum_{m\in\mathbb{Z}_{+}^{2}{\setminus}E}{\mathbb{P}}_{\ell}(Z(1)=m)\\!\\!\\!\sum_{k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}}g_{E}(m,k)\left(x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}}\right)<+\infty.$

∎

######  Lemma 6.3.

Suppose that the conditions (A1)-(A3) are satisfied and let
$\tilde{x}\in]x^{*},x^{**}[$ and $\tilde{y}\in]y^{*},y^{**}[$ be such that

(6.13)
$\tilde{x}>X_{1}(\tilde{y})\quad\text{and}\quad\tilde{y}>Y_{1}(\tilde{x}).$

Then for any $y_{1}>Y_{1}(\tilde{x})$ closed enough to $Y_{1}(\tilde{x})$ and
any $x_{2}>X_{1}(\tilde{y})$ closed enough to $X_{1}(\tilde{y})$, the points
$(x_{1},y_{1})=(\tilde{x},y_{1})$ and $(x_{2},y_{2})=(x_{2},\tilde{y})$
satisfy the conditions of Lemma 6.1.

###### Proof.

Indeed, by Lemma 4.2, for $\tilde{x}\in]x^{*},x^{**}[$ and
$\tilde{y}\in]y^{*},y^{**}[$, each of the line segments
$[Y_{1}(\tilde{x}),\tilde{Y}_{2}(\tilde{x})]=\\{y>0:\leavevmode\nobreak\
(\tilde{x},y)\in D\cap D_{1}\\}$ and
$[X_{1}(\tilde{y}),\tilde{X}_{2}(\tilde{y})]=\\{x>0:\leavevmode\nobreak\
(x,\tilde{y})\in D\cap D_{2}\\}$ has a non-zero length, and for any $y_{1}$
and $x_{2}$ such that $Y_{1}(\tilde{x})<y_{1}<\tilde{Y}_{2}(\tilde{x})$ and
$X_{1}(\tilde{y})<x_{2}<\tilde{X}_{2}(\tilde{y})$, one has

$(\tilde{x},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}},\quad\text{and}\quad(x_{2},\tilde{y})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}.$

Using (6.13), this proves that for any $y_{1}$ and $x_{2}$ such that
$Y_{1}(\tilde{x})<y_{1}<\min\\{\tilde{Y}_{2}(\tilde{x}),\tilde{y}\\}$ and
$X_{1}(\tilde{y})<x_{2}<\min\\{\tilde{x},\tilde{X}_{2}(\tilde{y})\\}$, the
points $(x_{1},y_{1})=(\tilde{x},y_{1})$ and $(x_{2},y_{2})=(x_{2},\tilde{y})$
satisfy the conditions of Lemma 6.1. ∎

Now we are ready to get

######  Lemma 6.4.

Suppose that the conditions (A1)-(A4) are satisfied and let

(6.14)
$\Theta_{0}=\begin{cases}\bigl{(}[0,x_{d}[\times[0,Y_{1}(x_{d})[\bigr{)}\cup\bigl{(}[0,X_{1}(y_{d})[\times[0,y_{d}[\bigr{)}&\text{if
either (B0), or (B1) or (B2) holds}\\\
[0,x_{d}[\times[0,Y_{2}(x_{d})[&\text{if either (B3) or (B4) holds}\\\
[0,X_{1}(y_{d})[\times[0,y_{d}[&\text{if either (B5) or (B6) holds}.\\\
\end{cases}$

Then for any $(x,y)\in\Theta_{0}$, there are two points $(x_{1},y_{1})$ and
$(x_{2},y_{2})$ satisfying the conditions of Lemma 6.1 and relations (6.10).

###### Proof.

Suppose first that one of the cases (B0), (B1) or (B2) occurs and let
$(x,y)\in[0,x_{d}[\times[0,Y_{1}(x_{d})[$. The definitions of the cases (B0),
(B1), (B2) and of the points $x_{d}$ and $y_{d}$ (see Proposition 2.2 and
relations (2.33), (2.34)), give

$\max\\{X_{1}(y_{d}),x\\}<x_{d}=x^{**}\quad\text{and}\quad
y<Y_{1}(x_{d})<y_{d}=y^{**},$

and by (2.9),

$x^{*}_{P}\leqslant x^{*}<x^{**}=x_{d}\quad\text{and}\quad y^{*}_{P}\leqslant
y^{*}<y^{**}=y_{d}.$

Since the functions $X_{1}:[y^{*}_{P},y^{**}_{P}]\to[x^{*}_{P},x^{**}_{P}]$
and $Y_{1}:[x^{*}_{P},x^{**}_{P}]\to[y^{*}_{P},y^{**}_{P}]$ are continuous, it
follows that for any $\tilde{x}\in]\max\\{x^{*},x\\},x_{d}[$ and
$\tilde{y}\in]y^{*},y_{d}[$ closed enough respectively to $x_{d}$ and $y_{d}$
one has

(6.15) $0\leqslant x<\tilde{x},\quad\quad 0\leqslant y<Y_{1}(\tilde{x}),$
(6.16) $X_{1}(\tilde{y})<\tilde{x}\quad\text{and}\quad
Y_{1}(\tilde{x})<\tilde{y}.$

By Lemma 6.3, from (6.16) it follows that for $y_{1}>Y_{1}(\tilde{x})$ and
$x_{2}>X_{1}(\tilde{y})$ closed enough respectively to $Y_{1}(\tilde{x})$ and
$X_{1}(\tilde{y})$, the points $(x_{1},y_{1})=(\tilde{x},y_{1})$ and
$(x_{2},y_{2})=(x_{2},\tilde{y})$ satisfy the conditions of Lemma 6.1, and
using (6.15) one gets (6.10).

When one of the cases (B0), (B1) or (B2) holds and
$(x,y)\in[0,x_{d}[\times[0,Y_{1}(x_{d})[$, our lemma is therefore proved. For
$(x,y)\in[0,X_{1}(y_{d})[\times[0,y_{d}[$, the proof is quite similar.

Suppose now that either (B3) or (B4) holds. The definitions of the cases (B3)
and (B4) and of the points $x_{d}$ and $y_{d}$ (see Proposition 2.2 and
relations (2.33), (2.34)), and by (2.9), give

(6.17) $x^{*}_{P}\leqslant x^{*}<x^{**}=x_{d}\leqslant X_{1}(y^{**}),$ (6.18)
$y^{*}\leqslant 1<y_{d}=Y_{2}(x_{d})\leqslant y^{**},$ (6.19)
$(X_{1}(y^{**}),y^{**}),\,(x^{**},Y_{2}(x^{**}))=(x_{d},y_{d})\in{\mathcal{S}}_{12},$

and

(6.20) $Y_{1}(x_{d})<Y_{2}(x_{d}).$

From the definition of ${\mathcal{S}}_{12}$ and relation (6.19), it follows
that $X_{1}(y^{**})\leqslant X_{1}(y^{**}_{P})$ and consequently, by (6.17),

(6.21) $x^{*}_{P}\leqslant x^{*}<x_{d}\leqslant X_{1}(y^{**})\leqslant
X_{1}(y^{**}_{P}).$

Consider now a point $(x,y)\in\Theta_{0}$. Then by (6.14) and using (6.21),
(6.18) and (6.20),

(6.22) $0\leqslant\max\\{x^{*},x\\}<x_{d}\leqslant
X_{1}(y^{**}_{P})\quad\text{and}\quad
0\leqslant\max\\{Y_{1}(x_{d}),y^{*},y\\}<y_{d}=Y_{2}(x_{d})\leqslant y^{**}$

Since the function $Y_{2}$ is strictly increasing on the line segment
$[x^{*}_{P},X_{1}(y^{**}_{P})]$ and the functions $Y_{1}$ and $Y_{2}$ are
continuous on $[x^{*}_{P},x^{**}_{P}]$, it follows that for any $\hat{x}$ and
$\tilde{x}$ closed enough to $x_{d}$ and such that

(6.23) $\max\\{x^{*},x\\}<\hat{x}<\tilde{x}<x^{**}=x_{d},$

one has

$\max\\{Y_{1}(\tilde{x}),y^{*},y\\}<Y_{2}(\hat{x})<Y_{2}(\tilde{x})<Y_{2}(x_{d})=y_{d}\leqslant
y^{**},$

Remark that because of (6.21) and (6.23), the points $\hat{x}$ and $\tilde{x}$
belong to the line segment $[x^{*}_{P},X_{1}(y^{**}_{P})]$. Since the
functions
$X_{1}:[Y_{1}(x^{*}_{P}),y^{**}_{P}]\to[x^{*}_{P},X_{1}(y^{**}_{P})]$ and
$Y_{2}:[x^{*}_{P},X_{1}(y^{**}_{P})]\to[Y_{1}(x^{*}_{P}),y^{**}_{P}]$ are
inverse to each other, letting $\tilde{y}=Y_{2}(\hat{x})$ we get therefore
$X_{1}(\tilde{y})=\hat{x}$ and using the above relations we obtain

(6.24) $y^{*}<\tilde{y}=Y_{2}(\hat{x})<y^{**},\quad
Y_{1}(\tilde{x})<Y_{2}(\hat{x})=\tilde{y},\quad
X_{1}(\tilde{y})=\hat{x}<\tilde{x},$ (6.25)
$x<\hat{x}=X_{1}(\tilde{y})\quad\text{and }\quad y<Y_{2}(\hat{x})=\tilde{y}.$

By Lemma 6.3, from (6.23) and (6.24) it follows that for any
$y_{1}>Y_{1}(\tilde{x})$ closed enough to $Y_{1}(\tilde{x})$ and
$x_{2}>X_{1}(\tilde{y})$ closed enough to $X_{1}(\tilde{y})$, the points
$(x_{1},y_{1})=(\tilde{x},y_{1})$ and $(x_{2},y_{2})=(x_{2},\tilde{y})$
satisfy the conditions of Lemma 6.1, and using moreover (6.25), we get that
for any $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$,

$x^{j_{1}}y^{j_{2}}\leqslant(X_{1}(\tilde{y}))^{j_{1}}\tilde{y}^{j_{2}}\leqslant
x_{2}^{j_{1}}y_{2}^{j_{2}}\leqslant
x_{1}^{j_{1}}y_{1}^{j_{2}}+x_{2}^{j_{1}}y_{2}^{j_{2}}.$

Hence, in the case when either (B3) or (B4) holds, Lemma 6.3 is also proved.
To prove this lemma the case when either (B5) or (B6) holds, it is sufficient
to exchange the roles of $x$ and $y$. ∎

####  6.2.3.  Proof of Proposition 6.1

This proposition is a consequence of Lemma 6.2 and Lemma 6.3: by Lemma 6.3,
for any $(x,y)\in\Theta_{0}$, there are two points $(x_{1},y_{1})$ and
$(x_{2},y_{2})$ for which the conditions of Lemma 6.1 are satisfied and
relations (6.10) hold. By Lemma Lemma 6.2 and using (6.10), this proves that

$\sum_{k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}}g(j,k)x^{k_{1}}y^{k_{2}}\leqslant\sum_{k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}}g(j,k)\left(x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}}\right)<+\infty.$

Hence, for any $(x,y)\in\Theta_{0}$, the series (6.1) converge. Since the set
$\Theta$ is the logarithmically convex hull of the set $\Theta_{0}$, and the
domain of convergence of power series with center $0$ is always
logarithmically convex, this proves that the series (6.1) converge in
$\Omega(\Theta)$.

###  6.3. Proof of Proposition 6.3

Consider first the case when $j=(j_{1},j_{2})\not=(0,0)$. By Proposition 6.1,
the series (6.1) converge on the set $\Omega(\Theta)$. Hence, for any
$(x,y)\in\Omega(\Theta)$ with non-zero $x$ and $y$, by the Fubini theorem and
using the Markov property, one gets

$\displaystyle H_{j}(x,y)$
$\displaystyle=x^{j_{1}}y^{j_{2}}+\sum_{k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}}\sum_{n=1}^{\infty}{\mathbb{P}}_{j}(Z(n)=k,\;\tau_{0}>n)x^{k_{1}}y^{k_{2}}$
(6.26)
$\displaystyle=x^{j_{1}}y^{j_{2}}+\sum_{\ell\in\mathbb{Z}^{2}_{+}{\setminus}\\{(0,0)\\}}g(j,\ell){\mathbb{E}}_{\ell}\left(x^{Z_{1}(1)}y^{Z_{2}(1)},\;\tau_{0}>1\right)$

Because of Assumptions (A1)-(A3), for $(x,y)\in\Omega(\Theta)$ with $x\not=0$
and $y\not=0$, we have

${\mathbb{E}}_{\ell}\left(x^{Z_{1}(1)}y^{Z_{2}(1)},\;\tau_{0}>1\right)=\begin{cases}x^{\ell_{1}}y^{\ell_{2}}P(x,y)-\mu(-1,-1)\mathbb{1}_{\\{(1,1)\\}}(\ell_{1},\ell_{2})&\text{if
$\ell_{1}>0$ and $\ell_{2}>0$,}\\\
x^{\ell_{1}}\,\phi_{1}(x,y)-\mu_{1}(-1,0)\mathbb{1}_{\\{(1,0)\\}}(\ell_{1},\ell_{2})&\text{if
$\ell_{1}>0$ and $\ell_{2}=0$,}\\\
y^{\ell_{2}}\,\phi_{2}(x,y)-\mu_{2}(0,-1)\mathbb{1}_{\\{(0,1)\\}}(\ell_{1},\ell_{2})&\text{if
$\ell_{1}=0$ and $\ell_{2}>0$.}\end{cases}$

Using these relations in (6.26), for $(x,y)\in\Omega(\Theta)$ with non-zero
$x$ and $y$, one gets therefore

$\displaystyle H_{j}(x,y)$
$\displaystyle=x^{j_{1}}y^{j_{2}}-\mu(-1,-1)g(j,(1,1))-\mu_{1}(-1,0)g(j,(1,0))-\mu_{2}(0,-1)g(j,(0,1))$
$\displaystyle+P(x,y)\sum_{\begin{subarray}{c}\ell=(\ell_{1},\ell_{2})\in\mathbb{Z}^{2}_{+}:\\\
\ell_{1}>0,\;\ell_{2}>0\end{subarray}}g(j,\ell)x^{\ell_{1}}y^{\ell_{2}}+\phi_{1}(x,y)\\!\\!\sum_{\ell_{1}=1}^{\infty}g(j,(\ell_{1},0))x^{\ell_{1}}+\;\phi_{2}(x,y)\\!\\!\sum_{\ell_{2}=1}^{\infty}g(j,(0,\ell_{2}))y^{\ell_{2}}$

or equivalently,

$H_{j}(x,y)=L_{j}(x,y)+xyP(x,y)h_{j}(x,y)+x\phi_{1}(x,y)h_{1j}(x)+y\phi_{2}(x,y)h_{2j}(y)\\\
$

with

$\displaystyle L_{j}(x,y)$
$\displaystyle=x^{j_{1}}y^{j_{2}}-\mu(-1,-1)g(j,(1,1))-\mu_{1}(-1,0)g(j,(1,0))-\mu_{2}(0,-1)g(j,(0,1))$
$\displaystyle=x^{j_{1}}y^{j_{2}}-{\mathbb{P}}_{j}(\tau_{0}<+\infty).$

Since clearly $H_{j}(x,y)=xyh_{j}(x,y)+xh_{1j}(x)+yh_{2j}(y)$, the last
relation proves (2.30) for any $(x,y)\in\Omega(\Theta)$ and
$j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}{{\setminus}}\\{(0,0)\\}$. To get (2.30)
for $(j_{1},j_{2})=(0,0)$, it is sufficient now to notice that for any
$(x,y)\in\Omega(\Theta)$,

$H_{(0,0)}(x,y)=\sum_{(j_{1},j_{2})\in\mathbb{Z}_{+}^{2}{\setminus}\\{(0,0)\\}}{\mathbb{P}}_{(0,0)}\bigl{(}Z(1)=(j_{1},j_{2})\bigr{)}H_{j}(x,y)$

and

$\displaystyle\sum_{(j_{1},j_{2})\in\mathbb{Z}_{+}^{2}{\setminus}\\{(0,0)\\}}{\mathbb{P}}_{(0,0)}\bigl{(}Z(1)=(j_{1},j_{2})\bigr{)}L_{j}(x,y)$
$\displaystyle={\mathbb{E}}_{(0,0)}\left(x^{Z_{1}(1)}y^{Z_{2}(1)};\;\tau_{0}>1\right)-{\mathbb{P}}_{(0,0)}(1<\tau_{0}<+\infty)$
$\displaystyle=\phi_{0}(x,y)-{\mathbb{P}}_{(0,0)}(\tau_{0}<+\infty).$

###  6.4. Proof of Proposition 6.4

Suppose that one of the cases (B0)-(B2) holds. Then by Proposition 2.2 and the
definition of the points $x_{d}$, $y_{d}$, the following relations hold

(6.27) $x_{d}=x^{**}>X_{1}(y_{d}),\quad\quad y_{d}=y^{**}>Y_{1}(x_{d}),$

and the set $\Theta$ is defined as a logarithmically convex hull of the set

$([0,x_{d}[\times[0,Y_{1}(x_{d})[)\cup([0,X_{1}(y_{d})[\times[0,y_{d}[).$

The points $(x_{1},y_{2})=(x_{d},Y_{1}(x_{d}))$ and
$(x_{2},y_{2})=(X_{1}(y_{d}),y_{d})$ are therefore on the boundary of the set
$\Theta$ and also on the boundary of the set $D$. Since the set $\Theta$ is
logarithmically convex and the set $D$ is strictly logarithmically convex, it
follows that for any $0<\theta<1$, the point $(x_{\theta},y_{\theta})$, with
$x_{\theta}=x_{1}^{\theta}x_{2}^{1-\theta}$ and
$y_{\theta}=y_{1}^{\theta}y_{2}^{1-\theta}$, belongs to the set
$\overline{\Theta}\cap\overset{\circ}{D}$, and consequently,

$Y_{1}(x_{\theta})<y_{\theta}<Y_{2}(x_{\theta}).$

By the definition of the set $\Theta$, for any $0<\theta<1$ and
$y\in[0,y_{\theta}[$, the point $(x_{\theta},y)$ is in $\Theta$, and for any
$y\in]Y_{1}(x_{\theta}),Y_{2}(x_{\theta})[$ and the point $(x_{\theta},y)$ is
in $\overset{\circ}{D}$. Consequently, for any $0<\theta<1$ and
$y\in[0,Y_{2}(x_{\theta})]$, the point $(x_{\theta},y)$ belongs to the set
$\overline{\Theta}\cup\overset{\circ}{D}$, or equivalently, that for any
$y\geqslant 0$,

(6.28) $y\geqslant
Y_{2}(x_{\theta})\quad\text{whenever}\quad(x_{\theta},y)\not\in\overline{\Theta}\cup\overset{\circ}{D}.$

Consider now a point $(\hat{x},\hat{y})\in[0,+\infty[^{2}$ with
$\hat{x}<x_{d}$ and $\hat{y}<y_{d}$ and such that
$(\hat{x},\hat{y})\not\in(\Theta\cup\overset{\circ}{D})$. Then
$\hat{x}\geqslant X_{1}(y_{d})$ (because otherwise
$(\hat{x},\hat{y})\in[0,X_{1}(y_{d})[\times[0,y_{d})[\subset\Theta$), and
consequently for some $0<\theta\leqslant 1$,
$\hat{x}=x_{\theta}=x_{d}^{\theta}(X_{1}(y_{d}))^{1-\theta}$. Hence, by
(6.28), we get $\hat{y}\geqslant Y_{2}(\hat{x})$, and with similar arguments
(it is sufficient to exchange the roles of $x$ and $y$) we obtain
$\hat{x}\geqslant X_{2}(\hat{y})$. By Lemma 5.2, these inequalities show that
$(\hat{x},Y_{2}(\hat{x})),(X_{2}(\hat{y}),\hat{y})\in{\mathcal{S}}_{22}$.
Hence, if we suppose that for some $(x^{\prime},y^{\prime})\in D$, the
inequalities $\hat{x}\leqslant x^{\prime}$ and $\hat{y}\leqslant y^{\prime}$
hold, then using the similar arguments as in the proof of Lemma 5.4, we will
obtain that $\hat{x}=x^{\prime}=X_{2}(\hat{y})$ and
$\hat{y}=y^{\prime}=Y_{2}(\hat{x})$. With the definition of the set
${\Gamma}$, this proves that $(\hat{x},\hat{y})\not\in{\Gamma}$.

###  6.5. Proof of Theorem 1

Now, we are ready to complete the proof of Theorem 1. In the cases (B3)-(B6),
Theorem 1 follows from Corollary 6.2 and Proposition 6.3, and in the cases
(B0)-(B2), the first assertion of Theorem 1 follows from Corollary 6.2 as
well.To complete the proof of Theorem 1 we have therefore to prove its second
assertion in the cases (B0)-(B2).

