$\mathbb{Z}_{+}$ and $\pi_{1}(0)=1\not=0$. It follows that $\pi_{1}(n)>0$ for
any $n\in\mathbb{Z}_{+}$, and consequently also
$\nu_{1}(n)=\pi_{1}(n)(Y_{1}(x_{d}))^{-n}>0$ for any $n\in\mathbb{Z}_{+}$. ∎

From this result it follows that to get (8.1), it is sufficient to show that
for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and
$k_{2}\in{\mathbb{N}}^{*}$, the sequence
$\bigl{(}g(j,(n,k_{2}))/g(j,(n,0))\bigr{)}$ converges and the limits
$\lim_{n\to\infty}g(j,(n,k_{2}))/g(j,(n,0))\bigr{)}$, $k_{2}\in{\mathbb{N}}$,
satisfy the system of the equations (8.5). To prove the convergence of these
sequences, the following preliminary results will be needed.

######  Corollary 8.2.

Under the hypotheses (A1)-(A3), if one of the conditions (B0)-(B6) holds, then
for any $j\in\mathbb{Z}^{2}_{+}$

(8.6) $\lim_{n}g(j,(k_{1}+n,0))/g(j,(n,0))=1,\quad\forall
k_{1}\in\mathbb{Z}_{+},$

and there are constants $c_{j}>0$ and $c^{\prime}_{j}>0$ such that for any
$k_{1}>0$,

(8.7) $c_{j}k_{1}^{\gamma}x_{d}^{-k_{1}}\leqslant g(j,(k_{1},0))\leqslant
c^{\prime}_{j}k_{1}^{\gamma}x_{d}^{-k_{1}},$

where

$\gamma=\begin{cases}-1/2&\text{ if $x_{d}=x^{**}_{P}$ and either (B2) holds
with $\phi_{1}(x_{d},Y_{1}(x_{d}))=1$,}\\\ \,&\text{\quad or (B5) holds with
$\phi_{1}(x_{d},Y_{1}(x_{d}))<1$,},\\\ -3/2&\text{if (B2) holds with
$x_{d}=x^{**}_{P}$ and $\phi_{1}(x_{d},Y_{1}(x_{d}))<1$},\\\ 1&\text{if (B5)
holds with $x_{d}<x^{**}_{P}$},\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

This result is a straightforward consequence of the asymptotics (2.56)-(2.62)
with $k_{2}=0$. These asymptotics follow from Theorem 2 and a Tauberian-like
theorem (see Corollary VI.1 of Flajolet and Sedgevick [10]) in a
straightforward way. ∎

######  Lemma 8.2.

Under the hypotheses (A1)-(A3), for any $j\in\mathbb{Z}^{2}_{+}$ and
$k_{2}\in\mathbb{Z}_{+}$ there are three constants $c_{1}>0$, $c_{2}>0$ and
$N(k_{2})>0$, such that

(8.8) $c_{1}\,g(j,(k_{1},0))\leqslant g(j,(k_{1},k_{2}))\leqslant
c_{2}\,g(j,(k_{1},0)),$

for any $k_{1}\in\mathbb{Z}_{+}$ such that $k_{1}\geqslant N(k_{2})$.

###### Proof.

Remark that for any $j,k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$ and
$N\in{\mathbb{N}}$, by using the Markov property one gets

$\displaystyle g(j,(k_{1},k_{2}))$
$\displaystyle\geqslant\sum_{n=N}^{\infty}{\mathbb{P}}_{j}(Z(n)=(k_{1},k_{2}),\,Z(N)=(k_{1},0),\;\tau_{0}>n)$
(8.9) $\displaystyle\geqslant
g(j,(k_{1},0)){\mathbb{P}}_{(k_{1},0)}(Z(N)=(k_{1},k_{2}),\;\tau_{0}>0).$

Moreover, similar arguments as in the proof of Lemma 7.2 shows that for any
$k_{2}\in\mathbb{Z}_{+}$, there is $N\in{\mathbb{N}}^{*}$ and a sequence
$\bigl{(}\ell_{1}^{(0)},\ell_{2}^{(0)}\bigr{)},\ldots,\bigl{(}\ell_{1}^{(N)},\ell_{2}^{(N)}\bigr{)}\in\mathbb{Z}^{2}$
such that