We know that, for any $j\in\mathbb{Z}^{2}_{+}$, the function $(x,y)\to
L_{j}(x,y)$ is analytic in $\Omega({\Gamma})$, the functions $(x,y)\to
Q(x,y)=xy(1-P(x,y)$, $(x,y)\to\psi_{1}(x,y)=x(\phi_{1}(x,y)-1)$ and
$(x,y)\to\psi_{2}(x,y)=y(1-\phi_{2}(x,y))$ can be analytically continued to
the set $\Omega({\Gamma})$, and by Corollary 6.2, the functions $h_{1j}$ and
$h_{2j}$ are analytic respectively in the discs $B(0,x_{d})$ and $B(0,y_{d})$.
Hence for any $j\in\mathbb{Z}^{2}_{+}$, the function $(x,y)\to
L_{j}(x,y)+\psi_{1}(x,y)h_{1j}(x)+\psi_{2}(x,y)h_{2j}(y)$ is analytic in the
set $\\{(x,y)\in\Omega({\Gamma}):\leavevmode\nobreak\
|x|<x_{d},\;|y|<y_{d}\\}$. Since by Proposition 6.3, on the set
$\Omega(\Theta)$,

$Q(x,y)h_{j}(x,y)=L_{j}(x,y)+\psi_{1}(x,y)h_{1j}(x)+\psi_{2}(x,y)h_{2j}(y),$

and since clearly $\Theta\subset\\{(x,y)\in{\Gamma}:\leavevmode\nobreak\
x<x_{d},y<y_{d}\\}$, we conclude therefore that the functions $(x,y)\to
Q(x,y)h_{j}(x,y)$ can be analytically continued to the set

$\Omega_{d}(\Gamma)\stackrel{{\scriptstyle\text{\rm
def.}}}{{=}}\\{(x,y)\in\Omega({\Gamma}):\leavevmode\nobreak\
|x|<x_{d},\;|y|<y_{d}\\}$

and the function $(x,y)\to h_{j}(x,y)$ can be continued as a meromorphic
function

$h_{j}(x,y)=\frac{L_{j}(x,y)+\psi_{1}(x,y)h_{1j}(x)+\psi_{2}(x,y)h_{2j}(y)}{Q(x,y)}.$

to the set $\Omega_{d}(\Gamma)$. Since

  * –

by Proposition 6.1, for any $j\in\mathbb{Z}^{2}_{+}$, the function $h_{j}$ is
analytic in $\Omega(\Theta)$;

  * –

for any $(x,y)\in\Omega(\overset{\circ}{D})$, by the definition of the set
$D$, we have $|P(x,y)|\leqslant P(|x|,|y|)<1$ and, consequently,
$Q(x,y)\not=0$;

  * –

and by Proposition 6.4, the set $\Omega_{d}(\Gamma)$ is included to the union
of the open sets $\Omega(\Theta)$ and $\Omega(\overset{\circ}{D})$,

we conclude therefore that the function $h_{j}$ can be analytically continued
to the set $\Omega_{d}(\Gamma)$.

##  7\. Singularity Analysis of Generating Functions

This section is devoted to the proof of Theorem 2. Recall that the case (B7)
will not be considered. See Section 2.2. Throughout this section we will
assume therefore that

Assumption (A4) one of the cases (B0)-(B6) holds.

###  7.1. The main ideas and the sketch of the proof

The main steps of our proof are the following. First, we prove the light
version of the assertion (i)-(v) of this theorem, i.e. we get, in each of the
corresponding cases, relations (2.40), (2.44) (2.42), (2.46), (2.48), (2.50),
(2.52), and (2.54) with positive constants given by (2.41), (2.43), (2.45),
(2.47), (2.49), (2.51), (2.53). This is a subject of Propositions 7.10 and
7.12 and Lemma 7.6 below.

With these results we will be able to show that in each of the assertions
(i)-(vi) of our theorem, the corresponding function $\varkappa_{i}$,
$\tilde{\varkappa}_{i}$ or $\varkappa_{(\hat{x},\hat{y})}$ is non-negative on
$\mathbb{Z}^{2}_{+}$.

Second, in Proposition 7.13 below, we prove that in each of the assertions
(i)-(vi), the corresponding function $\varkappa_{i}$, $\tilde{\varkappa}_{i}$
or $\varkappa_{(\hat{x},\hat{y})}$ is harmonic for the random walk
$({Z}_{\tau_{0}}(n))$ and positive everywhere in the set
$\mathbb{Z}^{2}_{+}{\setminus}E_{0}$. With this statement, the proof of our
theorem will be completed.

The first step of the proof of the last assertion of Theorem 2 is given by
Lemma 7.6. This result is obtained as a traightforward consequence of Theorem
1.

The first step of the proof of the assertions (i)-(v) of Theorem 2 proof is
the most difficult. Its main idea is the following: we extend the function
$Y_{1}$ as an analytic function to some domain of $\mathbb{C}$ large enough
and we inject next $y=Y_{1}(x)$ to the functional equation (2.30) in order to
get the identity

(7.1)
$L_{j}(x,Y_{1}(x))+\psi_{1}(x,Y_{1}(x))h_{1j}(x)+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))=0.$

Using this result we will be able to extend beyond the point $x_{d}$ first the
function $x\to\psi_{1}(x,Y_{1}(x))h_{1j}(x)=(\phi_{1}(x,Y_{1}(x)-1)H_{j}(x,0)$
and next the function $x\to H_{j}(x,0)$.

Remark that in a difference with nearest neighbor random walks, we have no
explicit form of the function $Y_{1}$. Moreover, by the implicit function
theorem, this function can be extended only to some neighborhood of the
interval $]x^{*}_{P},x^{**}_{P}[$ in $\mathbb{C}$ which is clearly not
sufficient for our purpose. The first difficulty of our proof is therefore to
extend the function $Y_{1}$ to some sufficient for our analysis domain of
$\mathbb{C}$. We perform this first step of our proof by using the
probabilistic representation of the function
$Y_{1}:[x^{*}_{P},x^{**}_{P}]\to[y^{*}_{P},y^{**}_{P}]$ obtained in [13]. This
is a subject of Proposition 7.1 below. In this statement, for some
${\varepsilon}>0$, we extend the function $Y_{1}$ as an analytic function to
the set

$\overset{\circ}{U}_{{\varepsilon}}=C(x^{*}_{P},x^{**}_{P}+{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}[,$

and a continuous function on the set

(7.2) $U_{{\varepsilon}}\leavevmode\nobreak\ =\leavevmode\nobreak\
C(x^{*}_{P},x^{**}_{P}+{\varepsilon}){\setminus}]x^{**}_{P},x^{**}_{P}+{\varepsilon}[\leavevmode\nobreak\
=\leavevmode\nobreak\ \overset{\circ}{U}_{{\varepsilon}}\cup\\{x^{**}_{P}\\}.$

It is proved moreover that the extended function $Y_{1}$ satisfies the
identity

(7.3) $Q(x,Y_{1}(x))=0,$

on the set $U_{\varepsilon}$, and that, on the closed annulus
$\overline{C}(x^{*}_{P},x^{**}_{P})$, the following relations holds

(7.4) $|Y_{1}(x)|<Y_{1}(|x|)\quad\text{whenever}\quad x\not=|x|.$

Another difficulty is that by Theorem 1, we know only that the function $y\to
h_{2j}(y)$ is analytic in $B(0,y_{d})$, and the function $(x,y)\to h_{j}(x,y)$
is analytic in $\\{(x,y)\in\Omega({\Gamma}):\leavevmode\nobreak\
|x|<x_{d},\;|y|<y_{d}\\}$. Hence, we can inject $y=Y_{1}(x)$ to the functional
equation (2.30) only for those $x\in B(0,x_{d})$ for which $|Y_{1}(x)|<y_{d}$.
To overcome this difficulty, we use the inequality (7.4). With this
inequality, we are able to show that for some $\delta>0$, the function
$x\to(x,Y_{1}(x))$ maps the annulus
$C(x_{d}-\delta,x_{d})=\\{x\in\mathbb{C}:x_{d}-\delta<|x|<x_{d}\\}$ to the set
$\\{(x,y)\in\Omega({\Gamma}):\leavevmode\nobreak\ |x|<x_{D},\,|y|<y_{d}\\}$
where the functions $(x,y)\to H_{j}(x,y)$, $(x,y)\to
L_{j}(x,y)+\psi_{2}(x,y)h_{2j}(y)$ and $(x,y)\to\psi_{1}(x,y)h_{1j}(x)$ are
analytic. This is a subject of Proposition 7.4 below.

In this way, on the annulus $C(x_{d}-\delta,x_{d})$, by letting in the
functional equation (2.30) $y=Y_{1}(x)$ and using (7.3) we obtain the identity
(7.1) with analytic in $C(x_{d}-\delta,x_{d})$ functions
$x\to\psi_{1}(x,Y_{1}(x))h_{1j}(x)$ and $x\to L_{j}(x,Y_{1}(x))$ and
$x\to\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$. Moreover, when one of the cases
(B0)-(B4) holds, using again the inequality (7.4), we will be able to show
that for some $\delta>0$, the function $(x,y)\to
L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in
$C(x_{d}-\delta,x_{d}+\delta)\cap\overset{\circ}{U}_{\varepsilon}$ and
continuous on $C(x_{d}-\delta,x_{d}+\delta)\cap{U}_{\varepsilon}$. Using this
result together with the identity (7.1), we will extend the function
$x\to-\psi_{1}(x,Y_{1}(x))h_{1j}(x)=(1-\phi_{1}(x,Y_{1}(x))H_{j}(x,0)$ as an
analytic function to the set
$C(x_{d}-\delta,x_{d}+\delta)\cap\overset{\circ}{U}_{\varepsilon}$ and a
continuous function to the set
$C(x_{d}-\delta,x_{d}+\delta)\cap{U}_{\varepsilon}$. This is a subject of
Corollary 7.6 below.

Next, in Proposition 7.7, we investigate the function $x\to
1-\phi_{1}(x,Y_{1}(x)))$, and finally, in Proposition 7.10 and Proposition
7.11 below, when one of the cases (B0)-(B4) holds, we extend the function
$x\to H_{j}(x,0)$ beyond the point $x_{d}$.

In the case when either (B5) or (B6) holds, we begin our analysis by
investigating the function $h_{2j}$. With the same arguments as in the
previous cases (it is sufficient to exchange the roles of $x$ and $y$) we
obtain that the function $h_{2j}$ can be extended as an analytic function to
the set $B(0,y_{d}+\delta_{0}){\setminus}\\{y_{d}\\}$. Next we show that for
some $\delta>0$, the function $x\to h_{2j}(Y_{1}(x))$ is analytic in the set
$C(x_{d}-\delta,x_{d}+\delta)\cap U_{\varepsilon}{\setminus}\\{x_{d}\\}$. And
finally, using again the identity (7.1) we extend first the function
$x\to\psi_{1}(x,Y_{1}(x))h_{1j}(x)$ and next the function $x\to h_{1j}(x)$
beyond the point $x_{d}$. This is a subject of Proposition 7.12 below.

###  7.2. Analytic continuation and properties of the function $x\to Y_{1}(x)$

We begin our analysis with the following result.

######  Proposition 7.1.

Analytic continuation of the function $Y_{1}$. Suppose the conditions (A1) are
satisfied and let $\mu(j)=0$ for all $j\in\mathbb{Z}^{2}$ with $j_{2}<-1$
(remark that we do not need the whole condition (A2) to be satisfied). Then

  1. i)

the function $Y_{1}$ is strictly convex on the line segment
$[x^{*}_{P},x^{**}_{P}]$ and for some ${\varepsilon}>0$, it can be extended to
the set $U_{\varepsilon}$ as an analytic function in the set
$\overset{\circ}{U}_{\varepsilon}$ and a continuous function on the set
$U_{\varepsilon}$ satisfying there the identity (7.3).

  2. ii)

relation (7.4) holds on the set $\overline{C}(x^{*}_{P},x^{**}_{P})$;

  3. iii)

for any $\hat{x}\in]X_{1}(y^{*}_{P}),x^{**}_{P}[$, the function
$x\to(Y_{1}(x)-Y_{1}(\hat{x})$ does not vanish in the set
$C(\hat{x}-\delta,\hat{x}+\delta){\setminus}\\{\hat{x}\\}$ for some
$\delta>0$, and has at the point $\hat{x}$ a simple zero with

(7.5)
$\frac{d}{dx}Y_{1}(\hat{x})=-\left.\partial_{x}P(x,y)/\partial_{y}P(x,y)\right|_{(x,y)=(\hat{x},Y_{1}(\hat{x}))}>0;$

  4. iv)

the function $x\to Y_{1}(x)-Y_{1}(x^{**}_{P})$ does not vanish in the set
$C(x^{**}_{P}-\delta,x^{**}_{P}+\delta){\setminus}[x^{**}_{P},x^{**}_{P}+\delta[$
for some $\delta>0$, and as $x\to x^{**}_{P}$,

(7.6) $Y_{1}(x)-Y_{1}(x^{**}_{P})\sim-c\sqrt{x^{**}_{P}-x}\quad\;\text{and
}\;\quad\frac{d}{dx}Y_{1}(x)\sim\frac{c}{2\sqrt{x^{**}_{P}-x}}$

with

$c=\left.\sqrt{\partial_{x}P(x,y)/\partial^{2}_{yy}P(x,y)}\,\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0.$

####  7.2.1. The main ideas and the sketch of the proof of Proposition 7.1

To prove this result, we first get a probabilistic representation of the
function $Y_{1}$. As a consequence, we obtain that the function $Y_{1}$ is
strictly convex on the line segment $[x^{*}_{P},x^{**}_{P}]$ and can be
continued as an analytic function to the open annulus
$C(x^{*}_{P},x^{**}_{P})$ and as a continuous function to the closed annulus
$\overline{C}(x^{*}_{P},x^{**}_{P})$ satisfying there the inequality (7.4) and
the identity (7.3). This is a subject of the results of Subsection 7.2.2.

Next, in Subsection 7.2.3, we extend the functions $Y_{1}$ and $Y_{2}$ to a
neighborhood of $x^{**}_{P}$ as two branches of a two-valued analytic function
having a branching point $x^{**}_{P}$ and we get (7.6). This result is
obtained by using the implicit function theorem and the Morse lemma.

Finally, in Subsection 7.2.4, the proof of Proposition 7.1 is completed: by
using the implicit function theorem and relations (7.4) and (7.3) on the
circle $\\{x\in\mathbb{C}:|x|=x^{**}_{P}\\}$, we extend the function $Y_{1}$
and the identity (7.3) to the whole set
$U_{\varepsilon}=C(x^{*}_{P},x^{**}_{P}+{\varepsilon}){\setminus}]x^{**}_{P},x^{**}_{P}+{\varepsilon}[$
for some ${\varepsilon}>0$, and we prove the two last assertions of
Proposition 7.1.

####  7.2.2. Probabilistic representation of the functions $Y_{1}$ its
consequences

Consider the homogeneous random walk $(S(n)=(S_{1}(n),S_{2}(n))$ on
$\mathbb{Z}^{2}$ with transition probabilities

${\mathbb{P}}_{j}(S(1)=k)=\mu(k-j),\quad\forall k,j\in\mathbb{Z}^{2},$

and the first time $\tau_{1}$ when the random walk $(S(n))$ hits the set
$\mathbb{Z}\times\\{0\\}$ :

$\tau_{1}=\inf\\{n\geqslant 1\leavevmode\nobreak\ :\leavevmode\nobreak\
S(n)\in\mathbb{Z}\times\\{0\\}\\},$

By Lemma 2.2 of [13]

######  Lemma 7.1.

Under the hypotheses of Proposition 7.1, for any
$j=(j_{1},j_{2})\in\mathbb{Z}\times{\mathbb{N}}^{*}$ and
$x\in[x^{*}_{P},x^{**}_{P}]$,

${\mathbb{E}}_{j}\left(x^{S_{1}(\tau_{1})};\;\tau_{1}<+\infty\right)=x^{j_{1}}Y_{1}(x)^{j_{2}}.$

As a straightforward consequence of this statement one gets the following
probabilistic representation of the function $Y_{1}$ :

######  Corollary 7.2.

Under the hypotheses (A1) and (A2), for any $x\in[x^{*}_{P},x^{**}_{P}]$,

(7.7)
$Y_{1}(x)={\mathbb{E}}_{(0,1)}\left(x^{S_{1}(\tau_{1})};\;\tau_{1}<+\infty\right)$

With this result we get

######  Lemma 7.2.

Under the hypotheses of Proposition 7.1, the function $Y_{1}$ is strictly
convex on $[x^{*}_{P},x^{**}_{P}]$ and can be extended to the set
$\overline{C}(x^{*}_{P},x^{**}_{P})$ as a function

(7.8)
$Y_{1}(x)=\sum_{k_{1}\in\mathbb{Z}}{\mathbb{P}}_{(0,1)}(S(\tau_{1})=(k_{1},0),\;\tau_{1}<+\infty)x^{k_{1}}$

which is continuous on $\overline{C}(x^{*}_{P},x^{**}_{P})$, analytic in
${C}(x^{*}_{P},x^{**}_{P})$ and, on the set
$\overline{C}(x^{*}_{P},x^{**}_{P})$, satisfies the inequality (7.4) and the
identity (7.3).

###### Proof.

Indeed, by Corollary 7.2, the series

(7.9)
$\sum_{k_{1}\in\mathbb{Z}}{\mathbb{P}}_{0,1)}(S(\tau_{1})=(k_{1},0),\;\tau_{1}<+\infty)x^{k_{1}}={\mathbb{E}}_{(0,1)}\left(x^{S_{1}(\tau_{1})};\;\tau_{1}<+\infty\right)$

converges and the identity (7.8) holds for any real
$x\in[x^{*}_{P},x^{**}_{P}]$. Since the coefficients of the series (7.9) are
real and non-negative, it follows that this series converges also on
$\overline{C}(x^{*}_{P},x^{**}_{P})$. Hence, by letting (7.8) for
$x\in\overline{C}(x^{*}_{P},x^{**}_{P})$, the function $Y_{1}$ can be extended
to the set $\overline{C}(x^{*}_{P},x^{**}_{P})$ as a function which is
continuous on $\overline{C}(x^{*}_{P},x^{**}_{P})$ and analytic in
${C}(x^{*}_{P},x^{**}_{P})$.

To get (7.4) and to show that the function $Y_{1}$ is strictly convex on
$[x^{*}_{P},x^{**}_{P}]$, we show that all coefficients of the series (7.9)
are strictly positive. For this we recall that under our hypotheses, the
random walk $(S(n))$ is irreducible on $\mathbb{Z}^{2}$ and hence, for any
$k_{1}\in\mathbb{Z}$, there are $n\in{\mathbb{N}}$ and a sequence of points
$\ell_{0},\ldots,\ell_{n}\in\mathbb{Z}^{2}$ with $\ell_{0}=(0,1)$ and
$\sum_{i=0}^{n}\ell_{i}=(k_{1},0)$ such that

$\mu(\ell_{i})>0\quad\text{for any $i\in\\{1,\ldots,n\\}$}.$

Moreover, without any restriction of generality, one can assume that for some
$j\in\\{1,\ldots,n\\}$, the second coordinate of each of the points
$\ell_{1},\ldots,\ell_{j}$ is either zero or positive, and the second
coordinate of each of the points $\ell_{j+1},\ldots,\ell_{n}$ strictly
negative. Then for any $j\in\\{0,\ldots,n-1\\}$, the second coordinate of the
point $\ell_{0}+\cdots+\ell_{n}$ is strictly positive and consequently,

${\mathbb{P}}_{0,1}(S(\tau_{1})=(k_{1},0),\;\tau_{1}<+\infty)\geqslant\prod_{i=1}^{n}\mu(\ell_{i})>0.$

All coefficients of f the series (7.9) are therefore strictly positive.

Since for any $k_{1}\in\mathbb{Z}$, the function
$x\to{\mathbb{P}}_{(2,0)}(S(\tau_{1})=(k_{1},0),\;\tau_{1}<+\infty)x^{k_{1}}$
is convex on $[x^{*}_{P},x^{**}_{P}]$ and the function
$x\to{\mathbb{P}}_{(2,0)}(S(\tau_{1})=(2,0),\;\tau_{1}<+\infty)x^{2}$ is
strictly convex on $[x^{*}_{P},x^{**}_{P}]$, this proves that the function
$Y_{1}$ is strictly convex on $[x^{*}_{P},x^{**}_{P}]$. And using moreover
Proposition P7.5 of [24] we get (7.4).

Remark finally that by relation (7.4), the function $x\to(x,Y_{1}(x))$ maps
the closed annulus $\overline{C}(x^{*}_{P},x^{**}_{P})$ to the set

$\overline{\Gamma}_{1}=\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\
x\in\overline{C}(x^{*}_{P},x^{**}_{P}),\;|y|\leqslant Y_{1}(|x|)\\}.$

Since under the hypotheses of Proposition 7.1, the function

$Q(x,y)=xy-\sum_{k=(k_{1},k_{2})\in{\mathbb{N}}^{2}}\mu(k_{1}-1,k_{2}-1)x^{k_{1}}y^{k_{2}}$

is analytic in a neighborhood of the set $\overline{\Gamma}_{1}$, and the
function $Y_{1}$ is already extended as an analytic function to the open
annulus $C(x^{*}_{P},x^{**}_{P})$ and as a continuous function to the closed
annulus $\overline{C}(x^{*}_{P},x^{**}_{P})$, it follows that the function
$x\to Q(x,Y_{1}(x))$ is analytic in $C(x^{*}_{P},x^{**}_{P})$ and continuous
on $\overline{C}(x^{*}_{P},x^{**}_{P})$. Since moreover the identity (7.3)
holds for any real $x\in[x^{*}_{P},x^{**}_{P}]$, by the uniqueness of the
analytic continuation to the set $C(x^{*}_{P},x^{**}_{P})$ and by continuity
of the function $x\to Q(x,Y_{1}(x))$ on $\overline{C}(x^{*}_{P},x^{**}_{P})$,
this proves that the identity (7.3) holds also on the whole set
$\overline{C}(x^{*}_{P},x^{**}_{P})$. ∎

####  7.2.3. Analytic continuation of the functions $Y_{1}$ and $Y_{2}$ to a
neighborhood of the branching point $x^{**}_{P}$

Now, we extend the functions $Y_{1}$ and $Y_{2}$ to a neighborhood of
$x^{**}_{P}$ as two branches of a two-valued analytic function having a
branching point $x^{**}_{P}$ and we get (7.6).