$\ell^{(0)}+\ldots+\ell^{(n)}\in\mathbb{Z}\times\mathbb{Z}_{+},\quad\forall
n\in\\{0,\ldots,N\\},$
$\sum_{n=0}^{N}\ell^{(n)}=(0,k_{2})\quad\quad\text{and}\quad\quad\mu_{1}\bigl{(}\ell^{(0)}\bigr{)}\prod_{n=1}^{N}\mu\bigl{(}\ell^{(n)}\bigr{)}>0.$

Hence, using Assumption (A2) we conclude that for any
$(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$ with $k_{1}>N$,

${\mathbb{P}}_{(k_{1},0)}\bigl{(}Z(n(k_{2}))=(k_{1},k_{2}),\;\tau_{0}>0\bigr{)}\geqslant\mu_{1}(\ell^{(0)})\prod_{n=1}^{N}\mu(\ell^{(n)})>0.$

When combined with (8.9) the last relation proves the first inequality of
(8.8) with

$c_{1}=\mu_{1}(\ell^{(0)})\prod_{n=1}^{N}\mu(\ell^{(n)}).$

The proof of the second inequality of (8.8) is quite similar. ∎

Now we are ready to complete proof of Theorem 3. Throughout our proof, the
starting point $j{\in}\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ will be given.

Since by Lemma 8.2 for any $k_{2}\in\mathbb{Z}_{+}$, the sequence
$(g(j,(n,k_{2}))/g(j,(n,0)),\,n\geqslant N_{n})$ is bounded below and above by
some positive constants, to get (8.1) it is sufficient to show that for any
subsequence $(N_{n})$ of the sequence of non-zero natural numbers $(n)$, for
which the sequence of functions

(8.10) $k_{2}\to f_{n}(k_{2})=g(j,(N_{n},k_{2}))/g(j,(N_{n},0))$

converges point-wise in $\mathbb{Z}_{+}$, one has

(8.11) $\lim_{n}f_{n}(k_{2})=\nu_{1}(k_{2}),\quad\forall
k_{2}\in\mathbb{Z}_{+}.$

Suppose now that for a subsequence $(N_{n})$, the sequence of functions
$(f_{n})$ defined by (8.10) converges point-wise in $\mathbb{Z}_{+}$ and let
$f_{\infty}=\lim_{n}f_{n}$. By Corollary 8.1, to get (8.11) it is sufficient
to show that the limit function $f_{\infty}$ satisfies the system of equations

(8.12)
$\sum_{\ell_{1}=1}^{\infty}\mu_{1}(\ell_{1},k_{2})+\sum_{\ell_{2}=1}^{n+1}f_{\infty}(\ell_{2})\sum_{\ell_{1}=1}^{\infty}\mu(\ell_{1},k_{2}-\ell_{2})=f_{\infty}(k_{2}),\quad
k_{2}\in\mathbb{Z}_{+}.$

To get this result, we consider the sequence of functions
$F_{n}:\mathbb{Z}\times\mathbb{Z}_{+}\to\mathbb{R}_{+}$ defined for any
$(k_{1},k_{2})\in\mathbb{Z}\times\mathbb{Z}_{+}$ by

$F_{n}(k_{1},k_{2})=\begin{cases}g(j,(k_{1}+N_{n},k_{2}))/g(j,(N_{n},0))&\text{if
$k_{1}+N_{n}\geqslant 0$,}\\\ 0&\text{otherwise.}\end{cases}$

Remark that by (8.6), the sequence of functions $(F_{n})$ also converges
point-wise in $\mathbb{Z}\times\mathbb{Z}_{+}$, and for any
$(k_{1},k_{2})\in\mathbb{Z}\times\mathbb{Z}_{+}$,

(8.13) $\lim_{n}F_{n}(k_{1},k_{2})=\lim_{n}f_{n}(k_{2})=f_{\infty}(k_{2}).$

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

$\displaystyle g(j,(k_{1},k_{2}))$
$\displaystyle=\sum_{(\ell_{1},\ell_{2})\in\mathbb{Z}^{2}_{+}}g(j,(\ell_{1},\ell_{2})){\mathbb{P}}_{(\ell_{1},\ell_{2})}(Z(1)=(k_{1},k_{2}),\,\tau_{0}>1)$
$\displaystyle=\sum_{\ell_{2}=1}^{k_{2}+1}g(j,(0,\ell_{2}))\mu_{2}(k_{1},k_{2}-\ell_{2})+\sum_{\ell_{1}=1}^{k_{1}+1}g(j,(\ell_{1},0))\mu_{1}(k_{1}-\ell_{1},k_{2})$
$\displaystyle\hskip
85.35826pt+\sum_{\ell_{1}=1}^{k_{1}+1}\sum_{\ell_{2}=1}^{k_{2}+1}g(j,(\ell_{1},\ell_{2}))\mu(k_{1}-\ell_{1},k_{2}-\ell_{2}).$