######  Lemma 7.3.

Under the hypotheses of Proposition 7.1, for some ${\varepsilon}>0$ small
enough, the functions $Y_{1}$ and $Y_{2}$ can be continued to the disk
$B(x^{**}_{P},{\varepsilon})$ as two branches of two-valued analytic in
$B(x^{**}_{P},{\varepsilon}){\setminus}\\{x^{**}_{P}\\}$ function such that
for any $x\in
B(x^{**}_{P},{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}]$,

(7.10) $P(x,Y_{1}(x))=P(x,Y_{2}(x))=1,$

and there is an analytic in $B(0,\sqrt{{\varepsilon}})$ function function
$F_{Y}$ such that

(7.11) $Y_{1}(x)=F_{Y}(-\sqrt{x^{**}_{P}-x}),\quad\quad\quad
Y_{2}(x)=F_{Y}(\sqrt{x^{**}_{P}-x})$

and

(7.12)
$\left.\frac{d}{du}F_{Y}(u)\right|_{u=0}=\left.\sqrt{\frac{\partial_{x}P(x,y)}{\partial^{2}_{yy}P(x,y)}}\,\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0.$

###### Proof.

By the definition of the point $x^{**}_{P}$ and the functions $Y_{1}$,
$Y_{2}$, we have

$Y_{1}(x^{**}_{P}))=Y_{2}(x^{**}_{P}),\quad\quad
P(x^{**}_{P},Y_{1}(x^{**}_{P}))=1,$

and by Lemma 4.1,

(7.13)
$\left.\partial_{y}P(x,y)\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}=0,\quad\text{and}\quad\left.\partial_{x}P(x,y)\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0.$

Hence, by the implicit function theorem, there are a neighborhood
${\mathcal{V}}$ of the point $x^{**}_{P}$ and an analytic in a neighborhood
$U$ of $Y_{1}(x^{**}_{P})$ function $y\to\psi(y)$ such that for any $(x,y)\in
U\times V$,

$P(x,y)=1\quad\Leftrightarrow\quad x=\psi(y),$
$\psi(Y_{1}(x^{**}_{P}))=x^{**}_{P},\quad\quad\quad\left.\frac{d}{dy}\psi(y)\right|_{y=Y_{1}(x^{**}_{P})}=0,$

and

$\left.\frac{d^{2}}{dy^{2}}\psi(y)\right|_{y=Y_{1}(x^{**}_{P})}=-\left.\frac{\partial^{2}_{yy}P(x,y)}{\partial_{x}P(x,y)}\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}<0,$

where the last relation follows from the second relation of (7.13) because
under the hypotheses (A1), the real valued function $y\to P(x^{**}_{P},y)$ is
strictly convex in a neighborhood of $Y_{1}(x^{**}_{P})$. Using the Morse
lemma we conclude therefore that for some neighborhood $\tilde{U}\subset U$ of
$Y_{1}(x^{**}_{P})$, there is a $\mathbb{C}$-diffeomorphism $\omega$ from
$\tilde{U}$ to a neighborhood $\tilde{V}$ of $0$, with

(7.14)
$\omega(Y_{1}(x^{**}_{P}))=0\quad\text{and}\quad\left.\frac{d}{dy}\omega(y)\right|_{y=Y_{1}(x^{**}_{P})}=\left.\sqrt{\frac{\frac{\partial^{2}}{\partial
y^{2}}P(x,y)}{\partial_{x}P(x,y)}}\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}$

such that $\tilde{V}+x^{**}_{P}\subset V$ and for any
$(x,y)\in(\tilde{V}+x^{**}_{P})\times\tilde{U}$,

$P(x,y)=1\quad\Leftrightarrow\quad x=x^{**}_{P}-\omega^{2}(y).$

Without any restriction of generality, one can assume that
$\omega(\tilde{U})=\tilde{V}=B(0,\sqrt{{\varepsilon}})$ with ${\varepsilon}>0$
small enough. Then for $x\in
B(x^{**}_{P},{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}]$,
and $y\in\tilde{U}$, from the above relation it follows that

$\displaystyle P(x,y)=1$
$\displaystyle\quad\Leftrightarrow\quad\bigl{(}\omega(y)=\sqrt{x^{**}_{P}-x}\quad\text{or}\quad\omega(y)=-\sqrt{x^{**}_{P}-x}\bigr{)}$
$\displaystyle\quad\Leftrightarrow\quad\bigl{(}y=\omega^{-1}(\sqrt{x^{**}_{P}-x})\quad\text{or}\quad
y=\omega^{-1}(-\sqrt{x^{**}_{P}-x})\bigr{)}$

with the square root function $\sqrt{\;}$ analytic in
$\mathbb{C}{\setminus}]-\infty,0]$. Since the inverse to $\omega$ function
$\omega^{-1}$ is analytic in $B(0,\sqrt{{\varepsilon}})$, and since for real
$x\in[x^{*}_{P},x^{**}_{P}]$ and $y>0$,

$P(x,y)=1\quad\Leftrightarrow\quad\bigl{(}y=Y_{1}(x)\quad\text{or}\quad
y=Y_{2}(x)\bigr{)}$

we conclude therefore that the functions $Y_{1}$ and $Y_{2}$ can be extended
to the set
$B(x^{**}_{P},{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}]$
as analytic functions such that either

$Y_{1}(x)=\omega^{-1}(\sqrt{x^{**}_{P}-x})\;\text{ and
}\;Y_{2}(x)=\omega^{-1}(-\sqrt{x^{**}_{P}-x}),\quad\forall x\in
B(x^{**}_{P},{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}],$

or

$Y_{1}(x)=\omega^{-1}(-\sqrt{x^{**}_{P}-x})\;\text{and}\;Y_{2}(x)=\omega^{-1}(\sqrt{x^{**}_{P}-x}),\quad\quad\forall
x\in
B(x^{**}_{P},{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}].$

These relations show that the function $\omega^{-1}$ is real valued on the
interval $]-\sqrt{{\varepsilon}},\sqrt{{\varepsilon}}[$, and hence, by the
second relation of (7.14), it is strictly increasing in a neighborhood of $0$.
Since the function $Y_{1}$ is increasing on the interval
$[X_{1}(y^{*}),x^{**}_{P}]$ and the function $Y_{2}$ is decreasing on the
interval $[X_{1}(y^{*}),x^{**}_{P}]$, it follows that (7.10) and (7.11) hold
with $F_{Y}=\omega^{-1}$. Relation (7.12) follows from (7.14). ∎

####  7.2.4. Analytic continuation of the function $Y_{1}$ to the set
$C(x^{*}_{P},x^{**}_{P}+{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}]$

Now we extend the function $Y_{1}$ and the identity (7.3) to the set
$C(x^{*}_{P},x^{**}_{P}+{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}]$
for some small ${\varepsilon}>0$.

######  Lemma 7.4.

Under the hypotheses of Proposition 7.1, for some ${\varepsilon}>0$ small
enough, the function $x\to Y_{1}(x)$ can be analytically continued to the set
$C(x^{*}_{P},x^{**}_{P}+{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}[$,
and satisfies there the identity (7.3).

###### Proof.

By Lemma 7.2, the function $Y_{1}$ is already extended to the closed annulus
$\overline{C}(x^{*}_{P},x^{**}_{P})$ as an analytic function in
${C}(x^{*}_{P},x^{**}_{P})$ and a continuous function on
$\overline{C}(x^{*}_{P},x^{**}_{P})$ satisfying there the identity (7.3), and
that by Lemma 7.3, for some ${\varepsilon}>0$, the function $Y_{1}$ is also
already analytically continued to the set
$B(x^{**}_{P},{\varepsilon}){\setminus}[x^{**}_{P},x^{**}+{\varepsilon}]$.
Hence, to prove this lemma, it is sufficient to show that the function $Y_{1}$
can be analytically continued to a neighborhood of the set
$\\{x\in\mathbb{C}:\leavevmode\nobreak\
|x|=x^{**}_{P},\;|x|\not=x^{**}_{P}\\}$. In order to get this result, we use
the implicit function theorem.

By Lemma 7.2, $Q(\tilde{x},Y_{1}(\tilde{x}))=0$ for any point
$\tilde{x}\in\mathbb{C}$ with $|x|=x^{**}_{P}$ and remark that under the
hypotheses of Proposition 7.1, the function $Q$ is analytic in a neighborhood
of $(\tilde{x},Y_{1}(\tilde{x}))$. Hence, by the implicit function theorem, it
is sufficient to show that for any such a point $\tilde{x}$,

(7.15)
$\partial_{y}Q(\tilde{x},Y_{1}(\tilde{x}))\not=0\quad\text{whenever}\quad\tilde{x}\not=x^{**}_{P}.$

To get this relation, we let us remark that according to the definition (2.24)
of the function $Q$ and using (7.4), for any point $\tilde{x}\in\mathbb{C}$
with $|\tilde{x}|=x^{**}_{P}$ and such that $\tilde{x}\not=x^{**}_{P}$, one
has

(7.16)
$\left|\tilde{x}-\partial_{y}Q(\tilde{x},Y_{1}(\tilde{x}))\right|\leqslant\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}=0}^{\infty}(k_{2}+1)\mu(k_{1},k_{2})(x^{**}_{P})^{k_{1}+1}|Y_{1}(\tilde{x})|^{k_{2}}\\\
\\\
<\sum_{k_{1}\in\mathbb{Z}}\sum_{k_{2}=0}^{\infty}(k_{2}+1)\mu(k_{1},k_{2})(x^{**}_{P})^{k_{1}+1}Y_{1}(x^{**}_{P})^{k_{2}}=x^{**}_{P}-\partial_{y}Q(x^{**}_{P},Y_{1}(x^{**}_{P}))$

where

$\partial_{y}Q(x^{**}_{P},Y_{1}(x^{**}_{P}))=x^{**}_{P}-x^{**}_{P}P(x^{**}_{P},Y_{1}(x^{**}_{P}))-x^{**}_{P}Y_{1}(x^{**}_{P})\partial_{y}Px^{**}_{P},Y_{1}(x^{**}_{P}))=0$

because $P(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$ and by Lemma 4.1,
$\partial_{y}P(x^{**}_{P},Y_{1}(x^{**}_{P}))=0$. Hence, for any
$\tilde{x}\in\mathbb{C}$ with $|\tilde{x}|=x^{**}_{P}$ and such that
$\tilde{x}\not=x^{**}_{P}$,

$\left|\partial_{y}Q(\tilde{x},Y_{1}(\tilde{x}))\right|\geqslant|\tilde{x}|-\left|\tilde{x}-\partial_{y}Q(\tilde{x},Y_{1}(\tilde{x}))\right|=x^{**}_{P}-\left|\tilde{x}-\partial_{y}Q(\tilde{x},Y_{1}(\tilde{x}))\right|>0$

and consequently, (7.15) holds. ∎

####  7.2.5. Proof of Proposition 7.1

The first two assertions of Proposition 7.1 are proved by Lemma 7.2, Lemma 7.3
and Lemma 7.4.

To prove the third assertion of Proposition 7.1, recall that by Lemma 7.2, the
function $Y_{1}$ is analytic in the annulus $C(x^{*}_{P},x^{**}_{P})$ and
satisfies there the inequality (7.4) and remark that any point
$\hat{x}\in]X_{1}(y^{*}_{P}),x^{**}_{P}[$ belongs to the annulus
$C(x^{*}_{P},x^{**}_{P})$ because $x^{*}_{P}<X_{1}(y^{*}_{P})<x^{**}_{P}$.
Hence, for any $\hat{x}\in]X_{1}(y^{*}_{P}),x^{**}_{P}[$, the function
$x\to(Y_{1}(x)-Y_{1}(\hat{x})$ is analytic in a neighborhood of the circle
$\\{x\in\mathbb{C}:\leavevmode\nobreak\ |x|=\hat{x}$, and by (7.4), the point
$\hat{x}$ is an only zero of this function in the circle
$\\{x\in\mathbb{C}:\leavevmode\nobreak\ |x|=\hat{x}$. Since the zeros of
analytic functions are always isolated, this proves that for any
$\hat{x}\in]X_{1}(y^{*}_{P}),x^{**}_{P}[$, there is $\delta>0$ for which the
function $x\to(Y_{1}(x)-Y_{1}(\hat{x})$ does not vanish in the set
$C(\hat{x}-\delta,\hat{x}+\delta){\setminus}\\{\hat{x}\\}$.

Moreover, we have

  * –

by Lemma 4.1, the real valued function $Y_{1}$ is strictly increasing on the
interval $]X_{1}(y^{*}_{P}),x^{**}_{P}[$,

  * –

by Lemma 7.2, it is also strictly convex on $]X_{1}(y^{*}_{P}),x^{**}_{P}[$,

  * –

and by Lemma 4.1, $P(\hat{x},Y_{1}(\hat{x}))=1$ and
$\partial_{y}P(\hat{x},Y_{1}(\hat{x}))<0$.

Hence, for any $\hat{x}\in]X_{1}(y^{*}_{P}),x^{**}_{P}[$, using the implicit
function theorem, one gets (7.5). From this relation, it follows moreover that
for any $\hat{x}\in]X_{1}(y^{*}_{P}),x^{**}_{P}[$, the point the point
$\hat{x}$ is a simple zero of the function $x\to Y_{1}(x)-Y_{1}(\hat{x})$. The
third assertion of Proposition 7.1 is therefore also proved.

To get the last assertion of Proposition 7.1, recall that by Lemma 7.3 and
Lemma 7.4 , the function $Y_{1}$ was already extended to the set
$U_{\varepsilon}=C(x^{*}_{P},x^{**}+{\varepsilon}){\setminus}]x^{**}_{P},x^{**}_{P}+{\varepsilon}[$
as an analytic function in
$\overset{\circ}{U}_{\varepsilon}=C(x^{*}_{P},x^{**}+{\varepsilon}){\setminus}[x^{**}_{P},x^{**}_{P}+{\varepsilon}[$
and a continuous function on $U_{\varepsilon}$ satisfying there the inequality
(7.4). The function $x\to Y_{1}(x)-Y_{1}(x^{**}_{P})$ is therefore analytic in
$\overset{\circ}{U}_{\varepsilon}$ and continuous on $U_{\varepsilon}$, and
moreover by (7.4), the point $x^{**}_{P}$ is its only zero in the circle
$\\{x\in\mathbb{C}:|x|=x^{**}_{P}\\}$. To complete the proof of the fourth
assertion of Proposition 7.1, it is therefore sufficient to get (7.6) and to
show that the point $x^{**}_{P}$ is an isolated zero of the function $x\to
Y_{1}(x)-Y_{1}(x^{**}_{P})$. For this, we remark that by Lemma 7.3, as $x\to
x^{**}_{P}$,

(7.17)
$Y_{1}(x)-Y_{1}(x^{**}_{P})=F_{Y}(-\sqrt{x^{**}_{P}-x})-F_{Y}(0)\leavevmode\nobreak\
\sim\leavevmode\nobreak\ -c\sqrt{x^{**}_{P}-x}$

and

$\frac{d}{dx}Y_{1}(x)\sim\frac{c}{2\sqrt{x^{**}_{P}-x}}$

with

$c=\left.\frac{d}{du}F_{Y}(0)\right|_{u=0}=\left.\sqrt{\partial_{x}P(x,y)/\partial^{2}_{yy}P(x,y)}\,\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0.$

Relations (7.6) are therefore proved, and since the constant $c$ is strictly
positive, from (7.17) it follows that the point $x^{**}_{P}$ is an isolated
zero of the function $x\to Y_{1}(x)-Y_{1}(x^{**}_{P})$.

###  7.3. The properties of the mapping $x\to(x,Y_{1}(x))$ and their
consequences

Remark that by (7.4), the function $x\to(x,Y_{1}(x))$ maps the closed annulus
to the set $\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\ \;x\in
C(x^{*}_{P},x^{**}_{P}),\;|y|\leqslant Y_{1}(|x|)\\}$, and under the
hypotheses (A1)-(A3), for any $j\in\mathbb{Z}^{2}_{+}$, the functions
$(x,y)\to\phi_{1}(x,y)$, $(x,y)\to\psi_{2}(x,y)$ and $(x,y)\to L_{j}(x,y)$ are
analytic in a neighborhood of this set. Hence, as a straightforward
consequence of Proposition 7.1 one gets

######  Corollary 7.3.

Under the hypotheses (A1)-(A3), for some $\delta>0$, the functions
$x\to\phi_{1}(x,Y_{1}(x))$, $x\to\psi_{2}(x,Y_{1}(x))$ and $x\to
L_{j}(x,Y_{1}(x))$ for any $j\in\mathbb{Z}^{2}_{+}$, are analytic in the set
$\overset{\circ}{U}_{\delta}$ and continuous on the set $U_{\delta}$.

Another consequence of Proposition 7.1 is the following property of the
mapping $x{\to}(x,Y_{1}(x))$.

######  Proposition 7.4.

Under the hypotheses (A1)-(A4), for some $\delta>0$, the function
$x\to(x,Y_{1}(x))$ maps the annulus $C(x_{d}-\delta,x_{d})$ to the set
$\\{(x,y)\in\Omega({\Gamma}):\,|x|<x_{d},\,|y|<y_{d}\\}$. Moreover, for any
neighborhood $V$ of the set $\Omega(\overline{\Gamma})$ in $\mathbb{C}^{2}$
and $\hat{y}>Y_{1}(x_{d})$, there is $\delta>0$, for which the function
$x\to(x,Y_{1}(x))$ maps the set $C(x_{d}-\delta,x_{d}+\delta)\cap
U_{\varepsilon}$ to the set $\\{(x,y)\in V:\,|y|<\hat{y}\\}$.

###### Proof.

Remark that by (7.4), for any $x\in]x^{*}_{P},x^{**}_{P}[$,

$|Y_{1}(x)|\leqslant Y_{1}(|x|)<Y_{2}(|x|).$

Hence, the function $x\to(x,Y_{1}(x))$ maps the set $C(x^{*}_{P},x^{**}_{P})$
to the set

${\Gamma}_{1}=\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\ \;x\in
C(x^{*}_{P},x^{**}_{P}),\;|y|<Y_{2}(|x|)\\}.$

For any $x\in]x^{*}_{P},x^{**}_{P}[$, $Y_{1}(x)$ and $Y_{2}(x)$ are the only
real and positive solution of the equation $P(x,y)=1$, $Y_{1}(x)<Y_{2}(x)$ and
for any $Y_{1}(x)<y<Y_{2}(x)$, the point $(x,y)$ belongs to the interior of
the set $D=\\{(x,y)\in[0,+\infty[^{2}:P(x,y)\leqslant 1\\}$ (see Lemma 4.1 for
more details). Hence for any $(x,y)\in{\Gamma}_{1})$, there is a point
$(x^{\prime},y^{\prime})\in D$ such that $|x|<x^{\prime}$ and $|x|<y^{\prime}$
and consequently, the set ${\Gamma}_{1}$ is included to the set ${\Gamma}$.
The mapping $x\to(x,Y_{1}(x))$ maps therefore the set
$C(x^{*}_{P},x^{**}_{P})$ to the set ${\Gamma}$.

Consider now the case when one of the cases (B0)-(B4) holds. In this case,
(see Proposition 2.2 and relations (2.33), (2.34)), we have
$Y_{1}(x_{d})<y_{d}$. Since the function $x\to(x,Y_{1}(x))$ maps the annulus
$C(x^{*}_{P},x^{**}_{P})$ to the set ${\Gamma}$ and by Proposition 7.1, the
function $Y_{1}$ is continuous on $U_{\varepsilon}$, it follows that for some
$\delta>0$, the function $x\to(x,Y_{1}(x))$ maps the annulus
$C(x_{d}-\delta,x)$ to the set
$\\{(x,y)\in\Omega({\Gamma}):\,|x|<x_{d},\,|y|<y_{d}\\}$.

When one of the cases (B0)-(B4) holds, the first assertion of Proposition 7.4
is therefore proved.

Consider now the case when (B5) or (B6) holds. In this case (see Proposition
2.2 and relations (2.33), (2.34)), the point
$(x_{d},y_{d})=(X_{2}(y^{**}),y^{**})$ belongs to the curve
${\mathcal{S}}_{21}=\\{(x,Y_{1}(x)):\leavevmode\nobreak\
x\in[X_{1}(y^{*}_{P}),x^{**}_{P}]\\}=\\{(X_{2}(y),y):\leavevmode\nobreak\
y\in[y^{*}_{P},Y_{1}(x^{**}_{P})]\\}$ and $y^{*}_{P}\leqslant
y^{*}<y_{d}=y^{**}<y^{**}_{P}$. Since by lemma 4.1, the functions
$X_{2}:[y^{*}_{P},Y_{1}(x^{**}_{P})]\to[X_{1}(y^{*}_{P}),x^{**}_{P}]$ and
$Y_{1}:[X_{1}(y^{*}_{P}),x^{**}_{P}]\to[y^{*}_{P},Y_{1}(x^{**}_{P})]$ are
strictly increasing and inverse to each other, it follows that in this case,
we have $x^{*}_{P}<X_{2}(y_{d})=x_{d}$. Hence, for any $x\in
C(x^{*}_{P},x_{d})$, using (7.4) one gets

$|Y_{1}(x)|\leqslant Y_{1}(|x|)<Y_{1}(x_{d}).$

Since we have already proved that the function $x\to(x,Y_{1}(x))$ maps the
annulus $C(x^{*}_{P},x^{**}_{P})$ to the set ${\Gamma}$, the last relations
prove that for $\delta=x_{d}-x^{*}_{P}>0$, this function maps the annulus
$C(x_{d}-\delta,x_{d})$ to the set
$\\{(x,y)\in\Omega({\Gamma}):\,|x|<x_{d},\,|y|<y_{d}\\}$.