Using this relation with $k_{1}=N_{n}$, for any $n\in{\mathbb{N}}$ such that
$N_{n}>j_{1}$, one gets

$\displaystyle
F_{n}(0,k_{2})=\sum_{\ell_{2}=1}^{k_{2}+1}F_{n}(-N_{n},\ell_{2})$
$\displaystyle\mu_{2}(N_{n},k_{2}-\ell_{2})+\sum_{\ell_{1}=-1}^{N_{n}}F_{n}(-\ell_{1},0))\mu_{1}(\ell_{1},k_{2})$
$\displaystyle+\sum_{\ell_{1}=-1}^{N_{n}}\sum_{\ell_{2}=1}^{k_{2}+1}F_{n}(-\ell_{1},\ell_{2})\mu(\ell_{1},k_{2}-\ell_{2})\quad\quad\forall
k_{2}\in\mathbb{Z}_{+},$

and consequently, using (8.13) we will obtain (8.12) if we prove that for any
$k_{2}\in\mathbb{Z}_{+}$ and $\ell_{2}\in\\{1,\ldots,k_{2}+1\\}$, the
following relations hold

(8.14) $\lim_{n}F_{n}(-N_{n},\ell_{2})\mu_{2}(N_{n},k_{2}-\ell_{2})=0,$ (8.15)
$\lim_{n}\sum_{\ell_{1}=-1}^{N_{n}}F_{n}(-\ell_{1},0))\mu_{1}(\ell_{1},k_{2})=\sum_{\ell_{1}=-1}^{\infty}\lim_{n}F_{n}(-\ell_{1},0))\mu_{1}(\ell_{1},k_{2})$

and

(8.16)
$\lim_{n}\sum_{\ell_{1}=-1}^{N_{n}}F_{n}(-\ell_{1},\ell_{2})\mu(\ell_{1},k_{2}-\ell_{2})=\sum_{\ell_{1}=-1}^{\infty}\lim_{n}F_{n}(-\ell_{1},\ell_{2})\mu(\ell_{1},k_{2}-\ell_{2}).$

To get (8.14) we remark that by Corollary 8.2, there is a constant $C>0$ such
that for any $N_{n}$ large enough,

$F_{n}(-N_{n},\ell_{2})=g(j,(0,\ell_{2}))/g(j,(N_{n},0))\leqslant
g(j,(0,\ell_{2}))C(N_{n})^{\gamma}x_{d}^{N_{n}}$

and consequently, for any ${\varepsilon}>0$, there is a constant
$C_{\varepsilon}>0$ such that for any $N_{n}$ large enough,

$F_{n}(-N_{n},\ell_{2})\leqslant
g(j,(0,\ell_{2}))C_{\varepsilon}(1+{\varepsilon})^{N_{n}}x_{d}^{N_{n}}.$

Since by Assumption (A3)(ii), the generating function $\phi_{1}$ is finite in
a neighborhood of the point $(x_{d},Y_{1}(x_{d}))$, there is ${\varepsilon}>0$
such that for any $k_{2}\in\mathbb{Z}_{+}$,

$\lim_{n\to\infty}\mu_{1}(n,k_{2})(1+{\varepsilon})^{n}x_{d}^{n}=0.$

and consequently, for any $k_{2}\in\mathbb{Z}_{+}$ and
$\ell_{2}\in\\{1,\ldots,k_{2}+1\\}$, (8.14) holds.