In the case when one of the cases (B5) or (B6) holds, the first assertion of
Proposition 7.4 is therefore also proved.

The second assertion of Proposition 7.4 holds because the function
$x\to(x,Y_{1}(x))$ is continuous in the set $U_{\varepsilon}$ including the
closed annulus $\overline{C}(x^{*}_{P},x^{**}_{P})$ and maps
$\overline{C}(x^{*}_{P},x^{**}_{P})$ to the closure of the set ${\Gamma}$.

∎

Remark that under the hypotheses (A1)(ii), (A2) and (A3) (ii), (iv), the
functions $(x,y)\to Q(x,y)=xy(1-P(x,y)$,
$(x,y)\to\psi_{1}(x,y)=x(\phi_{1}(x,y)-1)$ and
$(x,y)\to\psi_{2}(x,y)=y(1-\phi_{2}(x,y))$ can be continued as analytic
functions to some neighborhood of the set $\Omega(\overline{\Gamma})$. Hence,
from Theorem 1 it follows

######  Corollary 7.5.

Under the hypotheses (A1)-(A4), there is a neighborhood
${\mathcal{V}}\subset\mathbb{C}^{2}$ of the set $\Omega(\overline{\Gamma})$
such that for any $j\in\mathbb{Z}^{2}$,

  * –

the function $(x,y)\to\psi_{1}(x,y)h_{1j}(x)$ is analytic in the set
$\\{(x,y)\in{\mathcal{V}}:\leavevmode\nobreak\ |x|<x_{d}\\}$ ;

  * –

the function $(x,y)\to\psi_{2}(x,y)h_{2j}(y)$ is analytic in the set
$\\{(x,y)\in{\mathcal{V}}:\leavevmode\nobreak\ |y|<y_{d}\\}$

  * –

the function $(x,y)\to R_{j}(x,y)=Q(x,y)h_{j}(x,y)$ can be extended as an
analytic function to the set $\\{(x,y)\in{\mathcal{V}}:\leavevmode\nobreak\
|x|<x_{d},\;|y|<y_{d}\\}$ by letting

(7.18) $R_{j}(x,y)=L_{j}(x,y)+\psi_{1}(x,y)h_{1j}(x)+\psi_{2}(x,y)h_{2j}(y).$

When combined together, Proposition 7.4, Corollary 7.3 and Corollary 7.5 imply
the following statement.

######  Corollary 7.6.

Under the hypotheses (A1)-(A4) for some $\delta>0$, and any
$j\in\mathbb{Z}^{2}_{+}$,

i) the functions
$x\to-\psi_{1}(x,Y_{1}(x))h_{1j}(x)=(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ and
$x\to\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ are analytic in the annulus
$C(x_{d}-\delta,x_{d})$ and satisfy there the identity

(7.19)
$(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)=L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x));$

ii) moreover, if one of the cases (B0)-(B4) holds with $x_{d}<x^{**}_{P}$
(i.e. if either one of the cases (B0), (B1), (B3) or (B4) holds or (B2) and
$x_{d}<x^{**}_{P}$ hold), then for some $\delta>0$, the function $x\to
L_{J}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in the
annulus $C(x_{d}-\delta,x_{d}+\delta)$ and by (7.19), the function
$x\to\eta_{j}(x)=(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ can be extended as an
analytic function to $C(x_{d}-\delta,x_{d}+\delta)$.

iii) if (B2) holds with $x_{d}=x^{**}_{P}$, then for some $\delta>0$, the
function $x\to L_{j}(x,Y_{1}(x))+\psi(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic
in the set $C(x_{d}-\delta,x_{d}+\delta)\cap\overset{\circ}{U}_{\varepsilon}$
and continuous on the set $C(x_{d}-\delta,x_{d}+\delta)\cap{U}_{\varepsilon}$,
and by (7.19), the function
$x\to\eta_{j}(x)=(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ can be extended to the
set $C(x_{d}-\delta,x_{d}+\delta)\cap{U}_{\varepsilon}$ as an analytic
function in $C(x_{d}-\delta,x_{d}+\delta)\cap\overset{\circ}{U}_{\varepsilon}$
and a continuous function on
$C(x_{d}-\delta,x_{d}+\delta)\cap{U}_{\varepsilon}$.

With this result, when one of the cases (B0)-(B4) holds, the function
$x\to(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ is therefore already extended beyond
the point $x_{d}$. To extend the function $x\to H_{j}(x,0)$ beyond the point
$x_{d}$ we need to investigate the function $x\to\phi_{1}(x,Y_{1}(x))$. This
is a subject of the next section.

###  7.4. Analytic continuation and properties of the function
$x\to\phi_{1}(x,Y_{1}(x))$

By Corollary 7.3, for some $\delta>0$, the function $x\to\phi_{1}(x,Y_{1}(x))$
is already extended to the set $U_{\delta}$ as an analytic function in the set
$\overset{\circ}{U}_{\delta}$ and a continuous function on the set
$U_{\delta}$. We will need moreover the following properties of the function
$x\to\phi_{1}(x,Y_{1}(x))$.

######  Proposition 7.7.

Under the hypotheses (A1)- (A3), the following assertions hold:

i) the function $x\to\phi_{1}(x,Y_{1}(x))$ is strictly convex on the line
segment $[x^{*}_{P},x^{**}_{P}]$.

ii) for any $x\in C(x^{*},x^{**})$,

(7.20) $|\phi_{1}(x,Y_{1}(x))|\leqslant\phi(|x|,Y_{1}(|x|))<1;$

iii) if $x^{**}<x^{**}_{P}$, then for some $\hat{\delta}>0$, the point
$x^{**}$ is an only and simple zero of the function $x\to
1-\phi_{1}(x,Y_{1}(x))$ in the annulus $C(x^{*},x^{**}+\hat{\delta})$ and

(7.21) $\left.\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\right|_{x=x^{**}}>0;$

iv) if $x^{**}=x^{**}_{P}$, then
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))\leqslant 1$;

v) if $x^{**}=x^{**}_{P}$ and $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$, then
for some $\hat{\delta}>0$, the function $x\to 1-\phi_{1}(x,Y_{1}(x))$ has no
zeros in
$U_{\hat{\delta}}=C(x^{*},x^{**}_{P}+\hat{\delta}){\setminus}]x^{**}_{P},x^{**}_{P}+\hat{\delta}[$;

vi) if $x^{**}=x^{**}_{P}$ and $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$,
then for some $\hat{\delta}>0$, the point $x^{**}_{P}$ is an only zero of the
function $x\to 1-\phi_{1}(x,Y_{1}(x))$ in $U_{\hat{\delta}}$ and as $x\to
x^{**}_{P}$,

(7.22) $1-\phi_{1}(x,Y_{1}(x))\sim
c\sqrt{x^{**}_{P}-x}\quad\;\text{with}\;\quad
c=\left.\partial_{y}\phi_{1}(x,y)\sqrt{\frac{\partial_{x}P(x,y)}{\partial^{2}_{yy}P(x,y)}}\,\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0.$

####  7.4.1. Outline of the proof of Proposition 7.7

To prove this proposition, we first get a probabilistic representation of the
function $x\to\phi_{1}(x,Y_{1}(x))$ similar to those of the function $x\to
Y_{1}(x)$. This is a subject of Corollary 7.8. In Section 7.4.3, as a
straightforward consequence of this result, we get the first and the second
assertion of Proposition 7.7. The proof of the third assertion of this
statement is given in Section 7.4.4 and the proofs of the last tree assertions
are completed in Section 7.4.5.

####  7.4.2. Probabilistic representation of the function
$x\to\phi_{1}(x,Y_{1}(x))$

The following probabilistic representation of the function
$x\to\phi_{1}(x,Y_{1}(x))$ is another useful for our purpose consequence of
Lemma 7.1 :

######  Corollary 7.8.

Let $(\hat{S}(n))$ be a random walk on the half-plane
$\mathbb{Z}\times{\mathbb{N}}$ with transition probabilities

(7.23) ${\mathbb{P}}_{k}(\hat{S}(1)=j)\leavevmode\nobreak\
=\leavevmode\nobreak\ \begin{cases}\mu(j-k)&\text{for all
$j,k=(k_{1},k_{2})\in\mathbb{Z}\times{\mathbb{N}}$ with $k_{2}>0$}\\\
\mu_{1}(j-k)&\text{for all $j,k=(k_{1},k_{2})\in\mathbb{Z}\times{\mathbb{N}}$
with $k_{2}=0$}\\\ \end{cases}$

and let $T_{1}$ be the first time when the random walk
$\bigl{(}\hat{S}(n)=(\hat{S}_{1}(n),\hat{S}_{2}(n))\bigr{)}$ hits the boundary
$\mathbb{Z}\times\\{0\\}$ :

$T_{1}=\inf\\{n>0:\leavevmode\nobreak\
\hat{S}(n)\in\mathbb{Z}\times\\{0\\}\\}.$

Then under the hypotheses (A1) - (A3), for any
$x\in\overline{C}(x^{*}_{P},x^{**}_{P})$

(7.24)
$\phi_{1}(x,Y_{1}(x))={\mathbb{E}}_{(0,0)}\left(x^{\hat{S}_{1}(T_{1})};\;T_{1}<+\infty\right).$

###### Proof.

To get (7.24) from Lemma 7.1 it is sufficient to notice that by the Markov
property, and according to the definition of the random walks $(S(n))$ and
$(\hat{S}(n))$,

$\displaystyle{\mathbb{E}}_{(0,0)}\left(x^{\hat{S}_{1}(\tau_{1})};\;\tau_{1}<+\infty\right)$
$\displaystyle=\sum_{k\in\mathbb{Z}\times\\{0\\}}\mu_{1}(k)x^{k_{1}}+\sum_{k=(k_{1},k_{2})\in\mathbb{Z}\times{\mathbb{N}}:\leavevmode\nobreak\
k_{2}\not=0}\mu_{1}(k){\mathbb{E}}_{k}(x^{S_{1}(T_{1})},\;T_{1}<+\infty)$
$\displaystyle=\sum_{k\in\mathbb{Z}\times\\{0\\}}\mu_{1}(k)x^{k_{1}}+\sum_{k=(k_{1},k_{2})\in\mathbb{Z}\times{\mathbb{N}}:\leavevmode\nobreak\
k_{2}\not=0}\mu_{1}(k)x^{j_{1}}Y_{1}(x)^{j_{2}}$
$\displaystyle=\phi_{1}(x,Y_{1}(x)).$

∎

####  7.4.3. Proof of the first and the second assertions of Proposition 7.7

With Corollary 7.8, by using the same arguments as in the proof of Lemma 7.2
we get

######  Corollary 7.9.

Under the hypotheses (A1) - (A3), the function $x\to\phi_{1}(x,Y_{1}(x))$ is
strictly convex on $[x^{*}_{P},x^{**}_{P}]$ and satisfies on the set
$\overline{C}(x^{*}_{P},x^{**}_{P})$ the following relation

(7.25) $|\phi_{1}(x,Y_{1}(x))|<\phi_{1}(|x|,Y_{1}(|x|)),\quad\forall
x\not=|x|.$

Remark that this statement proves the first assertion of Proposition 7.7.
Moreover, since by Corollary 4.1,

(7.26) $\phi_{1}(x,Y_{1}(x))<1\quad\text{for any}\quad x\in]x^{*},x^{**}[,$

using (7.25) one gets (7.20), and consequently, the second assertion of
Proposition 7.7 also holds.

####  7.4.4. Proof of the third assertion of Proposition 7.7

Suppose now that $x^{**}<x^{**}_{P}$. In this case, by Corollary 4.1,

(7.27) $\phi_{1}(x^{**},Y_{1}(x^{**}))=1\quad\text{and
}\quad\phi_{1}(x,Y_{1}(x))>1\quad\text{for all}\quad x\in]x^{**},x^{**}_{P}].$

Using this relation together with (7.20) one get that the point $x^{**}$ is an
only zero of the function $x\to\phi_{1}(x,Y_{1}(x))-1$ in the set
$\\{x\in\mathbb{C}\leavevmode\nobreak\ :\leavevmode\nobreak\
x^{*}<|x|\leqslant x^{**}\\}$, and since the function
$x\to\phi_{1}(x,Y_{1}(x))-1$ is analytic in the neighborhood
$\overset{\circ}{U}_{\delta}$ of this set, it follows that for some
$\hat{\delta}>0$, the point $x^{**}$ is an only zero of this function in the
annulus $C(x^{*},x^{**}+\hat{\delta})$. Moreover, since the function
$x\to\phi_{1}(x,Y_{1}(x))$ is strictly convex on the line segment
$[x^{*}_{P},x^{**}_{P}]$ and by (2.9), in the case when $x^{**}<x^{**}_{P}$,
the point $x^{**}$ belongs to the interior of this line segment, from (7.26)
and (7.27) it follows that

$\left.\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\right|_{x=x^{**}}>0,$

and consequently, the point $x^{**}$ is in this case a simple zero of the
function $x\to\phi_{1}(x,Y_{1}(x))-1$. The third assertion of Proposition 7.7
is therefore also proved.

####  7.4.5. Proof of the last three assertions of Proposition 7.7

Suppose now that $x^{**}=x^{**}_{P}$. Then by Corollary 4.1,
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))\leqslant 1$, and consequently, the
fourth assertion of Proposition 7.7 holds.

Moreover, if $x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$, using (7.25) we get

$|\phi_{1}(x,Y_{1}(x))]\leqslant\phi_{1}(|x|,Y_{1}(|x|))<1,\quad\text{for any
$x\in\mathbb{C}$ with $x^{*}<|x|\leqslant x^{**}_{P}$},$

and consequently, the function $x\to\phi_{1}(x,Y_{1}(x))-1$ has no zeros in
the set $\\{x\in\mathbb{C}\leavevmode\nobreak\ :\leavevmode\nobreak\
x^{*}<|x|\leqslant x^{**}_{P}\\}$. Since the function
$x\to\phi_{1}(x,Y_{1}(x))$ is continuous on the set $U_{\delta}$, it follows
that, for some $\hat{\delta}>0$, it has no zeros in the set
$U_{\hat{\delta}}$.

An finally, if $x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$, using again (7.25) one gets that
the point $x^{**}_{P}$ is an only zero of the function
$x\to\phi_{1}(x,Y_{1}(x))-1$ in the set $\\{x\in\mathbb{C}\leavevmode\nobreak\
:\leavevmode\nobreak\ x^{*}<|x|\leqslant x^{**}_{P}\\}$. Moreover, since under
our hypotheses, the function $(x,y)\to\phi_{1}(x,y)$ is analytic in a
neighborhood of the point $(x^{**}_{P},Y_{1}(x^{**}_{P}))$ and since because
of Assumption (A3)(iv),

$\partial_{y}\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))>0,$

using (7.6) one gets that as $x\to x^{**}_{P}$,

$\displaystyle\phi_{1}(x,Y_{1}(x))-1$ $\displaystyle\leavevmode\nobreak\
\sim\leavevmode\nobreak\
(x-x^{**}_{P})\partial_{x}\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))-c\sqrt{x^{**}_{P}-x}\partial_{y}\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))$
$\displaystyle\leavevmode\nobreak\ \sim\leavevmode\nobreak\
-c\sqrt{x^{**}_{P}-x}\partial_{y}\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))$

with

$c=\left.\sqrt{\partial_{x}P(x,y)/\partial^{2}_{yy}P(x,y)}\,\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0.$

This proves that the point $x^{**}_{P}$ is in this case an isolated zero of
the function $x\to\phi_{1}(x,Y_{1}(x))-1$ in $U_{\delta}$. Since this function
is continuous in a neighborhood $U_{\delta}$ of the set
$\\{x\in\mathbb{C}\leavevmode\nobreak\ :\leavevmode\nobreak\
x^{*}<|x|\leqslant x^{**}_{P}\\}$, and has no zeros in
$\\{x\in\mathbb{C}\leavevmode\nobreak\ :\leavevmode\nobreak\
x^{*}<|x|\leqslant x^{**}_{P}\\}$ aside of the point $x^{**}_{P}$, we conclude
therefore that for some $\hat{\delta}>0$, the point $x^{**}_{P}$ is an only
zero of the function $x\to\phi_{1}(x,Y_{1}(x))-1$ in $U_{\hat{\delta}}$. The
last assertion of Proposition 7.7 is therefore also proved.

###  7.5. Analytic continuation of the function $x{\to}H_{j}(x,0)$, cases
(B0)-(B4)

In this section, we extend the function $x\to H_{j}(x,0)$ beyond the point
$x_{d}$ when one of the cases (B0)-(B4) holds. The cases (B0), (B1), (B3),
(B4), and the case (B2) with $x^{**}<x^{**}_{P}$ are considered in Proposition
7.10. The case (B2) with $x_{d}=x^{**}_{P}$ is considered in Proposition 7.11
below.

######  Proposition 7.10.

Suppose that the conditions (A1)-(A3) are satisfied and let either one of the
cases (B0), (B1), (B3), (B4) holds or (B2) and $x^{**}<x^{**}_{P}$ hold. Then
for some $\delta>0$ and any $j\in\mathbb{Z}^{2}_{+}$, the function $x\to
H_{j}(x,0)$ can be extended as an analytic function to the set
$B(0,x_{d}+\delta){\setminus}\\{x_{d}\\}$, and (2.40) holds with
$\varkappa_{1}(j)$ defined by (2.35) and $a_{1}>0$ defined by (2.41).

###### Proof.

Under the hypotheses of this proposition, by Corollary 7.6, for some
$\delta_{1}>0$, the function $x\to
L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in the
annulus $C(x_{d}-\delta_{1},x_{d}+\delta_{1})$, and using the identities

(7.28) $\displaystyle(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$
$\displaystyle=-\psi_{1}(x,Y_{1}(x))h_{1j}(x)$ (7.29)
$\displaystyle=L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$

the function $x\to(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ can be analytically
continued to the open annulus $C(x_{d}-\delta_{1},x_{d}+\delta_{1})$. Since by
Proposition 7.7, for some $\delta_{2}>0$, the function
$x\to(1-\phi_{1}(x,Y_{1}(x)))^{-1}$ is analytic in the set
$C(x_{d}-\delta_{2},x_{d}+\delta_{2}){\setminus}\\{x_{d}\\}$ and has at the
point $x_{d}$ a simple pole with

$\lim_{x\to
x_{d}}\frac{x_{d}-x}{1-\phi_{1}(x,Y_{1}(x))}=\left.\left(\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\right)^{-1}\right|_{x=x_{d}}>0,$

we conclude therefore that the function $x\to H_{j}(x,0)$ can be extended as
an analytic function to the set $B(0,x_{d}+\delta){\setminus}\\{x_{d}\\}$, and
that (2.40) holds with $\varkappa_{1}(j)$ given by (2.35) , and $a_{1}>0$
given by (2.41). ∎

######  Proposition 7.11.

Suppose that the conditions (A1)-(A3) are satisfied and let (B2) and
$x_{d}=x^{**}_{P}$ hold. Then the following assertions holds.

  * –

If $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$, then for some $\delta>0$, the
function $x\to H_{j}(x,0)$ can be extended as an analytic function to the set
$B(0,x^{**}_{P}+\delta){\setminus}[x^{**},x^{**}_{P}+\delta[$ satisfying
(2.42) with $\varkappa_{1}(j)$ defined by (2.35) with $x_{d}=x^{**}_{P}$ and
$a_{2}>0$ defined by (2.43).

  * –

If $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$, then for some $\delta>0$, the
function $x\to H_{j}(x,0)$ can be extended to the set
$B(0,x^{**}_{P}+\delta){\setminus}]x^{**}_{P},x^{**}_{P}+\delta[$ as an
analytic function in
$B(0,x^{**}_{P}+\delta){\setminus}[x^{**}_{P},x^{**}_{P}+\delta[$ and a
continuous function on
$B(0,x^{**}_{P}+\delta){\setminus}]x^{**}_{P},x^{**}_{P}+\delta[$, satisfying
(2.44) with $\tilde{\varkappa}_{1}(j)$ defined by (2.36) and $\tilde{a}_{2}>0$
defined by (2.45).