To get (8.15) we use the implicit function theorem and Corollary 8.2. By
Corollary 8.2, there is a constant $C_{j}>0$ such that for any $N_{n}>0$ and
$\ell_{1}\in\mathbb{Z}$ such that $-1\leqslant\ell_{1}\leqslant N_{n}$,

(8.17)
$F_{n}(-\ell_{1},0)=\frac{g(j,(N_{n}-\ell_{1},0))}{g(j,(N_{n},0))}\leqslant
C_{1}\frac{(N_{n}-\ell_{1})^{\gamma}}{N_{n}^{\gamma}}x_{d}^{\ell_{1}}.$

In the case when $\gamma\geqslant 0$, it follows that for any $N_{n}>0$ and
$\ell_{1}\in\mathbb{Z}$ such that $-1\leqslant\ell_{1}\leqslant N_{n}$,

$F_{n}(-\ell_{1},0)\leqslant C_{1}x_{d}^{\ell_{1}},$

and consequently, since $F_{n}(-\ell_{1},0)=0$ for all $\ell_{1}\geqslant
N_{n}$, and since under our hypotheses,

$\sum_{\ell_{1}=-1}^{\infty}x_{d}^{\ell_{1}}\mu_{1}(\ell_{1},k_{2})\leqslant(Y_{1}(x_{d}))^{-k_{2}}\sum_{\ell=(\ell_{1},\ell_{2})\in\mathbb{Z}\times\mathbb{Z}_{+}}^{\infty}x_{d}^{\ell_{1}}(Y_{1}(x_{d}))^{\ell_{2}}\mu_{1}(\ell_{1},\ell_{2})=\phi_{1}(x_{d},Y_{1}(x_{d}))<+\infty,$

by the implicit function theorem one gets (8.15). In the case when
$\gamma\geqslant 0$, (8.15) is therefore proved.

Suppose now that $\gamma<0$. In this case, using (8.17), one gets that for any
$N_{n}>0$ and $\ell_{1}\in\mathbb{Z}$ such that $-1\leqslant\ell_{1}<N_{n}$,

$F_{n}(-\ell_{1},0)\leqslant
C_{1}\frac{N_{n}^{|\gamma|}}{(N_{n}-\ell_{1})^{|\gamma|}}x_{d}^{\ell_{1}}\leqslant\begin{cases}C_{1}N_{n}^{|\gamma|}x_{d}^{\ell_{1}}\leqslant
C_{1}(2\ell_{1})^{|\gamma|}x_{d}^{\ell_{1}}&\text{if $\ell_{1}>N_{n}/2$}\\\
C_{1}2^{|\gamma|}x_{d}^{\ell_{1}}&\text{if $\ell_{1}\leqslant
N_{n}/2$}\end{cases}$

and consequently, for any ${\varepsilon}>0$ there is a constant
$C_{\varepsilon}>0$ such that $N_{n}>0$ and
$\ell_{1}\in\\{-1,\ldots,N_{n}-1\\}$, one has

(8.18) $F_{n}(-\ell_{1},0)\leqslant
C_{\varepsilon}(1+{\varepsilon})^{\ell_{1}}x_{d}^{\ell_{1}}.$

Since $F_{n}(-\ell_{1},0)=0$ for all $\ell_{1}\geqslant N_{n}$, and since
under our hypotheses, for some ${\varepsilon}>0$ small enough,

$\displaystyle\sum_{\ell_{1}=-1}^{\infty}(1+{\varepsilon})^{\ell_{1}}x_{d}^{\ell_{1}}\mu_{1}(\ell_{1},k_{2})$
$\displaystyle\leqslant(Y_{1}(x_{d}))^{-k_{2}}\\!\\!\\!\sum_{\ell=(\ell_{1},\ell_{2})\in\mathbb{Z}\times\mathbb{Z}_{+}}^{\infty}(1+{\varepsilon})^{\ell_{1}}x_{d}^{\ell_{1}}(Y_{1}(x_{d}))^{\ell_{2}}\mu_{1}(\ell_{1},\ell_{2})$
(8.19)
$\displaystyle\leqslant(Y_{1}(x_{d}))^{-k_{2}}\phi_{1}((1+{\varepsilon})x_{d},Y_{1}(x_{d}))<+\infty,$

by the implicit function theorem, it follows (8.15). Relation (8.15) is
therefore proved.

To get (8.16) we use first Lemma 8.2. By Lemma 8.2, there are two constants
$N(\ell_{2})$ and $C_{1}>0$ such that for any $N_{n}\geqslant 0$ and
$\ell_{1}\in\\{-1,\ldots,N_{n}-N(\ell_{2})\\}$,

$F_{n}(-\ell_{1},\ell_{2})=\frac{g(j,(N_{n}-\ell_{1},\ell_{2}))}{g(j,(N_{n},0))}\leqslant
C_{1}\frac{g(j,(N_{n}-\ell_{1},0))}{g(j,(N_{n},0))}$

and for $\ell_{1}\in\\{N_{n}-N(\ell_{2})+1,\ldots,N_{n}-1\\}$,

$F_{n}(-\ell_{1},\ell_{2})=\frac{g(j,(N_{n}-\ell_{1},\ell_{2}))}{g(j,(N_{n},0))}\frac{C_{2}}{g(j,(N_{n},0))}$

with $C_{2}=\max\\{g(j,(1,0)),\ldots,g(j,(N(\ell_{2}),0)\\}$. Hence with the
similar arguments as above one gets that for any ${\varepsilon}>0$ there is a
constant $C_{\varepsilon}>0$ such that $N_{n}>0$ and
$\ell_{1}\in\\{-1,\ldots,N_{n}-1\\}$, (8.18) holds, and consequently, using
(8.19) we conclude that (8.16) holds.

Theorem 3 is therefore proved.

##  9\. Asymptotics along Directions of ${\mathbb{S}}^{1}_{+}$

This section is devoted to the proof of Theorem 4.

###  9.1. Main ideas and the sketch of the proof

The main ideas of the proof of Theorem 4 are the following:

By using Theorem 1, we first get an integral representation

(9.1) $g(j,k)=\\\
\int_{|x|=\hat{x}}\int_{|y|=\hat{y}}\frac{L_{j}(x,y)+(\phi_{1}(x,y){-}P(x,y))H_{j}(x,0){+}(\phi_{2}(x,y){-}P(x,y))H_{j}(0,y)}{(2\pi
i)^{2}\,x^{k_{1}+1}y^{k_{2}+1}(1{-}P(x,y))}dxdy$

for any $(\hat{x},\hat{y})\in{\Gamma}$ with $\hat{x}<x_{d}$ and
$\hat{y}<y_{d}$. Next we show that the set
$\\{(x,y)\in\overset{\circ}{D}:x<x_{d},y<y_{d}\\}$ is non empty. By the
definition of the set ${\Gamma}$, the set $\overset{\circ}{D}$ is included to
the set ${\Gamma}$, with this result we will be able to consider the integral
representation (9.1) with $\hat{x}<x_{d}$, $\hat{y}<y_{d}$ such that
$(\hat{x},\hat{y})\in\overset{\circ}{D}$.

To prove the first tree assertions of Theorem 4, the integral representation
(9.1) is next modified in the following way: since the functions $(x,y)\to
L_{j}(x,y)$ and $(x,y)\to(1-P(x,y))^{-1}$ are analytic in the polycircular set
$\Omega(\overset{\circ}{D})$, and, by Theorem 1,

  * –

the function $(x,y){\to}(\phi_{1}(x,y){-}P(x,y))H_{j}(x,0)$ is analytic in the
polycircular set $\\{(x,y){\in}\Omega(\overset{\circ}{D}){:}|x|{<}x_{d}\\}$,

  * –

the function $(x,y){\to}(\phi_{2}(x,y){-}P(x,y))H_{j}(0,y)$ is analytic in the
polycircular set $\\{(x,y){\in}\Omega(\overset{\circ}{D}){:}|y|{<}y_{d}\\}$,

and since the set $\Omega(\overset{\circ}{D})$ does not contain zeros of the
function $(x,y){\to}x^{k_{1}+1}y^{k_{2}+1}(1{-}P(x,y))$, we can write

(9.2) $g(j,k)=I_{0}(j,k)+I_{1}(j,k)+I_{2}(j,k)$

with

(9.3) $I_{0}(j,k)=\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}_{0}}\int_{|y|=\hat{y}_{0}}\frac{L_{j}(x,y)}{x^{k_{1}+1}y^{k_{2}+1}(1-P(x,y))}dx\,dy,$
(9.4) $I_{1}(j,k)=\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}_{1}}\int_{|y|=\hat{y}_{1}}\frac{(\phi_{1}(x,y)-P(x,y))H_{j}(x,0)}{x^{k_{1}+1}y^{k_{2}+1}(1-P(x,y))}dx\,dy$

and

(9.5) $I_{2}(j,k)=\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}_{2}}\int_{|y|=\hat{y}_{2}}\frac{(\phi_{2}(x,y)-P(x,y))H_{j}(0,y)}{x^{k_{1}+1}y^{k_{2}+1}(1-P(x,y))}dx\,dy$

for any $(\hat{x}_{0},\hat{y}_{0})\in\overset{\circ}{D}$,
$(\hat{x}_{1},\hat{y}_{1})\in\\{(x,y)\in\overset{\circ}{D}:x<x_{d}\\}$, and
$(\hat{x}_{2},\hat{y}_{2})\in\\{(x,y)\in\overset{\circ}{D}:y<y_{d}\\}$.