###### Proof.

Indeed, in this case, by Corollary 7.6, for some
$0<\delta_{1}\leqslant{\varepsilon}$, the function $x\to
L_{j}(x,Y_{1}(x))+\psi(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in
$C(x_{d}-\delta_{1},x_{d}+\delta_{1})\cap\overset{\circ}{U}_{\varepsilon}$ and
continuous on $C(x_{d}-\delta_{1},x_{d}+\delta_{1})\cap{U}_{\varepsilon}$, and
using the identities (7.28) end (7.29), the function
$x\to\eta_{j}(x)=(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ can be extended to the
set $C(x_{d}-\delta_{1},x_{d}+\delta_{1})\cap{U}_{\varepsilon}$ as an analytic
function in
$C(x_{d}-\delta_{1},x_{d}+\delta_{1})\cap\overset{\circ}{U}_{\varepsilon}$ and
a continuous function on
$C(x_{d}-\delta_{1},x_{d}+\delta_{1})\cap{U}_{\varepsilon}$. Since by
Proposition 7.7, for some $0<\delta\leqslant\delta_{1}$, the function $x\to
1/(1-\phi_{1}(x,Y_{1}(x)))$ is analytic in the set
$C(x_{d}-\delta,x_{d}+\delta)\cap\overset{\circ}{U}_{\varepsilon}$, we get
therefore that the function $x\to H_{j}(x,0)$ can be extended as an analytic
function to the set
$C(x_{d}-\delta,x_{d}+\delta)\cap\overset{\circ}{U}_{\varepsilon}$ by letting

$\displaystyle H_{j}(x,0)$
$\displaystyle=\eta_{j}(x)\bigl{(}1-\phi_{1}(x,Y_{1}(x))\bigr{)}^{-1}$ (7.30)
$\displaystyle=\left(L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))\right)\bigl{(}1-\phi_{1}(x,Y_{1}(x))\bigr{)}^{-1}.$

Recall moreover that by Proposition 7.7, when
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$, the function $x\to
1/(1-\phi_{1}(x,Y_{1}(x)))$ is continuous on
$C(x_{d}-\delta,x_{d}+\delta)\cap{U}_{\varepsilon}$, and when
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$,

(7.31) $\lim_{x\to
x^{**}_{P}}\frac{\sqrt{x^{**}_{P}-x}}{1-\phi_{1}(x,Y_{1}(x))}=\left.\left(\partial_{y}\phi_{1}(x,y)\sqrt{\partial_{x}P(x,y)/\partial^{2}_{yy}P(x,y)}\right)^{-1}\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0.$

Hence, in the case when $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$, the
function $x\to H_{j}(x,0)$ can be extended as a continuous function to the set
$C(x_{d}-\delta,x_{d}+\delta)\cap{U}_{\varepsilon}$, and in the case when
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$, using (7.30) (7.31), one gets
(2.42) with $\varkappa_{1}(j)$ defined by (2.35), and $a_{2}>0$ given by
(2.43).

Remark finally that when $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$, using
(7.30) , for $x\in B(0,x^{**}_{P})$ closed enough to $x^{**}_{P}$, one gets

$\displaystyle\frac{d}{dx}$ $\displaystyle
H_{j}(x,0)=\frac{d}{dx}\left(r_{j}(x,Y_{1}(x))(1-\phi_{1}(x,Y_{1}(x)))^{-1}\right)$
$\displaystyle=\frac{d}{dx}\left((L_{j}(x,Y_{1}(x))+(\phi_{2}(x,Y_{1}(x))-1)H_{j}(0,Y_{1}(x)))(1-\phi_{1}(x,Y_{1}(x)))^{-1}\right)$
$\displaystyle=\partial_{x}\left.\left((L_{j}(x,y)+(\phi_{2}(x,y)-1)H_{j}(0,y))(1-\phi_{1}(x,y))^{-1}\right)\right|_{y=Y_{1}(x)}$
$\displaystyle\;+\partial_{y}\left.\left((L_{j}(x,Y_{1}(x))+(\phi_{2}(x,Y_{1}(x))-1)H_{j}(0,Y_{1}(x)))(1-\phi_{1}(x,Y_{1}(x)))^{-1}\right)\right|_{y=Y_{1}(x)}\times\frac{d}{dx}Y_{1}(x)$

where by (7.6), as $x\to x^{**}_{P}$,

$\frac{d}{dx}Y_{1}(x)\sim\frac{c}{2\sqrt{x^{**}_{P}-x}}\quad\;\text{with}\;\quad
c=\left.\frac{1}{2}\sqrt{\partial_{x}P(x,y)/\partial^{2}_{yy}P(x,y)}\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0.$

Since in the case when $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$, the
function

$(x,y)\to(L_{j}(x,y)+(\phi_{2}(x,y)-1)H_{j}(0,y))(1-\phi_{1}(x,y))^{-1}$

is analytic in a neighborhood of the point $(x^{**}_{P},Y_{1}(x^{**}_{P}))$,
it follows (2.44) with $\tilde{a}_{2}>0$ given by (2.45). ∎

###  7.6. Analytic continuation of the function $x{\to}H_{j}(x,0)$, cases (B5)
and (B6)

Now we are ready to extend the function $x\to H_{j}(x,0)$ beyond the point
$x_{d}$ when either (B5) or (B6) holds. This is a subject of Proposition 7.12
below.

######  Proposition 7.12.

Under the hypotheses (A1)-(A3), there is $\delta>0$ such that for any
$j\in\mathbb{Z}^{2}_{+}$, the following assertions hold

i) If (B6) holds, then the function $x\to H_{j}(x,0)$ can be extended as an
analytic function to the set $B(0,x_{d}+\delta){\setminus}\\{x_{d}\\}$ and
(2.52) holds with $a_{5}>0$ given by (2.53).

ii) If (B5) holds and $x_{d}=x^{**}<x^{**}_{P}$, then the function $x\to
H_{j}(x,0)$ can be extended as an analytic function to the set
$B(0,x_{d}+\delta){\setminus}\\{x_{d}\\}$ and (2.46) holds with $a_{3}>0$
given by (2.47).

iii) If (B5), $x_{d}=x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$ hold, the function $x\to
H_{j}(x,0)$ can be extended as an analytic function to the set
$B(0,x_{d}+\delta){\setminus}[x_{d},x_{d}+\delta[$ and (2.48) holds with
$a_{4}>0$ given by (2.49).

iv) If (B5), $x_{d}=x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$ hold, the function $x\to
H_{j}(x,0)$ can be extended as an analytic function to the set
$B(0,x_{d}+\tilde{\delta}){\setminus}[x_{d},x_{d}+\tilde{\delta}[$ and (2.50)
holds with $\tilde{a}_{4}$ given by (2.51).

Recall that under our hypotheses, all functions $(x,y){\to}L_{j}(x,y)$,
$(x,y){\to}\psi_{1}(x,y)$ and $(x,y){\to}\psi_{2}(x,y)$ are analytic in some
neighborhood ${\mathcal{V}}$ of the set $\Omega(\overline{\Gamma})$.
Throughout this section, the set ${\mathcal{V}}$ will be given.

####  7.6.1. Preliminary result

We begin the proof of this proposition with the following lemma.

######  Lemma 7.5.

Under the hypotheses (A1)-(A3) and if one of the cases (B5) or (B6) holds,
there is $\delta>0$ such that for any $j\in\mathbb{Z}^{2}_{+}$, the function
$x\to(1-\phi_{1}(x,Y_{1}(x)))H_{j}(0,Y_{1}(x))$ can be analytically continued
to the set $C(x_{d}-\delta,x_{d}+\delta)\cap
U_{\varepsilon}\setminus\\{x_{d}\\}$.

###### Proof.

Suppose that either (B5) or (B6) holds. Then by Proposition 2.2 and using
relations (2.33), (2.34)), one gets

(7.32) $X_{1}(y_{d})<x_{d}=X_{2}(y_{d})\leqslant x^{**},\quad
y^{*}<y_{d}=y^{**}=Y_{1}(x_{d})<y^{**}_{P}\quad\text{and}\quad(x_{d},y_{d})\in{\mathcal{S}}_{21},$

with $x_{d}=x^{**}$ in the case $(B5)$, and $x_{d}<x^{**}$ in the case (B6).
Hence, with the same arguments as in the proof of proposition 7.10, (it is
sufficient to exchange the roles of $x$ and $y$), one gets that there exists
$\delta_{0}>0$ such that for any $j\in\mathbb{Z}^{2}_{+}$, the functions
$h_{1j}$ and $y\to yh_{2j}(y)=H_{j}(0,y)$ and can be analytically continued to
the set $B(0,y_{d}+\delta_{0}){\setminus}\\{y_{d}\\}$, and

$\displaystyle\lim_{y\to y_{d}}(y_{d}-y)yh_{2j}(y)$ $\displaystyle=\lim_{y\to
y_{d}}(y_{d}-y)H_{j}(0,y)$ (7.33)
$\displaystyle=c_{1}\Bigl{(}L_{j}(X_{1}(y_{d}),y_{d})+(\phi_{1}(X_{1}(y_{d}),y_{d}))-1)H_{j}(X_{1}(y_{d}),0)\Bigr{)}$

with

(7.34)
$c_{1}=\left.\left(\frac{d}{dy}\phi_{2}(X_{1}(y),y)\right)^{-1}\right|_{y=y_{d}}>0.$

Since by (7.32), $y_{d}<y^{**}_{P}$, without any restriction of generality, we
will suppose throughout our proof that

(7.35) $y_{d}+\delta_{0}<y^{**}_{P}.$

Moreover, by using Proposition 7.1 and the first assertion of Corollary 7.6 we
can inject $y=Y_{1}(x)$ to the functional equation (2.30):

  * –

by Proposition 7.1, for some ${\varepsilon}>0$, the function $Y_{1}$ is
already analytically continued to $\overset{\circ}{U}_{\varepsilon}$ and
extended as a continuous function to the set $U_{\varepsilon}$,

  * –

by Corollary 7.6, there is $\delta_{1}{\in}]0,{\varepsilon}[$, such that any
$j\in\mathbb{Z}^{2}_{+}$, the functions $x\to
L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ and
$x\to-\psi_{1}(x,Y_{1}(x))h_{1j}(x)=(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ are
analytic in the set $C(x_{d}-\delta_{1},x_{d})$ and for any $x\in
C(x_{d}-\delta_{1},x_{d})$ the following relation holds

(7.36)
$(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)=L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x)).$

To complete the proof of Lemma 7.5, the following steps will be performed:

  1. step 1:

First we will show that for some $\delta_{2}\in]0,\delta_{1}[$, the function
$x\to L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in
the set

(7.37) $\\{x\in C(x_{d}-\delta_{2},x_{d}+\delta_{2})\cap
U_{\varepsilon}:\leavevmode\nobreak\ Y_{1}(x)\not=Y_{1}(x_{d})\\}.$

  2. step 2:

Next we will prove that for some $\delta_{3}\in]0,\delta_{2}[$, the point
$x_{d}$ is an only zero of the function $x\to Y_{1}(x)-Y_{1}(x_{d})$ in
$C(x_{d}-\delta_{3},x_{d}+\delta_{3})\cap U_{\varepsilon}$, and we will deduce
from our previous result that the function $x\to
L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in the set

(7.38) $C(x_{d}-\delta_{3},x_{d}+\delta_{3})\cap
U_{\varepsilon}\setminus\\{x_{d}\\}.$

With this result and using the identity (7.36) we will be able to complete our
proof.

Step 1: Since because of (7.32), $Y_{1}(x_{d})=y_{d}<y_{d}+\delta_{0}$, by
Proposition 7.4 applied with $\hat{y}=y_{d}+\delta_{0}$, we obtain that for
some $0<\delta_{2}<\delta_{1}$, the function $x\to(x,Y_{1}(x))$ maps the set
$C(x_{d}-\delta_{2},x_{d}+\delta_{2})\cap U_{\varepsilon}$ to the set
$\\{(x,y)\in{\mathcal{V}}:\,|y|<y_{d}+\delta_{0}\\}$. Since the function $y\to
H_{j}(0,y)$ is already analytically continued to the set
$B(0,y_{d}+\delta_{0})\setminus\\{y_{d}\\}$ and the functions
$(x,y){\to}\psi_{2}(x,y)$ and $(x,y)\to L_{j}(x,y)$ are analytic in
${\mathcal{V}}$, it follows that the function $x\to
L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in the set
(7.37).

Step 2: by (7.32) and (2.9), we have

$y^{*}_{P}\leqslant y^{*}<y_{d}=y^{**}=Y_{1}(x_{d}),$

and consequently, $(x_{d},y_{d})\not=(X_{1}(y^{*}_{P}),y^{*}_{P})$. Since by
(7.32), $(x_{d},y_{d})\in{\mathcal{S}}_{21}$, using the definition of the
curve ${\mathcal{S}}_{21}$ (see (2.17)) one gets therefore

$X_{1}(y^{*}_{P})<x_{d}\leqslant x^{**}_{P},$

and consequently, using the assertions (ii) and (iv) of Proposition 7.1, we
conclude that for some $0<\delta_{3}<\delta_{2}$, the point $x_{d}$ is an only
zero of the function $x\to Y_{1}(x)-Y_{1}(x_{d})$ in
$C(x_{d}-\delta_{3},x_{d}+\delta_{3})\cap U_{\varepsilon}$. Since with our
previous result, we have already proved that the function $x\to
L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in the set
(7.37), this proves that the function $x\to
L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$ is analytic in the set
(7.38). Finally, since on the annulus $C(x_{d}-\delta_{1},x_{d})$, the
identity (7.36) holds, this proves that the function
$x{\to}-\psi_{1}(x,Y_{1}(x))h_{1j}(x)=(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ can
be analytically continued to the set
$(C(x_{d}-\delta_{3},x_{d}+\delta_{3})\cap
U_{\varepsilon}){\setminus}\\{x_{d}\\}$. ∎

####  7.6.2. Proof of Proposition 7.12

To complete the proof of Proposition 7.12, we consider separately all possible
cases :

  * –

when (B6) holds;

  * –

when (B5) holds with $x_{d}=x^{**}<x^{**}_{P}$;

  * –

when (B5) holds with $x_{d}=x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$;

  * –

when (B5) holds with $x_{d}=x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$.

Suppose first that (B6) holds. Then by Proposition 2.2 and the definition of
$x_{d}$ (see (2.33)), the following relations hold

(7.39) $x^{*}_{P}\leqslant
x^{*}<X_{1}(x^{**})<x_{d}=X_{2}(y^{**})<x^{**}\leqslant x^{**}_{P}.$

Hence, by Lemma 7.5 and according to the definition of the set
$U_{\varepsilon}$ (see (7.2)), for some $\hat{\delta}>0$, the function
$x\to-\psi_{1}(x,Y_{1}(x))h_{1j}(x)=(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$ is
already analytically continued to the set
$C(x_{d}-\hat{\delta},x_{d}+\hat{\delta}){\setminus}\\{x_{d}\\}$ . By
Proposition 7.7, for any $x\in C(x^{*},x^{**})$,

$|\phi_{1}(x,Y_{1}(x))|\leqslant\phi_{1}(|x|,Y_{1}(|x|))<1.$

Since by (7.39), $x^{*}<x_{d}<x^{**}$, it follows that for
$0<\delta^{\prime}<\min\\{x^{**}-x_{d},x_{d}-x^{*}\\}$, the function $x\to
1/(1-\phi_{1}(x,Y_{1}(x))$ is analytic in
$C(x_{d}-\delta^{\prime},x_{d}+\delta^{\prime})$ and

(7.40) $1-\phi_{1}(x_{d},Y_{1}(x_{d}))>0.$

Since the function
$x\to\eta_{j}(x)=-\psi_{1}(x,Y_{1}(x))h_{1j}(x)=(1-\phi_{1}(x,Y_{1}(x)))H_{j}(x,0)$
is already extended as an analytic function to the set
$C(x_{d}-\hat{\delta},x_{d}+\hat{\delta}){\setminus}\\{x_{d}\\}$, it follows
that for $\delta=\min\\{\delta^{\prime},\hat{\delta}\\}$, the function the
function $x\to H_{j}(x,0)$ can be analytically continued to
$C(x_{d}-\delta,x_{d}+\delta){\setminus}\\{x_{d}\\}$ by letting

(7.41)
$H_{j}(x,0)=(1-\phi_{1}(x,Y_{1}(x)))^{-1}\bigl{(}L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))\bigr{)}.$

Remark finally that by (7.32), $y_{d}=y^{**}<y^{**}_{P}$. Hence, using exactly
the same arguments as in the proof of Proposition 7.7 (it is sufficient to
exchange the roles of $x$ and $y$), one gets
$\phi_{2}(X_{1}(y_{d}),y_{d})=\phi_{2}(X_{1}(y^{**}),y^{**})=1$. Since by
(7.32), $X_{1}(y_{d})<X_{2}(y_{d})=x_{d}$ and $Y_{1}(x_{d})=y_{d}$, and since
under our hypotheses (see Assumption (A3)(vi)) the real valued function
$x\to\phi_{2}(x,y_{d})$ is strictly increasing, it follows that

(7.42)
$\phi_{2}(x_{d},Y_{1}(x_{d}))=\phi_{2}(x_{d},y_{d})=\phi_{2}(X_{2}(y_{d}),y_{d})>\phi_{2}(X_{1}(y_{d}),y_{d})=1.$

Using these relations together with (7.41), (7.40), (7.33) and (7.5), we
obtain

$\displaystyle\lim_{x\to x_{d}}(x_{d}-x)$ $\displaystyle H_{j}(x,0)=\lim_{x\to
x_{d}}(x_{d}-x)(1-\phi_{1}(x,Y_{1}(x)))^{-1}\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$
$\displaystyle=\lim_{x\to
x_{d}}(x_{d}-x)(1-\phi_{1}(x,Y_{1}(x)))^{-1}(\phi_{2}(x,Y_{1}(x))-1)H_{j}(0,Y_{1}(x))$
$\displaystyle=a_{5}\Bigl{(}L_{j}(X_{1}(y_{d}),y_{d})+(\phi_{1}(X_{1}(y_{d}),y_{d})-1)H_{j}(X_{1}(y_{d}),0)\Bigr{)}=a_{5}\varkappa_{2}(j)$

with

$a_{5}=\left.(\phi_{2}(x,y)-1)\left((1-\phi_{1}(x,y))\frac{d}{dy}\phi_{2}(X_{1}(y),y)\frac{d}{dx}Y_{1}(x)\right)^{-1}\right|_{(x,y)=(x_{d},y_{d})}>0.$

The first assertion of Proposition 7.12 is therefore proved.

Suppose now that (B5) holds and let $x_{d}=x^{**}<x^{**}_{P}$. Then by
Proposition 2.2 and the definition of $x_{d}$ and $y_{d}$ (see (2.33) and
(2.34)), one has

$y_{d}=y^{**}=Y_{1}(x^{**})=Y_{1}(x_{d})\quad\text{and}\quad
x^{*}<X_{1}(y_{d})<x_{d}=X_{2}(y_{d})=x^{**}<x^{**}_{P}.$

and by Proposition 7.7, for some $\delta>0$, the function $x\to
1/(1-\phi_{1}(x,Y_{1}(x)))$ is meromorphic in the annulus
$C(x^{*},x_{d}+\delta)$, and has there a unique and simple pole at the point
$x_{d}=x^{**}$ with

(7.43) $\lim_{x\to
x_{d}}\frac{x_{d}-x}{1-\phi_{1}(x,Y_{1}(x))}=\left.\left(\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\right)^{-1}\right|_{x=x_{d}}>0;$

Here, the only difference with the previous case is that the function $x\to
1/(1-\phi_{1}(x,Y_{1}(x))$ has a simple pole at $x_{d}=x^{**}$. Using
therefore exactly the same arguments as in the previous case we obtain that
for some $\delta>0$, the function $x\to H_{j}(x,0)$ can be extended as an
analytic function to the set
$C(x_{d}-\delta,x_{d}+\delta){\setminus}\\{x_{d}\\}$. And using finally (7.41)
together with (7.33), (7.42), (7.5) and (7.43) we get

$\displaystyle\lim_{x\to x_{d}}$
$\displaystyle(x_{d}-x)^{2}H_{j}(x,0)=a_{3}\Bigl{(}L_{j}(X_{1}(y_{d}),y_{d})+(\phi_{1}(X_{1}(y_{d}),y_{d})-1)H_{j}(X_{1}(y_{d}),0)\bigr{)}=a_{3}\varkappa_{2}(j)$

with

$\displaystyle a_{3}$
$\displaystyle=\left.(\phi_{2}(x,Y_{1}(x))-1)\left(\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\frac{d}{dy}\phi_{2}(X_{1}(y),y)\frac{d}{dx}Y_{1}(x)\right)^{-1}\right|_{x=x_{d},y=y_{d}}$
$\displaystyle=\left.(\phi_{2}(x,y)-1)\left(\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\frac{d}{dy}\phi_{2}(X_{1}(y),y)\frac{d}{dx}Y_{1}(x)\right)^{-1}\right|_{x=x_{d},y=y_{d}}>0.$

The second assertion of Proposition 7.12 is therefore also proved.