To prove the first assertion of Theorem 4, it is sufficient to get the
asymptotic behavior (2.75) when $\min\\{k_{1},k_{2}\\}\to+\infty$ and
$w_{k}=k/\|k\|\to w$ for any $w\in{\mathcal{W}}_{1}$. In order to get this
result, we identify the asymptotic behavior of $I_{1}(j,k)$ by using the
residue theorem (applied first for the integral with respect to $x$ and next
for the integral with respect to $y$), and using next large deviation
estimates of $I_{0}(j,k)$ and $I_{2}(j,k)$ we prove that the terms
$I_{0}(j,k)$ and $I_{3}(j,k)$ are negligible with respect to $I_{1}(j,k)$.

The proof of the second assertion of Theorem 4 is exactly the same as the
proof of the first assertion, it is sufficient to exchange the roles of the
first and the second coordinates of the points $j,k\in\mathbb{Z}^{2}_{+}$.

To prove the assertions (iii)-(v) of our theorem, we show that the term
$I_{0}(j,k)$ is negligible with respect to $I_{1}(j,k)+I_{2}(j,k)$ and we
identify (in the same way as in the proof of the first assertion of our
theorem) the asymptotic behavior of each term $I_{1}(j,k)$ and $I_{2}(j,k)$.

Finally, the last assertion of Theorem 4, is obtained as a consequence of
Proposition 1 of the paper [11].

The proof of Theorem 4 is organized as follows:

The integral representations (9.1) and (9.2) of the Green function $g(j,k)$
are obtained respectively in Lemma 9.1 and Corollary 9.1 of Section 9.2.

Section 9.3 is devoted to the upper large deviation estimates for the
integrals $I_{0}(j,k)$, $I_{1}(j,k)$ and $I_{2}(j,k)$. These large deviation
estimates will be used in order to identify the dominant terms of the sum
$I_{0}(j,k){+}I_{0}(j,k){+}I_{0}(j,k)$ of the right-hand side of (9.2).

In Section 9.4 we obtain exact asymptotics of $I_{1}(j,k)$ (resp $I_{2}(j,k)$)
as $\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w$ for those
$w=(u,v)\in{\mathbb{S}}^{1}_{+}$ for which $u>u_{D}(x_{d},Y_{2}(x_{d}))$
(resp. for which $v>v_{D}(X_{2}(y_{d}),y_{d})$). This is a subject of Lemma
9.6 and Lemma 9.5 below. Remark that we do not need the exact asymptotics of
$I_{1}(j,k)$ (resp $I_{2}(j,k)$) when $k/\|k\|\to
w=(u,v)\in{\mathbb{S}}^{1}_{+}$ and $u\leqslant u_{D}(x_{d},Y_{2}(x_{d}))$
(resp. $v\leqslant v_{D}(X_{2}(y_{d}),y_{d})$): the large deviation
asymptotics obtained in Section 9.3 will be in this case sufficient.

In Section 9.5 the proof of the first assertion of our theorem is completed.
It will be proved there that for any $w\in{\mathcal{W}}_{1}$, the inequality
$u>u_{D}(x_{d},Y_{2}(x_{d}))$ holds, and consequently that the results of
Section 9.4 provide the exact asymptotic for $I_{1}(j,k)$ as
$\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w\in{\mathcal{W}}_{1}$. A
comparison of the exact asymptotics for $I_{1}(j,k)$ as
$\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w\in{\mathcal{W}}_{1}$ with
the asymptotics of $I_{2}(j,k)$ and $I_{0}(j,k)$ will show that the terms
$I_{0}(j,k)$ and $I_{2}(j,k)$ are negligible with respect to $I_{1}(j,k)$.

The second assertion of Theorem 4 is obtained by using the arguments of the
symmetry: to get this statement, the same arguments as in the proof of the
first assertion can be applied if one exchanges the roles of $x$ and $y$.