Suppose now that (B5), $x_{d}=x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$ hold. Then by Lemma 7.5 and
according to the definition of the set $U_{\varepsilon}$ (see (7.2)), by using
the identity (7.36), the function $x\to-\psi_{1}(x,Y_{1}(x))h_{1j}(x)$ was
already analytically continued to the set
$C(x^{**}_{P}-\hat{\delta},x^{**}_{P}+\hat{\delta}){\setminus}[x^{**}_{P},x^{**}_{P}+\hat{\delta}[$,
and by Proposition 7.7, for some $0<\delta<\hat{\delta}$, the function
$x\to(1-\phi_{1}(x,Y_{1}(x))^{-1}$ is analytic in
$C(x^{**}_{P}-\delta,x^{**}_{P}+\delta){\setminus}[x^{**}_{P},x^{**}_{P}+\delta[$
and

$H_{j}(x,0)=xh_{1j}(x)=x\eta_{j}(x)(\psi_{1}(x,Y_{1}(x)))^{-1}=\eta_{j}(x)(1-\phi_{1}(x,Y_{1}(x))^{-1},\quad\forall
x\in C(x^{**}_{P}-\delta,x^{**}_{P}).$

Hence, by using the identity (7.41), the function $x\to H_{j}(x,0)$ can be
analytically continued to the set
$C(x^{**}_{P}-\delta,x^{**}_{P}+\delta){\setminus}[x^{**}_{P},x^{**}_{P}[$,
and using finally (7.41) together with (7.33), (7.6) and (7.32) one gets

$\displaystyle\lim_{x\to x^{**}_{P}}$
$\displaystyle\sqrt{x^{**}_{P}-x}H_{j}(x,0)$ $\displaystyle=\lim_{x\to
x^{**}_{P}}\sqrt{x^{**}_{P}-x}(1-\phi_{1}(x,Y_{1}(x))^{-1}\bigl{(}L_{j}(x,Y_{1}(x))+\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))\bigr{)}$
$\displaystyle=\lim_{x\to
x^{**}_{P}}\sqrt{x^{**}_{P}-x}(1-\phi_{1}(x,Y_{1}(x))^{-1}\psi_{2}(x,Y_{1}(x))h_{2j}(Y_{1}(x))$
$\displaystyle=\lim_{x\to
x^{**}_{P}}\sqrt{x^{**}_{P}-x}(1-\phi_{1}(x,Y_{1}(x))^{-1}(\phi_{2}(x,Y_{1}(x))-1)H_{j}(0,Y_{1}(x))$
$\displaystyle=\tilde{a}_{4}\Bigl{(}L_{j}(X_{1}(y_{d}),y_{d})+(\phi_{1}(X_{1}(y_{d}),y_{d})-1)H_{j}(X_{1}(y_{d}),0)\bigr{)}=\tilde{a}_{4}\varkappa_{2}(j)$

with

$\tilde{a}_{4}=(\phi_{2}(x_{d},y_{d})-1)\sqrt{\partial^{2}_{yy}P(x_{d},y_{d})/\partial_{x}P(x_{d},y_{d})}\left((1-\phi_{1}(x_{d},y_{d}))\frac{d}{dy}\phi_{2}(X_{1}(y_{d}),y_{d})\right)^{-1}>0,$

Consider now the case when (B5), $x_{d}=x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$ hold. In this case, with exactly
the same arguments as above one gets that for some $\delta>0$, the function
$x\to H_{j}(x,0)$ can be continued as an analytic function to the set
$C(x^{**}_{P}-\delta,x^{**}_{P}+\delta){\setminus}[x^{**}_{P},x^{**}_{P}[$ by
using the identity (7.41). The only difference is here that now, by
Proposition 7.7,

$\frac{1}{1-\phi_{1}(x,Y_{1}(x))}\sim\frac{c_{1}}{\sqrt{x^{**}_{P}-x}}$

with

$c_{1}=\left.\left(\partial_{y}\phi_{1}(x,y))\sqrt{\partial_{x}P(x,y)/\partial^{2}_{yy}P(x,y)}\right)^{-1}\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))}>0$

and consequently, since in this case $x_{d}=x^{**}_{P}$ and
$y_{d}=Y_{1}(x^{**}_{P})$, using (7.41) together with (7.33) and (7.32), one
gets

$\lim_{x\to
x_{d}}(x_{d}-x)H_{j}(x,0)=a_{4}\Bigl{(}L_{j}(X_{1}(y_{d}),y_{d})+(\phi_{1}(X_{1}(y_{d}),y_{d})-1)H_{j}(X_{1}(y_{d}),0)\bigr{)}$

with

$a_{4}=\left.(\phi_{2}(x,y)-1)\partial^{2}_{yy}P(x,y)\left(\partial_{y}\phi_{1}(x,y)\partial_{x}P(x,y)\frac{d}{dy}\phi_{2}(X_{1}(y),y)\right)^{-1}\right|_{(x,y)=(x_{d},y_{d})}>0.$

Proposition 7.12 is therefore proved.

###  7.7. Analytic continuation of the function $(x,y)\to(1-P(x,y))H_{j}(x,y)$

The following Lemma is the first step in the proof of the last assertion of
Theorem 2.

######  Lemma 7.6.

If the conditions (A1)-(A3) are satisfied and (B2) holds, then

  * –

the set $\\{(x,y)\in{\mathcal{S}}_{22}:\;x<x_{d},\;y<y_{d}\\}$ is non-empty;

  * –

there exists a neighborhood ${\mathcal{V}}$ of the set ${\mathcal{S}}_{22}$ in
$\mathbb{R}^{2}_{+}$ such that, for any $j{\in}\mathbb{Z}^{2}_{+}$,
the function $x{\to}(1{-}P(x,y))H_{j}(x,y)$ can be analytically continued to
the set $\\{(x,y){\in}\Omega({\mathcal{V}}):|x|{<}x_{d},|y|{<}y_{d}\\}$;

  * –

for any
$(\hat{x},\hat{y}){\in}\\{(x,y){\in}{\mathcal{S}}_{22}:x{<}x_{d},y{<}y_{d}\\}$,
the function $\varkappa_{(\hat{x},\hat{y})}$ is non-negative on
$\mathbb{Z}^{2}_{+}$ and for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$,

(7.44) $\lim_{\begin{subarray}{c}(x,y)\to(\hat{x},\hat{y})\\\
(x,y)\in\overset{\circ}{D}\end{subarray}}(1-P(x,y))(H_{j}(x,y)-H_{j}(x,0)-H_{j}(0,y))=\varkappa_{(\hat{x},\hat{y})}$

###### Proof.

Indeed, by Theorem 1, for any $j\in\mathbb{Z}^{2}_{+}$ and
$(\hat{x},\hat{y})\in{\mathcal{S}}_{22}$ with $\hat{x}<x_{d}$ and
$\hat{y}<y_{d}$, there is a neighborhood $V(\hat{x},\hat{y})$ of the point
$(\hat{x},\hat{y})$ in $\mathbb{R}^{2}$, such that for any
$j\in\mathbb{Z}^{2}_{+}$, the function

$\displaystyle(x,y)\to L_{j}(x,y)+(\phi_{1}(x,y)-1)$ $\displaystyle
H_{j}(x,0)+(\phi_{2}(x,y)-1)H_{j}(0,y)$
$\displaystyle=L_{j}(x,y)+\psi_{1}(x,y)h_{1j}(x)+\psi_{2}(x,y)h_{2j}(y)$

is analytic in the polycircular set
$\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\ (|x|,|y|)\in
V(\hat{x},\hat{y})\\}$, and the function
$(x,y)\to(1-P(x,y))(H_{j}(x,y)-H_{j}(x,0)-H_{j}(0,y))=Q(x,y)h_{j}(x,y)$ can be
continued as an analytic function to
$\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\ (|x|,|y|)\in
V(\hat{x},\hat{y})\\}$ by letting

$(1-P(x,y))(H_{j}(x,y)-H_{j}(x,0)-H_{j}(0,y))=L_{j}(x,y)+(\phi_{1}(x,y)-1)H_{j}(x,0)+(\phi_{2}(x,y)-1)H_{j}(0,y).$

Hence, for any $j\in\mathbb{Z}^{2}_{+}$ the quantity

$\varkappa_{(\hat{x},\hat{y})}=L_{j}(\hat{x},\hat{y})+(\phi_{1}(\hat{x},\hat{y})-1)H_{j}(\hat{x},0)+(\phi_{2}(\hat{x},\hat{y})-1)H_{j}(0,\hat{y})$

is well defined and as $(x,y)\to(\hat{x},\hat{y})$ for
$(x,y)\in\overset{\circ}{D}$, (7.44) holds. Moreover, since by Theorem 1, the
function $(x,y)\to H_{j}(x,y)$ is analytic in the set
$\Omega_{d}(\Gamma)=\\{(x,y)\in\mathbb{C}^{2}:(|x|,|y|)\in\Gamma,\;|x|<x_{d},\;|y|<y_{d}\\}$,
and since the set $\Omega_{d}(\Gamma)$ is a union of the poly-discs centered
at the origin in $\mathbb{C}^{2}$, the power series

$H_{j}(x,y)-H_{j}(x,0)-H_{j}(0,y)=\sum_{k_{1}=1}^{\infty}\sum_{k_{2}=1}^{\infty}g(j,k)x^{k_{1}}y^{k_{2}}$

converge on the set $\Omega_{d}(\Gamma)$, and consequently, for
$(x,y)\in\Gamma$ such that $x<x_{d}$ and $y<y_{d}$,

$H_{j}(x,y)-H_{j}(x,0)-H_{j}(0,y)\geqslant 0.$

Since for $(x,y)\in\overset{\circ}{D}$, $P(x,y)<1$, it follows that for any
$j\in\mathbb{Z}^{2}_{+}$, the left hand side of (7.44) is non-negative and
consequently, the function $\varkappa_{(\hat{x},\hat{y})}$ is non-negative on
$\mathbb{Z}^{2}_{+}$. ∎

###  7.8. Harmonic functions $\varkappa_{i}$, $\tilde{\varkappa}_{i}$ and
$\varkappa_{(x,y)}$ and their properties

In this section, the second step of the proof of Theorem 2 is performed (see
Section 7.1). This is a subject of the following statement.

######  Proposition 7.13.

Under the hypotheses (A1) - (A3), the following assertions hold:

i) The function $\varkappa_{1}$ defined by (2.35) is non-negative on
$\mathbb{Z}^{2}_{+}$, positive on $\mathbb{Z}_{+}^{2}{\setminus}E_{0}$ and
harmonic for $(Z_{\tau_{0}}(n))$ in each of the following cases :

  * –

if one of the assertions (B0), (B1), (B3) or (B4) is valid;

  * –

if (B2) holds and $x^{*}<x^{**}_{P}$;

  * –

if (B2) holds, $x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$.

ii) The function $\varkappa_{2}$ defined by (2.37) is non-negative on
$\mathbb{Z}^{2}_{+}$, positive on $\mathbb{Z}_{+}^{2}{\setminus}E_{0}$ and
harmonic for $(Z_{\tau_{0}}(n))$ in each of the following cases :

  * –

if one of the assertions (B0), (B1), (B5) or (B6) is valid;

  * –

if (B2) holds and $y^{**}<y^{**}_{P}$;

  * –

if (B2) holds, $y^{**}=y^{**}_{P}$ and
$\phi_{1}(X_{1}(y^{**}_{P}),y^{**}_{P})=1$.

iii) If (B2) is valid with $x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$, then the function
$\tilde{\varkappa}_{1}$ defined by (2.36) is non-negative on
$\mathbb{Z}^{2}_{+}$, positive on $\mathbb{Z}_{+}^{2}{\setminus}E_{0}$ and
harmonic for $(Z_{\tau_{0}}(n))$.

iv) If (B2) is valid, then for any $(\hat{x},\hat{y})\in{\mathcal{S}}_{22}$
with $\tilde{x}<x_{d}$ and $\hat{y}<y_{d}$, the function
$\varkappa_{(\hat{x},\hat{y})}$ defined by (2.39) is non-negative on
$\mathbb{Z}^{2}_{+}$, positive on $\mathbb{Z}_{+}^{2}{\setminus}E_{0}$ and
harmonic for $(Z_{\tau_{0}}(n))$.

To get this result, we first show there that the left hand sides of (2.40),
(2.42), (2.44), (2.46), (2.48), (2.50), (2.52) and (2.54) are non-negative.
With these arguments we conclude that each of the functions defined by (2.35),
(2.36), (2.37), (2.39) is non-negative on $\mathbb{Z}^{2}_{+}$. Next we show
that each of these functions (in the corresponding cases i)-iv). of our
proposition) is harmonic for $(Z_{\tau_{0}}(n))$, and by using suitable
Lyapunov functions, we prove that each of them is not identically zero on the
set $\\{j\in\mathbb{Z}^{2}_{+};\;\|j\|\geqslant N_{0}\\}$. With this results
and using Lemma 2.1 we will be able to show that each of the functions defined
by (2.35), (2.36), (2.37), (2.39) (in the corresponding cases i)-iv). of our
proposition) is positive throughout the set
$\mathbb{Z}^{2}_{+}{\setminus}E_{0}$.

####  7.8.1. Preliminary estimates.

We begin the proof of this proposition with the following preliminary result

######  Lemma 7.7.

Suppose that the conditions (A1)-(A3) are satisfied and let two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfy (6.6).
Then for some constant $C>0$ (depending on the points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ but do not
depending on $j\in\mathbb{Z}^{2}_{+}$),

(7.45) ${\mathbb{P}}_{j}(\tau_{0}<+\infty)\leqslant
C\left(x_{1}^{j_{1}}y_{1}^{j_{2}}+x_{2}^{j_{1}}y_{2}^{j_{2}}\right),\quad\forall
j\in\mathbb{Z}^{2}_{+}$

and

(7.46)
$\sum_{k\in\mathbb{Z}^{2}_{+}}g(j,k)\left(x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}}\right)\leqslant
C\left(x_{1}^{j_{1}}y_{1}^{j_{2}}+x_{2}^{j_{1}}y_{2}^{j_{2}}\right),\quad\forall
j\in\mathbb{Z}^{2}_{+}.$

###### Proof.

Indeed, if two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfy (6.6),
then by Lemma 6.1, for some $0<\theta<1$ and some finite subset $E$ of
$\mathbb{Z}^{2}_{+}$ such that $(0,0)\in E$ and the function
$f:\mathbb{Z}^{2}_{+}\to\mathbb{R}_{+}$ defined by
$f(j_{1},j_{2}){=}x_{1}^{j_{1}}y_{1}^{j_{2}}{+}x_{2}^{j_{1}}y_{2}^{j_{2}}$,
for all $(j_{1},j_{2}){\in}\mathbb{Z}^{2}_{+}$ satisfies the inequality

${\mathbb{E}}_{j}(f(Z(1));\;\tau_{E}>1)\leqslant\theta f(j),\quad\forall
j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+},$

with $\tau_{E}=\inf\\{n>0\leavevmode\nobreak\ :\leavevmode\nobreak\ Z(n)\in
E\\}$. By Lemma 6.2, it follows that for any $j\in\mathbb{Z}^{2}_{+}$, the
series

(7.47)
$\sum_{k\in\mathbb{Z}^{2}}g(j,k)f(k)=\sum_{k\in\mathbb{Z}^{2}}g(j,k)\left(x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}}\right)$

converges, and moreover, (see (6.12) in the proof of Lemma 6.2), for

$g_{E}(j,k)=\sum_{n=0}^{\infty}{\mathbb{P}}_{j}(Z(n)=k;\;\tau_{E}\geqslant
n),\quad j\in\mathbb{Z}^{2}_{+}{\setminus}E,\;k\in\mathbb{Z}^{2}_{+},$

the following relation holds :

(7.48)
$\sum_{k\in\mathbb{Z}^{2}_{+}}g_{E}(j,k)f(k)\leqslant\frac{f(j)}{1-\theta}=\frac{x_{1}^{j_{1}}y_{1}^{j_{2}}+x_{2}^{j_{1}}y_{2}^{j_{2}}}{1-\theta}.$

Since $(0,0)\in E$, from the last relation it follows in particular that for
any $j\in\mathbb{Z}^{2}_{+}{\setminus}E$,

$\displaystyle{\mathbb{P}}_{j}(\tau_{0}<+\infty)\leqslant{\mathbb{P}}_{j}(\tau_{E}<+\infty)$
$\displaystyle\leqslant\left(\min_{\ell\in
E}f(\ell)\right)^{-1}\sum_{k\in\mathbb{Z}^{2}_{+}}g_{E}(j,k)f(k)$ (7.49)
$\displaystyle\leqslant\left(\min_{\ell\in
E}f(\ell)\right)^{-1}\frac{1}{1-\theta}\left(x_{1}^{j_{1}}y_{1}^{j_{2}}+x_{2}^{j_{1}}y_{2}^{j_{2}}\right)$

and consequently (7.45) is proved.

Remark finally that for any $j=(j_{1},0)\in\mathbb{Z}^{2}_{+}{\setminus}E$,

$\displaystyle\sum_{k\in\mathbb{Z}^{2}_{+}}g(j,k)f(k)$
$\displaystyle=\sum_{k\in\mathbb{Z}^{2}_{+}}g_{E}(j,k)f(k)+\sum_{\ell\in
E\setminus\\{(0,0)\\}}g_{E}(j,\ell)\sum_{k\in\mathbb{Z}^{2}_{+}}g(\ell,k)f(k),$
$\displaystyle\leqslant\sum_{k\in\mathbb{Z}^{2}_{+}}g_{E}(j,k)f(k)\leavevmode\nobreak\
+\leavevmode\nobreak\ {\mathbb{P}}_{j}(\tau_{E}<+\infty)\,\max_{\ell\in
E}\sum_{k\in\mathbb{Z}^{2}_{+}}g(\ell,k)f(k).$

Since the set $E$ is finite and for any $j\in\mathbb{Z}^{2}_{+}$, the series
(7.47) converge, this last relation combined with (7.48) and (7.49) prove
(7.46). ∎

######  Lemma 7.8.

Under the hypotheses (A1)-(A3), for any $0<y<y_{d}$ there are two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfying
(6.6) and such that for some $C>0$,

(7.50) $H_{j}(0,y)\leqslant
C\left(x_{1}^{k_{1}}y_{1}^{k_{2}}+x_{2}^{k_{1}}y_{2}^{k_{2}}\right),\quad\forall
j\in\mathbb{Z}^{2}_{+}.$

###### Proof.

Indeed, if $0<y<y_{d}$, then, by the definition of the set $\Theta$, the point
$(0,Y_{1}(x^{**}))$ is in $\Theta$ and, consequently, by Lemma 6.4 there are
two points $(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfying
relations (6.6) and (6.9) with $(x,y)=(0,Y_{1}(x^{**}))$. By Lemma 7.7, for
these two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$, one gets (7.46) with
$(x,y)=(0,y)$, and consequently (7.50) holds. ∎

####  7.8.2. Proof of the first two assertions of Proposition 7.13

Consider first the case when one of the following assertions holds:

  * –

one of the cases (B0), (B1), (B3), (B4) holds;

  * –

(B2) and $x_{d}{<}x^{**}_{P}$ hold;

  * –

(B2), $x_{d}{=}x^{**}_{P}$ and $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P})){=}1$
hold.

By Proposition 7.7 and the definition of $x_{d}$, we have always
$x_{d}=x^{**}$ and

(7.51) $\phi_{1}(x_{d},Y_{1}(x_{d}))=1.$

To prove Proposition 7.13 in each of the above cases, we first show that the
function $\varkappa_{1}$ is non-negative and harmonic for $(Z_{\tau_{0}}(n))$.
Next we will prove that $\varkappa_{1}(j_{1},0)>0$ for all $j_{1}>0$ large
enough, and using finally Lemma 2.1, we will conclude that the function
$\varkappa_{1}$ is strictly positive on the set
$\mathbb{Z}^{2}_{+}{\setminus}E_{0}$.

Suppose first that either, one of the cases (B0), (B1), (B3), (B4) holds, or
(B2) and $x_{d}<x^{**}_{P}$ hold. Then, by Proposition 7.10, for any
$j\in\mathbb{Z}^{2}_{+}$,

$\lim_{x\to x_{d}}(x_{d}-x)H_{j}(x,0)=c\varkappa_{1}(j)$

with $c>0$. Since $(x_{d}-x)H_{j}(x,0)\geqslant 0$ for any real
$x\in]0,x^{**}[$, it follows that $\varkappa_{1}(j)\geqslant 0$ for any
$j\in\mathbb{Z}^{2}_{+}$. When either one of the cases (B0), (B1), (B3), (B4)
occurs or (B2) and $x^{**}<x^{**}_{P}$ hold, the function $\varkappa_{1}$ is
therefore non-negative on $\mathbb{Z}^{2}_{+}$.

Suppose now that (B2), $x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))=1$ hold. In this case, by Proposition
7.11, for any $j\in\mathbb{Z}^{2}_{+}$,

$\lim_{x\to x_{d}}\sqrt{x_{d}-x}\,H_{j}(x,0)=c\,\varkappa_{1}(j)$

with $c>0$. Since $\sqrt{x_{d}-x}\,H_{j}(x,0)\geqslant 0$ for any real
$x\in]0,x_{d}[$, it follows that $\varkappa_{1}(j)\geqslant 0$ for any
$j\in\mathbb{Z}^{2}_{+}$. Hence, in this case, the function $\varkappa_{1}$ is
therefore also non-negative on $\mathbb{Z}^{2}_{+}$.

Harmonicity property of the function $\varkappa_{1}$ for the killed random
walk. See Definition 1. For $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$, by the
definition of the functions $(x,y)\to L_{j}(x,y)$ and
$\varkappa_{1}:\mathbb{Z}^{2}_{+}\to\mathbb{R}_{+}$, we have

$\varkappa_{1}(j)=\begin{cases}x_{d}^{j_{1}}(Y_{1}(x_{d}))^{j_{2}}-{\mathbb{P}}_{j}(\tau<+\infty)+(\phi_{2}(x_{d},Y_{1}(x_{d}))-1)H_{j}(0,Y_{1}(x_{d}))&\text{if
$j\not=(0,0)$},\\\
\phi_{0}(x_{d},Y_{1}(x_{d}))-{\mathbb{P}}_{(0,0)}(\tau<+\infty)+(\phi_{2}(x_{d},Y_{1}(x_{d}))-1)H_{(0,0)}(0,Y_{1}(x_{d}))&\text{if
$j=(0,0)$}.\end{cases}$

Remark that for any $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$,

$\displaystyle{\mathbb{E}}_{j}\left({\mathbb{P}}_{Z(1)}(\tau<+\infty);\;\tau>1\right)=$
$\displaystyle{\mathbb{P}}_{j}(\tau<+\infty)-\mu_{1}(-1,0)\mathbb{1}_{\\{(0,1)\\}}(j_{1},j_{2})-\mu_{2}(0,-1)\mathbb{1}_{\\{(1,0)\\}}(j_{1},j_{2})$
$\displaystyle-\mu(-1,-1)\mathbb{1}_{\\{(1,1)\\}}(j_{1},j_{2})-\mu_{0}(0,0)\mathbb{1}_{\\{(0,0)\\}}(j_{1},j_{2})$

and since the function $j\to H_{j}(0,Y_{1}(x_{d}))$ is potential for
$(Z_{\tau_{0}}(n))$ (see the properties of potential for the Markov chains
functions in [25]),

${\mathbb{E}}_{j}(H_{Z(1)}(0,Y_{1}(x_{d})),\;\tau>1)=H_{j}(0,Y_{1}(x_{d}))-(Y_{1}(x_{d}))^{j_{2}}\mathbb{1}_{\\{j_{1}=0,j_{2}>0\\}}(j_{1},j_{2}),$

where for $j,k\in\mathbb{Z}^{2}$ we denote

$\mathbb{1}_{\\{k\\}}(j)=\begin{cases}1&\text{if $j=k$},\\\
0&\text{otherwise},\end{cases}$

By the definition of the function $Y_{1}$, we have $P(x_{d},Y_{1}(x_{d}))=1$,
and because of (7.51), $\phi_{1}(x_{d},Y_{1}(x_{d}))=1$. Hence, for any
$j\in\mathbb{Z}^{2}_{+}$,

$\displaystyle{\mathbb{E}}_{j}\bigl{(}x_{d}^{Z_{1}(1)}$
$\displaystyle(Y_{1}(x_{d}))^{Z_{2}(1)};\;\tau>1\bigr{)}$
$\displaystyle=\begin{cases}\phi_{0}(x_{d},Y_{1}(x_{d}))-\mu_{0}(0,0)\mathbb{1}_{\\{(0,0)\\}}(j_{1},j_{2})&\text{if
$j=(0,0)$},\\\
x_{d}^{j_{1}}\phi_{1}(x_{d},Y_{1}(x_{d}))-\mu_{1}(-1,0)\mathbb{1}_{\\{(0,1)\\}}(j_{1},j_{2})&\text{if
$j_{1}>0$ and $j_{2}=0$},\\\
(Y_{1}(x_{d}))^{j_{2}}\phi_{2}(x_{d},Y_{1}(x_{d}))-\mu_{2}(0,-1)\mathbb{1}_{\\{(1,0)\\}}(j_{1},j_{2})&\text{if
$j_{1}=0$ and $j_{2}>0$},\\\
{x_{d}^{j_{1}}(Y_{1}(x_{d}))^{j_{2}}}P(x_{d},Y_{1}(x_{d}))-\mu(-1,-1)\mathbb{1}_{\\{(1,1)\\}}(j_{1},j_{2})&\text{if
$j_{1}>0$, $j_{2}>0$},\end{cases}$
$\displaystyle=\begin{cases}\phi_{0}(x_{d},Y_{1}(x_{d}))-\mu_{0}(0,0)\mathbb{1}_{\\{(0,0)\\}}(j_{1},j_{2})&\text{if
$j=(0,0)$},\\\
x_{d}^{j_{1}}-\mu_{1}(-1,0)\mathbb{1}_{\\{(0,1)\\}}(j_{1},j_{2})&\text{if
$j_{1}>0$ and $j_{2}=0$},\\\
(Y_{1}(x_{d}))^{j_{2}}\phi_{2}(x_{d},Y_{1}(x_{d}))-\mu_{2}(0,-1)\mathbb{1}_{\\{(1,0)\\}}(j_{1},j_{2})&\text{if
$j_{1}=0$ and $j_{2}>0$},\\\
{x_{d}^{j_{1}}(Y_{1}(x_{d}))^{j_{2}}}-\mu(-1,-1)\mathbb{1}_{\\{(1,1)\\}}(j_{1},j_{2})&\text{if
$j_{1}>0$, $j_{2}>0$},\end{cases}$

With these relations, for any $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with
$j_{1}>0$, we get

$\displaystyle{\mathbb{E}}_{j}(\varkappa_{1}(Z(1)),\;\tau>1)$
$\displaystyle=x_{d}^{j_{1}}(Y_{1}(x_{d}))^{j_{2}}-{\mathbb{P}}_{j}(\tau<+\infty)+(\phi_{2}(x_{d},Y_{1}(x_{d}))-1)H_{j}(0,Y_{1}(x_{d})),$
(7.52) $\displaystyle=\varkappa_{1}(j),$

for any $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with $j_{1}=0$ and $j_{2}>0$,
we get

$\displaystyle{\mathbb{E}}_{j}(\varkappa_{1}(Z(1))$
$\displaystyle={x_{d}^{j_{1}}(Y_{1}(x_{d}))^{j_{2}}}\phi_{2}(x_{d},Y_{1}(x_{d}))-{\mathbb{P}}_{j}(\tau<+\infty)$
$\displaystyle\hskip
56.9055pt+(\phi_{2}(x_{d},Y_{1}(x_{d}))-1)\bigl{(}-{x_{d}^{j_{1}}(Y_{1}(x_{d}))^{j_{2}}}+H_{j}(0,Y_{1}(x_{d}))\bigr{)}$
(7.53) $\displaystyle=\varkappa_{1}(j).$

and for $j=(0,0)$, we obtain

$\displaystyle{\mathbb{E}}_{j}(\varkappa_{1}(Z(1))$
$\displaystyle=\phi_{0}(x_{d},Y_{1}(x_{d}))-{\mathbb{P}}_{(0,0)}(\tau<+\infty)+(\phi_{2}(x_{d},Y_{1}(x_{d}))-1)H_{(0,0)}(0,Y_{1}(x_{d}))$
(7.54) $\displaystyle=\varkappa_{1}(0,0).$

Relations (7.52), (7.53) and (7.54) prove that function $\varkappa_{1}$ is
harmonic for $(Z_{\tau_{0}}(n))$.

To show that the function $\varkappa_{1}$ is strictly positive at some point
$j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$, we investigate an asymptotic behavior
of this function as $j_{2}=0$ and $j_{1}\to+\infty$. By Proposition 2.2, in
each of the cases (B0)-(B4), we have $0<Y_{1}(x_{d})<y_{d}$. Hence, by Lemma
7.8, there are two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfying
(6.6) for which (7.50) holds for $y=Y_{1}(x_{d})$ and consequently, by Lemma
7.7 we obtain (7.45). Using these relations for
$j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with $j_{2}=0$ one gets

(7.55) $\left|\varkappa_{1}(j_{1},0)-x_{d}^{j_{1}}\right|\leqslant
C_{1}\left(x_{1}^{j_{1}}+x_{2}^{j_{1}}\right),\quad\forall
j=(j_{1},0)\in\mathbb{Z}^{2}_{+},$

with some do not depending on $j$ constant $C_{1}>0$. Remark that
$0<x_{2}<x_{1}$ because the points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ satisfy
(6.6), and that $x_{1}<x_{d}$ because
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$. Hence, from
the last relation it follows that

$\varkappa_{1}(j_{1},0)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
x_{d}^{j_{1}}\quad\text{as $j_{1}\to\infty$},$

and consequently, there is $N_{1}>0$ such that

(7.56) $\varkappa_{1}(j_{1},0)>0\quad\text{ for any
$j=(j_{1},0)\in\mathbb{Z}^{2}_{+}$ with $j_{1}\geqslant N_{1}$}.$

Now we are ready to complete the proof of the first assertion of Proposition
7.13 : By Lemma 2.1, there are $N_{0}>0$ and a finite subset $E_{0}$ of
$\mathbb{Z}^{2}_{+}$ such that $(0,0)\not\in E_{0}$ and for any
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and $k\in\mathbb{Z}^{2}_{+}$ with
$\|k\|>N_{0}$,

$g(j,k)=\sum_{n=0}^{\infty}{\mathbb{P}}_{j}(Z(n)=k,\;\tau_{0}>n)>0.$

Hence, for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and
$k\in\mathbb{Z}^{2}_{+}$ with $\|k\|>N_{0}$, there is $n_{j,k}\in{\mathbb{N}}$
such that

${\mathbb{P}}_{j}(Z(n_{j,k})=k,\;\tau_{0}>n_{j,k})>0.$

For any $j\in\mathbb{Z}^{2}_{+}$, $k\in\mathbb{Z}^{2}_{+}$ and
$n\in{\mathbb{N}}$, since the function $\varkappa_{1}$ is non-negative and
harmonic for the killed random walk, one has

$\varkappa_{1}(j)={\mathbb{E}}_{j}(\varkappa_{1}(Z(n)),\;\tau_{0}>n)\geqslant{\mathbb{P}}_{j}(Z(n)=k,\;\tau_{0}>n)\varkappa_{1}(k).$

Using this relation together with (7.56), for
$k=(k_{1},0)\in\mathbb{Z}^{2}_{+}$ with $k_{1}\geqslant\max\\{N_{0},N_{1}\\}$,
one gets that for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$

$\varkappa(j)\geqslant{\mathbb{P}}_{j}(Z(n_{j,k})=k,\;\tau_{0}>n_{j,k})\varkappa_{1}(k)>0,$

and consequently, the function $\varkappa_{1}$ is positive on
$\mathbb{Z}^{2}_{+}{\setminus}E_{0}$.

The first assertion of Proposition 7.13 is therefore proved. To get the second
assertion of this proposition, it is sufficient to exchange the roles of $x$
and $y$.

####  7.8.3. Proof of the third assertion of Proposition 7.13

Suppose that (B2) holds and let $x^{**}=x^{**}_{P}$ and
$\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$. By the definition of the point
$x_{d}$, we have here

$x_{d}=x^{**}=x^{**}_{P},$

and remark that in this case,

$\tilde{\varkappa}_{1}(j)=\left.\partial_{y}\left(\frac{L_{j}(x,y)+\left(\phi_{2}(x,y)-1\right)H_{j}(0,y)}{1-\phi_{1}(x,y)}\right)\right|_{(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))},\quad\;j\in\mathbb{Z}^{2}_{+}.$

To show that the function $\tilde{\varkappa}_{1}$ is non-negative on
$\mathbb{Z}^{2}_{+}$, we recall that in this cases, by Proposition 7.11,

$\lim_{x\to
x_{d}}\sqrt{x_{d}-x}\,\frac{d}{dx}\,H_{j}(x,0)=c\,\tilde{\varkappa}_{1}(j)$

with $c>0$. Since for real $x\in]0,x^{**}_{P}[$, the function $x\to
H_{j}(0,x)$ is increasing on $]0,x^{**}_{P}[$, it follows that
$\tilde{\varkappa}_{1}(j)\geqslant 0$ for any $j\in\mathbb{Z}^{2}_{+}$.

Harmonicity property of the function $\tilde{\varkappa}_{1}$ for the killed
random walk. For $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$, we have

(7.57)
$\tilde{\varkappa}_{1}(j)=\frac{\partial_{y}\phi_{1}((x^{**}_{P},Y_{1}(x^{**}_{P})))}{(1-\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P})))^{2}}\varkappa_{1}(j)+\frac{1}{1-\phi_{1}((x^{**}_{P},Y_{1}(x^{**}_{P})))}\hat{\varkappa}_{1}(j)$

with $\varkappa_{1}(j)$ defined by (2.35) and

(7.58)
$\displaystyle\hat{\varkappa}_{1}(j)=\partial_{y}L_{j}(x^{**}_{P},Y_{1}(x^{**}_{P}))+$
$\displaystyle(\phi_{2}(x^{**}_{P},Y_{1}(x^{**}_{P}))-1)\partial_{y}H_{j}(0,Y_{1}(x^{**}_{P}))$
$\displaystyle\quad\quad+\partial_{y}\phi_{2}((x^{**}_{P},Y_{1}(x^{**}_{P})))H_{j}(0,Y_{1}(x^{**}_{P})),$

where

$\partial_{y}L_{j}(x^{**}_{P},Y_{1}(x^{**}_{P}))=\begin{cases}j_{2}\times(x^{**}_{P})^{j_{1}}(Y_{1}(x^{**}_{P}))^{j_{2}-1}&\text{if
$j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}{\setminus}\\{(0,0)\\}$,}\\\
\partial_{y}\phi_{0}(x^{**}_{P},Y_{1}(x^{**}_{P}))&\text{if
$j=(j_{1},j_{2})=(0,0)$}.\end{cases}$

Remark now that for any $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$, with the same
arguments as in the proof of the first assertion of our proposition, one gets

(7.59)
${\mathbb{E}}_{j}(\varkappa_{1}(Z(1)),\;\tau_{0}>1)=\varkappa_{1}(j)+(\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))-1)(x^{**}_{P})^{j_{1}}\mathbb{1}_{\\{j_{1}>0,j_{2}=0\\}}(j_{1},j_{2})$

(here, an only difference with the proof of the first assertion of our
proposition is that $\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))<1$). Moreover,
since the functions $j\to H_{j}(0,Y_{1}(x^{**}_{P}))$ and
$j\to\partial_{y}H_{j}(0,Y_{1}(x^{**}_{P}))$ are potential for
$(Z_{\tau_{0}}(n))$ (see the properties of potential functions for Markov
chains in Woess [25]), then for any $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$,
one has

(7.60)
${\mathbb{E}}_{j}(H_{Z(1)}(0,Y_{1}(x^{**}_{P})),\;\tau_{0}>1)=H_{j}(0,Y_{1}(x^{**}_{P}))-(Y_{1}(x^{**}_{P}))^{j_{2}}\mathbb{1}_{\\{j_{1}=0,j_{2}>0\\}}(j_{1},j_{2})$

and

(7.61)
${\mathbb{E}}_{j}(\partial_{y}H_{Z(1)}(0,Y_{1}(x^{**}_{P})),\;\tau_{0}>1)=\partial
H_{j}(0,Y_{1}(x^{**}_{P}))-j_{2}\times(Y_{1}(x^{**}_{P}))^{j_{2}-1}\mathbb{1}_{\\{j_{1}=0,j_{2}>0\\}}.$

Finally, straightforward calculation shows that

$\displaystyle{\mathbb{E}}_{j}($
$\displaystyle\partial_{y}L_{Z(1)}(x^{**}_{P},Y_{1}(x^{**}_{P})),\;\tau_{0}>1)={\mathbb{E}}_{j}\bigl{(}Z_{2}(1)(x^{**}_{P})^{Z_{1}(1)}(Y_{1}(x^{**}_{P}))^{Z_{2}(1)-1},\;\tau_{0}>1\bigr{)}$
$\displaystyle=\partial_{y}\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))(x^{**}_{P})^{j_{1}}\mathbb{1}_{\\{j_{1}>0,j_{2}=0\\}}(j_{1},j_{2})$
$\displaystyle+\left(Y_{1}(x^{**}_{P})\partial_{y}P(x^{**}_{P},Y_{1}(x^{**}_{P}))+j_{2}P(x^{**}_{P},Y_{1}(x^{**}_{P}))\right)j_{2}(x^{**}_{P})^{j_{1}}(Y_{1}(x^{**}_{P}))^{j_{2}-1}$
$\displaystyle+j_{2}(Y_{1}(x^{**}_{P}))^{j_{2}-1}\bigl{(}Y_{1}(x^{**}_{P})\partial_{y}\phi_{2}(x^{**}_{P},Y_{1}(x^{**}_{P}))+j_{2}\phi_{2}(x^{**}_{P},Y_{1}(x^{**}_{P}))\bigr{)}\mathbb{1}_{\\{j_{1}=0,j_{1}>0\\}}(j_{1},j_{2}).$

Since $\partial_{y}P(x^{**}_{P},Y_{1}(x^{**}))=0$ and
$P(x^{**}_{P},Y_{1}(x^{**}))=1$, from the last relation it follows that

(7.62) $\displaystyle{\mathbb{E}}_{j}($
$\displaystyle\partial_{y}L_{Z(1)}(x^{**}_{P},Y_{1}(x^{**}_{P})),\;\tau_{0}>1)$
$\displaystyle=\partial_{y}\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P}))(x^{**}_{P})^{j_{1}}\mathbb{1}_{\\{j_{1}>0,j_{2}=0\\}}(j_{1},j_{2})+j_{2}(x^{**}_{P})^{j_{1}}(Y_{1}(x^{**}_{P}))^{j_{2}-1}$
$\displaystyle+j_{2}(Y_{1}(x^{**}_{P}))^{j_{2}-1}\bigl{(}Y_{1}(x^{**}_{P})\partial_{y}\phi_{2}(x^{**}_{P},Y_{1}(x^{**}_{P}))+j_{2}\phi_{2}(x^{**}_{P},Y_{1}(x^{**}_{P}))\bigr{)}\mathbb{1}_{\\{j_{1}=0,j_{1}>0\\}}(j_{1},j_{2}).$

When combined together, relations (7.57) - (7.61) imply that
${\mathbb{E}}_{j}(\tilde{\varkappa}_{1}(Z(1),\;\tau_{0}>1)=\tilde{\varkappa}_{1}(j)$
for any $j\in\mathbb{Z}^{2}_{+}$, and hence, the function
$\tilde{\varkappa}_{1}$ is harmonic for $(Z_{\tau_{0}}(n))$.

To show that the function $\tilde{\varkappa}_{1}$ is strictly positive on
$\mathbb{Z}^{2}_{+}{\setminus}\\{(0,0)\\}$, we investigate an asymptotical
behavior of this function as $j_{2}=0$ and $j_{1}\to+\infty$. Remark that for
any $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}{\setminus}\\{(0,0)\\}$ with
$j_{2}=0$,

$\tilde{\varkappa}_{1}(j)=C_{1}(x^{**}_{P})^{j_{1}}+C_{2}{\mathbb{P}}_{j}(\tau_{0}<+\infty)+C_{3}H_{j}(0,Y_{1}(x^{**}_{P}))+C_{4}\partial_{y}H_{j}(0,Y_{1}(x^{**}_{P}))$

with

(7.63)
$C_{1}=\frac{\partial_{y}\phi_{1}((x^{**}_{P},Y_{1}(x^{**}_{P})))}{(1-\phi_{1}(x^{**}_{P},Y_{1}(x^{**}_{P})))^{2}}$

and some $C_{2},C_{3},C_{4}\in{\mathbb{R}}$ do not depending on $j_{1}$. When
(B2) and $x^{**}=x^{**}_{P}$ hold, we have

$x_{d}=x^{**}=x^{**}_{P},\quad\text{and}\quad
Y_{1}(x^{**}_{P})<y_{d}=y^{**}\leqslant y^{**}_{P},$

Hence by Lemma 7.8, for ${\varepsilon}>0$ such that
$Y_{1}(x^{**})+{\varepsilon}<y_{d}$, there are two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfying
(6.6) for which (7.50) holds with $y=Y_{1}(x^{**})+{\varepsilon}$, and by
Lemma 7.7 we get (7.45). Using these relations for
$j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with $j_{2}=0$ we obtain

(7.64) ${\mathbb{P}}_{(j_{1},0)}(\tau_{0}<+\infty)\leqslant
C_{5}\,\frac{x_{1}^{j_{1}}+x_{2}^{j_{1}}}{1-\theta},\quad\forall
j=(j_{1},0)\in\mathbb{Z}^{2}_{+},$

and

(7.65) $H_{(j_{1},0)}(0,Y_{1}(x^{**}_{P})+{\varepsilon})\leqslant
C_{6}(x_{1}^{j_{1}}+x_{2}^{j_{1}})$

with some $C_{5}>0$ and $C_{6}>0$ do not depending on $j_{1}$. Since the
function $y\to H_{j}(0,y)$ is equal on the disk of its analyticity
$B(0,y_{d})$ to its power series with the positive coefficients, we have
moreover

$\partial_{y}H_{j}(0,Y_{1}(x^{**}_{P}))\leqslant
C_{7}H_{j}(0,Y_{1}(x^{**}_{P})+{\varepsilon}_{2})$

with some do not depending on $j_{1}$ constant $C_{7}>0$, and consequently,
using (7.8.3), (7.64) and (7.65), we obtain

$\left|\tilde{\varkappa}_{1}(j_{1},0)-C_{1}(x^{**}_{P})^{j_{1}}\right|\leqslant
C_{9}(x_{1}^{j_{1}}+x_{2}^{j_{1}}),\quad\forall j_{1}>0,$

with some do not depending on $j_{1}$ constant $C_{8}>0$. Remark finally that
the constant $C_{1}$ defined by (7.63) is strictly positive because the
function $y\to\phi_{1}(x^{**}_{P},y)$ is convex and strictly increasing on
$]0,+\infty[$. Hence, with the same argument as in the proof of the first
assertion of Proposition 7.13 we obtain

$\tilde{\varkappa}_{1}(j_{1},0)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
C_{1}(x^{**}_{P})^{j_{1}}\quad\text{as $j_{1}\to+\infty$},$

and consequently, $\tilde{\varkappa}_{1}(j)>0$ for all $j_{1}>0$ large enough.

The function $\tilde{\varkappa}_{1}$ is therefore non-negative, harmonic for
$(Z_{\tau_{0}}(n))$ and non zero at the points $(j_{1},0)$ for all $j_{1}>0$
large enough. With this results and using exactly the same arguments as in the
proof of the first assertion of our proposition, we conclude that the function
$\tilde{\varkappa}_{1}$ is positive everywhere in
$\mathbb{Z}^{2}_{+}{\setminus}E_{0}$.

####  7.8.4. Proof of the last assertions of Proposition 7.13

Suppose now that (B2) holds, and let a point
$(\hat{x},\hat{y})\in{\mathcal{S}}_{22}$ be such that $\hat{x}<x_{d}$ and
$\hat{y}<y_{d}$. For such a point $(\hat{x},\hat{y})$, by Lemma 7.6, the
function $\varkappa_{(\hat{x},\hat{y})}$ is non-negative on
$\mathbb{Z}^{2}_{+}$.

The proof of the identity

${\mathbb{E}}_{j}(\varkappa_{(\hat{x},\hat{y})}(Z(1)),\;\tau_{0}>1)=\varkappa_{(\hat{x},\hat{y})}(j),\quad\forall
j\in\mathbb{Z}^{2}_{+},$

is straightforward, it uses the arguments quite similar to those of the proofs
of (7.52), (7.53) and (7.54). The function $\varkappa_{(\hat{x},\hat{y})}$ is
therefore harmonic for the killed random walk $(Z_{\tau_{0}}(n))$.

To prove that $\varkappa_{(\hat{x},\hat{y})}(j)>0$ for some
$j\in\mathbb{Z}^{2}_{+}$ we consider the point
$\hat{w}=w_{D}(\hat{x},\hat{y})=(u_{D}(\hat{x},\hat{y}),v_{D}(\hat{x},\hat{y}))$
on the unit circle ${\mathbb{S}}^{1}$ defined by (2.66) with $x=\hat{x}$ and
$y=\hat{y}$, and we investigate an asymptotical behavior of
$\varkappa_{(\hat{x},\hat{y})}(j)$ as $\|j\|\to+\infty$ and
$j/\|j\|\to\hat{w}$.

By the definitions of the functions $L_{j}$ and
$\varkappa_{(\hat{x},\hat{y})}$, for
$j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}{\setminus}\\{(0,0)\\}$,

$\varkappa_{(\hat{x},\hat{y})}(j)=\hat{x}^{j_{1}}\hat{y}^{j_{2}}-{\mathbb{P}}_{j}(\tau_{0}<+\infty)+\left(\phi_{1}(\hat{x},\hat{y})-1\right)H_{j}(\hat{x},0)+\left(\phi_{2}(\hat{x},\hat{y})-1\right)H_{j}(0,\hat{y})$

Since we assume that $\hat{y}<y_{d}$, by Lemma 7.8, there are two points
$(x_{1},y_{1})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{2},y_{2})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ satisfying
(6.6) for which (7.50) holds with $y=\hat{y}$, and by Lemma 7.7 we get (7.45).
With the same arguments (it is sufficient to exchange the roles of $x$ and
$y$, we get that there are two
$(x_{3},y_{3})\in\overset{\circ}{D}\cap\overset{\circ}{D_{1}}$ and
$(x_{4},y_{4})\in\overset{\circ}{D}\cap\overset{\circ}{D_{2}}$ for which

$H_{j}(\hat{x},0)\leqslant
C^{\prime}\left(x_{3}^{k_{1}}y_{3}^{k_{2}}+x_{4}^{k_{1}}y_{4}^{k_{2}}\right),\quad\forall
j\in\mathbb{Z}^{2}_{+},$

with some do not depending on $j\in\mathbb{Z}^{2}_{+}$ constant
$C^{\prime}>0$. Using the last relation together with (7.45) and (7.50), one
gets

(7.66)
$\bigl{|}\varkappa_{(\hat{x},\hat{y})}(j)-\hat{x}^{j_{1}}\hat{y}^{j_{2}}\bigr{|}\leqslant
C_{1}\sum_{i=1}^{4}x_{i}^{j_{1}}y_{i}^{j_{2}}.$

with some do not depending on $j$ constant $C_{1}>0$. For
$\hat{w}=w_{D}(\hat{x},\hat{y})=(u_{D}(\hat{x},\hat{y}),v_{D}(\hat{x},\hat{y}))$
defined by relation (2.66) with $x=\hat{x}$ and $y=\hat{y}$, the point
$(\hat{x},\hat{y})$ is an only point in the set $D$ where the function
$(x,y)\to x^{\hat{u}_{1}}y^{\hat{u_{2}}}$ achieves its maximum over $D$. Since
non of the points $(x_{1},y_{1}),\ldots,(x_{4},y_{4})$ is equal to
$(\hat{x},\hat{y})$ (recall that the points
$(x_{1},y_{1}),\ldots,(x_{4},y_{4})$ belong to the interior of the set $D$ and
the point $(\hat{x},\hat{y})$ belongs to the boundary of $D$), it follows that

$\frac{1}{\hat{x}^{j_{1}}\hat{y}^{j_{2}}}\sum_{i=1}^{4}x_{i}^{j_{1}}y_{i}^{j_{2}}\to
0\quad\text{as}\quad\|j\|\to+\infty\quad\text{and}\quad j/\|j\|\to\hat{u}$

and consequently, using (7.66), we get

$\varkappa_{(\hat{x},\hat{y})}(j)\sim\hat{x}^{j_{1}}\hat{y}^{j_{2}}\quad\text{as}\quad\|j\|\to+\infty\quad\text{and}\quad
j/\|j\|\to\hat{u}.$

This proves that $\varkappa_{(\hat{x},\hat{y})}(j)>0$ for any
$j\in\mathbb{Z}^{2}_{+}$ with $\|j\|$ large enough and $j\|j\|$ closed enough
to $\hat{u}$.

The function $\varkappa_{(\hat{x},\hat{y})}$ is therefore non-negative,
harmonic for $(Z_{\tau_{0}}(n))$, and non zero at some points
$j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with $\|j\|>N_{0}$. Hence, the similar
arguments as in the proof of the first assertion of our proposition, we
conclude that the function $\tilde{\varkappa}_{1}$ is positive everywhere on
$\mathbb{Z}^{2}_{+}{\setminus}E_{0}$.

###  7.9. Proof of Theorem 2

Now we summarize the above results in orther to get Theorem 2:

  * –

The first assertion of Theorem 2 follows from Proposition 7.10 and the first
assertion of Proposition 7.13.

  * –

The second assertion of Theorem 2 is a consequence of Proposition 7.11 and the
third assertion of Proposition 7.13.

  * –

Proposition 7.12, Proposition 7.13 and the second assertion of Proposition
7.13 prove the assertions iii)- v) of Theorem 2.

  * –

The last assertion of our theorem is proved by Lemma 7.6 and the last
assertion of Proposition 7.13.

##  8\. Asymptotics along the Axes

This section is devoted to the proof of Theorem 3.

For $k_{2}{=}0$, the asymptotics (2.56)-(2.62) follows from Theorem 2 and the
Tauberian-like theorem (see Corollary VI.1 of Flajolet and Sedgevick [10]) in
a straightforward way. Hence, to complete the proof of our theorem, i.e. to
get the asymptotics (2.56)-(2.62) for $k_{2}>0$, it is sufficient to show that
for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$,

(8.1)
$\lim_{n\to+\infty}\frac{g(j,(n,k_{2}))}{g(j,(n,0))}=\nu_{1}(k_{2})>0,\quad\forall
k_{2}\in\mathbb{Z}^{2}.$

Before getting this result, let us notice that under the hypotheses (A1)-(A3),
the function $y\to\psi_{1}(x_{d},y)/Q(x_{d},y)$ is analytic at the origin
$y=0$ because the functions $y\to\psi_{1}(x_{d},y)$ and $y\to Q(x_{d},y)$, see
definitions (2.25) and (2.24), are analytic in a neighborhood of the closed
disk $\overline{B}(0,Y_{2}(x_{d}))$ and with the definition of the function
$Q$ (see (2.24))

$Q(x_{d},0)={-}\sum_{\begin{subarray}{c}k=(k_{1},k_{2})\in\mathbb{Z}^{2}:\\\
k_{2}=-1\end{subarray}}x_{d}^{k_{1}}\mu(k)<0.$

For any given $n\in{\mathbb{N}}^{*}$, the quantity (2.55) is therefore well
defined and equal to the $(n-1)$-th coefficient of the Taylor expansion of the
function $y\to\psi_{1}(x_{d},y)/Q(x_{d},y)$ in a neighborhood of the origin
$y=0$ :

$\psi_{1}(x_{d},y)/Q(x_{d},y)=\sum_{n=1}^{\infty}\nu_{1}(n)y^{n-1}.$

To show that all coefficients $\nu_{1}(n),\,n\in{\mathbb{N}}$ are positive,
and to get (8.1) for $k_{2}>0$, we will need a probabilistic representation of
the quantities $\nu_{1}(n),n\in{\mathbb{N}}^{*}$, in terms of the invariant
measure of the following Markov chain on $\mathbb{Z}_{+}$: Define the twisted
positive measures $\tilde{\mu}$ on $\mathbb{Z}^{2}$ and $\tilde{\mu}_{1}$ on
$\mathbb{Z}\times\mathbb{Z}_{+}$ by letting

$\tilde{\mu}(k_{1},k_{2})=x_{d}^{k_{1}}(Y_{1}(x_{d}))^{k_{2}}\mu(k_{1},k_{2}),\quad\forall(k_{1},k_{2})\in\mathbb{Z}^{2},$
$\tilde{\mu}_{1}(k_{1},k_{2})=x_{d}^{k_{1}}(Y_{1}(x_{d}))^{k_{2}}\mu_{1}(k_{1},k_{2}),\quad\forall(k_{1},k_{2})\in\mathbb{Z}\times\mathbb{Z}_{+}.$

With the definitions of the point $x_{d}$ and the function $Y_{1}$, we have

$\tilde{\mu}(\mathbb{Z}^{2})=P(x_{d},Y_{1}(x_{d}))=1\quad\text{and}\quad\tilde{\mu}_{1}(\mathbb{Z}\times\mathbb{Z}_{+})=\phi_{1}(x_{d},Y_{1}(x_{d}))\leqslant
1,$

and consequently, the twisted measure $\tilde{\mu}$ is stochastic on
$\mathbb{Z}^{2}_{+}$ and the twisted measure $\tilde{\mu}_{1}$ is sub-
stochastic on $\mathbb{Z}\times\mathbb{Z}_{+}$. Consider now a twisted random
walk $(\tilde{Z}^{1}(n)=(\tilde{Z}^{(1)}_{1}(n),\tilde{Z}^{(1)}_{2}(n)))$ on
$\mathbb{Z}\times\mathbb{Z}_{+}$ with transition probabilities

${\mathbb{P}}_{(j_{1},j_{2})}\bigl{(}\tilde{Z}^{(1)}(1)=(k_{1},k_{2})\bigr{)}\leavevmode\nobreak\
=\leavevmode\nobreak\ \begin{cases}\tilde{\mu}(k-j)&\text{if $j_{2}>0$,}\\\
\tilde{\mu}_{1}(k-j)&\text{if $j_{2}=0$.}\end{cases}$

Since the transition probabilities of the twisted random walk
$(\tilde{Z}^{1}(n)=(\tilde{Z}^{(1)}_{1}(n),\tilde{Z}^{(1)}_{2}(n)))$ are
invariant with respect to the shifts on $(\ell,0)$ for any
$\ell\in\mathbb{Z}$, its second component $(\tilde{Z}^{(1)}_{2}(n))$ is a
Markov chain on $\mathbb{Z}_{+}$ (stochastic if
$\phi_{1}(x_{d},Y_{1}(x_{d}))=1$ and sub-stochastic if
$\phi_{1}(x_{d},Y_{1}(x_{d}))<1$) with transition probabilities

${\mathbb{P}}_{j_{2}}\bigl{(}\tilde{Z}^{(1)}_{2}(1)=k_{2}\bigr{)}\leavevmode\nobreak\
=\leavevmode\nobreak\
\begin{cases}\sum_{k_{1}=-1}^{\infty}\tilde{\mu}(k_{1},k_{2}-j_{2})&\text{if
$j_{2}>0$,}\\\ \\\
\sum_{k_{1}=-1}^{\infty}\tilde{\mu}_{1}(k_{1},k_{2})&\text{if
$j_{2}=0$.}\end{cases}$

Recall that a non-negative measure $\pi_{1}$ on $\mathbb{Z}_{+}$ is invariant
for $(\tilde{Z}^{(1)}_{2}(n))$ if, for any $\ell{\in}\mathbb{Z}_{+}$, the
relation

(8.2)
$\pi_{1}(\ell)=\sum_{\ell^{\prime}\in\mathbb{Z}_{+}}\pi_{1}(\ell^{\prime}){\mathbb{P}}_{\ell^{\prime}}(\tilde{Z}^{(1)}_{2}(n)=\ell)$

holds. The following lemma relates the vector
$\nu_{1}=(\nu_{1}(k_{2}),k_{2}\in\mathbb{Z}_{+})$ with the unique (up to a
multiplicative constant) invariant measure for $(\tilde{Z}^{(1)}_{2}(n))$.

######  Lemma 8.1.

Under the hypotheses (A1)-(A3),
$\pi_{1}=(\pi_{1}(n)=\nu_{1}(n)(Y_{1}(x_{d}))^{n},\,n\in{\mathbb{N}})$ is a
unique up to the multiplication by constants invariant measure of the Markov
chain $(\tilde{Z}^{(1)}_{2}(n))$.

###### Proof.

We prove this lemma in two steps: First we will show that the Markov chain
$(\tilde{Z}^{(1)}_{2}(n))$ has an invariant measure
$\pi_{1}=(\pi_{1}(n),\,n\in{\mathbb{N}})$ with $\pi_{1}(0)=1$, and next we
will prove that the generating function
$f(y)=\sum_{n=0}^{\infty}\pi_{1}(n)(y/Y_{1}(x_{d}))^{n}$ satisfies in a
neighborhood of the point $y=0$ in $\mathbb{C}$ the identity

$f(y)=\phi_{1}(x_{d},y)+P(x_{d},y)(f(y)-1)$

and we will deduce from this identity that
$\pi_{1}(n)=\nu_{1}(n)(Y_{1}(x_{d}))^{n}$ for any $n\in{\mathbb{N}}$.

To perform the first step of our proof, let us notice that under our
hypotheses, the jumps of the Markov chain $(\tilde{Z}^{(1)}_{2}(n))$ are
integrable and moreover, for any non-zero $\ell\in\mathbb{Z}_{+}$,

$\displaystyle{\mathbb{E}}_{\ell}(\tilde{Z}^{(1)}_{2}(1))$
$\displaystyle=\ell+\sum_{k_{2}=-1}^{\infty}k_{2}\sum_{k_{1}=-1}^{\infty}x_{d}^{k_{1}}(Y_{1}(x_{d}))^{k_{2}}\mu(k_{1},k_{2})=\ell+Y_{1}(x_{d})\partial_{y}P(x_{d},Y_{1}(x_{d}))$

with $\partial_{y}P(x_{d},Y_{1}(x_{d}))\leqslant 0$. The Markov chain
$(\tilde{Z}^{(1)}_{2}(n))$ is therefore recurrent if the measure
$\tilde{\mu}_{1}$ is stochastic, i.e. if $\phi_{1}(x_{d},Y_{1}(x_{d}))=1$, and
it is transient if $\phi_{1}(x_{d},Y_{1}(x_{d}))<1$.

In the case when the Markov chain $(\tilde{Z}^{(1)}_{2}(n))$ is recurrent, it
has a unique invariant measure $(\pi_{1}(\ell),\ell\in\mathbb{Z}_{+})$ with
$\pi_{1}(0)=1$ and

$\pi_{1}(\ell)=\sum_{n=1}^{\infty}{\mathbb{P}}_{0}(\tilde{Z}^{(1)}_{2}(n)=\ell,\;\tilde{Z}^{(1)}_{2}(k)\not=0,\,\forall
k<n),\quad\forall\ell>0.$

Suppose now that the Markov chain $(\tilde{Z}^{(1)}_{2}(n))$ is transient, and
let

$\pi_{1}^{(k)}(\ell)=\sum_{n=0}^{\infty}{\mathbb{P}}_{k}(\tilde{Z}^{(1)}_{2}(n)=\ell),\quad
k,\ell\in\mathbb{Z}_{+}.$

Then for any $k,\ell\in\mathbb{Z}_{+}$ such that $k>\ell$, with a
straightforward calculation one gets

(8.3)
$\sum_{\ell^{\prime}\in\mathbb{Z}_{+}}\pi_{1}^{(k)}(\ell^{\prime}){\mathbb{P}}_{\ell^{\prime}}(\tilde{Z}^{(1)}_{2}(1)=\ell)\leavevmode\nobreak\
=\leavevmode\nobreak\ \pi_{1}^{(k)}(\ell).$

Remark moreover that because of Assumption (A2), by the strong Markov
property,

${\pi_{1}^{(k)}(\ell)}/{\pi_{1}^{(k)}(0)}={\pi_{1}^{(\ell)}(\ell)}/{\pi_{1}^{(\ell)}(0)},\quad\forall
k\geqslant\ell,$

and ${\mathbb{P}}_{\ell^{\prime}}(\tilde{Z}^{(1)}_{2}(1)=\ell)=0$ for all
$\ell^{\prime}>\ell+1$. Using (8.3), it follows that the measure
$\pi_{1}=(\pi_{1}(\ell),\,\ell\in\mathbb{Z}_{+})$ with $\pi_{1}(0)=1$ and

$\pi_{1}(\ell)=\pi_{1}^{(\ell)}(\ell)/\pi_{1}^{(\ell)}(0),\quad\forall\ell\in{\mathbb{N}}^{*},$

is invariant for $(\tilde{Z}^{(1)}_{2}(n))$. The first step of our proof is
therefore completed.

Remark now that for any $n\in\mathbb{Z}_{+}$, using the identity

(8.4)
$\sum_{\ell^{\prime}=0}^{\ell+n}\pi_{1}(\ell^{\prime}){\mathbb{P}}_{\ell^{\prime}}(\tilde{Z}^{(1)}_{2}(n)=\ell)=\pi_{1}(\ell)$

with $\ell=0$, one gets

$\pi_{1}(n)\leqslant\pi_{1}(0)\left({\mathbb{P}}_{n}(\tilde{Z}^{(1)}_{2}(n)=0)\right)^{-1}\leqslant\pi_{1}(0)\left(\sum_{k_{1}=-1}^{\infty}\tilde{\mu}(k_{1},-1)\right)^{-n}$

with

$\sum_{k_{1}=-1}^{\infty}\tilde{\mu}(k_{1},-1)=\sum_{k_{1}=-1}^{\infty}x_{d}^{k_{1}}(Y_{1}(x_{d}))^{-1}\mu(k_{1},-1)=-Q(x_{d},0)>0.$

Hence, the generating function
$f(y)=\sum_{n=0}^{\infty}\pi_{1}(n)(y/Y_{1}(x_{d}))^{n}$ is analytic in a
neighborhood of the origin $y=0$ in $\mathbb{C}$, and by (8.2), for any non
zero $y$, satisfies there the identity

$f(y)=f(0)\Bigl{(}\sum_{(k_{1},k_{2})\in\mathbb{Z}\times\mathbb{Z}_{+}}\tilde{\mu}_{1}(k_{1},k_{2})(y/Y_{1}(x_{d}))^{k_{2}}\Bigr{)}\\\
+(f(y)-f(0))\Bigl{(}\sum_{(k_{1},k_{2})\in\mathbb{Z}\times\mathbb{Z}}\tilde{\mu}(k_{1},k_{2})(y/Y_{1}(x_{d}))^{k_{2}}\Bigr{)}.$

Since $f(0)=\pi_{1}(0)=1$ and with the definition of the twisted measures
$\tilde{\mu}$ and $\tilde{\mu}_{1}$, for non-zero $y$, we have

$\sum_{(k_{1},k_{2})\in\mathbb{Z}\times\mathbb{Z}}\tilde{\mu}(k_{1},k_{2})\frac{y^{k_{2}}}{(Y_{1}(x_{d}))^{k_{2}}}=P(x_{d},y)$

and

$\sum_{(k_{1},k_{2})\in\mathbb{Z}\times\mathbb{Z}_{+}}\tilde{\mu}_{1}(k_{1},k_{2})\frac{y^{k_{2}}}{(Y_{1}(x_{d}))^{k_{2}}}=\phi_{1}(x_{d},y),$

this implies that for any non-zero $y\in\mathbb{C}$ with $|y|$ small enough,

$(f(y)-1)(1-P(x_{d},y))=\phi_{1}(x_{d},y)-1.$

By the definition of the functions $(x,y)\to\psi_{1}(x,y)$ and $(x,y)\to
Q(x,y)$, from the last relation it follows that for any $y\in\mathbb{C}$ with
$|y|$ small enough,

$(f(y)-1)Q(x_{d},y)=y\psi_{1}(x_{d},y)$

and consequently, since $Q(x_{d},0)\not=0$, for any $y\in\mathbb{C}$ with
$|y|$ small enough, we have also

$f(y)=1+\frac{y\psi_{1}(x_{d},y)}{Q(x_{d},y)}.$

From the last relation, by the uniqueness of the coefficients of the Taylor
expansion at the point $y=0$, it follows that

$\pi_{1}(n)(Y_{1}(x_{d}))^{-n}=\nu_{1}(n),\quad\forall n\in\mathbb{Z}_{+},$

and consequently, the measure
$\pi_{1}=(\nu_{1}(n)Y_{1}(x_{d}))^{n},\,n\in\mathbb{Z}_{+})$ is the unique
invariant measure of the Markov chain $(\tilde{Z}^{(1)}_{2}(n))$ with
$\pi_{1}(0)=1$. ∎

As a straightforward consequence of Lemma 8.1 one gets the following property
of the coefficients $\nu_{1}(n),n\in{\mathbb{N}}$:

######  Corollary 8.1.

Under the hypotheses (A1)-(A3), all coefficients
$\nu_{1}(n),\,n\in\mathbb{Z}_{+}$, are positive and
$(\nu_{1}(n),n\in\mathbb{Z}_{+})$ is an only positive solution of the system

(8.5)
$\sum_{\ell_{1}=1}^{\infty}\mu_{1}(\ell_{1},n)+\sum_{\ell_{2}=1}^{n+1}\nu_{1}(\ell_{2})\sum_{\ell_{1}=1}^{\infty}\mu(\ell_{1},n-\ell_{2})=\nu_{1}(n),\quad
k_{2}\in\mathbb{Z}_{+}.$

###### Proof.

Indeed, since by Lemma 8.1, $\nu_{1}(n)=\pi_{1}(n)(Y_{1}(x_{d}))^{-n}$ for any
$n\in\mathbb{Z}_{+}$, the system (8.5) is equivalent to (8.2). Under our
hypotheses, the Markov chain $(\tilde{Z}^{(1)}_{2}(n))$ is irreducible on