In Section 9.6 and 9.7, we complete the proof of the assertions (iii)-(v) of
Theorem 4. It will be shown in these cases, the exact asymptotics of
$I_{1}(j,k)$ and $I_{2}(j,k)$ are given by the results of Section 9.4 and that
the term $I_{0}(j,k)$ is negligible with respect to the terms $I_{1}(j,k)$ and
$I_{2}(j,k)$. The exact asymptotic of the Green function $g(j,k)$ will be
obtained by comparing the exact asymptotics of $I_{1}(j,k)$ and $I_{2}(j,k)$.

###  9.2. Integral representation of the coefficients $g(j,k)$

######  Lemma 9.1.

Under the assumptions (A1)-(A4), the set
$\\{(x,y)\in\overset{\circ}{D}:x<x_{d},\;y<y_{d}\\}$ is non empty and for any
$(\hat{x},\hat{y})\in\overset{\circ}{D}$ such that $\hat{x}<x_{d}$ and
$\hat{y}<y_{d}$, (9.1) holds.

###### Proof.

By the first assertion of Theorem 1, the series (2.28) (and consequently also
the series (6.1)) converge on the polycircular set
$\\{(x,y){\in}\Omega({\Gamma}):|x|{<}x_{d},|y|{<}y_{d}\\}$. The function
$(x,y){\to}H_{j}(x,y)$ is therefore analytic in
$\\{(x,y){\in}\Omega({\Gamma}):|x|{<}x_{d},|y|{<}y_{d}\\}$ and for any
$(\hat{x},\hat{y}){\in}{\Gamma}$ with $\hat{x}{<}x_{d}$ and $\hat{y}{<}y_{d}$,
one has

(9.6) $g(j,k)=\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}}\int_{|y|=\hat{y}}\frac{H_{j}(x,y)}{x^{k_{1}+1}y^{k_{2}+1}}dx\,dy,\quad\forall
j,k{\in}\mathbb{Z}^{2}_{+}$

Remark now that under our assumptions, the set
$\\{(x,y)\in\overset{\circ}{D}:x<x_{d},y<y_{d}\\}$ is non empty:

  * –

when one of the assertions (B0)-(B2) is valid, this is a consequence of the
first assertion of Lemma 6.4

  * –

when one of the assertions (B3) or (B4) holds, the set
$\\{(x,y)\in\overset{\circ}{D}:x<x_{d},y<y_{d}\\}$ is non empty because in
this case and $x^{*}_{P}<x_{d}=x^{**}<x^{**}_{P}$, $y_{d}>Y_{1}(x_{d})$ and by
Lemma 4.1, any point of the line segment
$[(x_{d},Y_{1}(x_{d})),(x_{d},Y_{2}(x_{d}))]$, aside of the ends points
$(x_{d},Y_{1}(x_{d}))$ and $(x_{d},Y_{2}(x_{d}))$ belongs to the interior
$\overset{\circ}{D}$ of the set $D$;

  * –

similarly, when one of the assertions (B5) or (B6) holds, the set
$\\{(x,y){\in}\overset{\circ}{D}:x{<}x_{d},y{<}y_{d}\\}$ is non empty because
$y^{*}_{P}{<}y_{d}{=}y^{**}{<}y^{**}_{P}$, $x_{d}>X_{1}(y_{d})$ and by Lemma
4.1, any point of line segment $[(X_{1}(y_{d}),y_{d}),(X_{2}(y_{d}),y_{d})]$,
aside of the end points $(X_{1}(y_{d}),y_{d})$ and $(X_{2}(y_{d}),y_{d})$
belongs to the interior of the set $D$.

By the definition of the set ${\Gamma}$, the set $\overset{\circ}{D}$ is
included to ${\Gamma}$, it follows that (9.6) holds also for any
$(\hat{x},\hat{y})\in\overset{\circ}{D}$ with $\hat{x}<x_{d}$ and
$\hat{y}<y_{d}$.

Furthermore, by the second assertion of Theorem 1, for any
$j\in\mathbb{Z}^{2}_{+}$, on the set
$\\{(x,y){\in}\Omega({\Gamma}):\leavevmode\nobreak\
|x|{<}x_{d},\;|y|{<}y_{d}\\}$, the function $h_{j}$ satisfies the identity
(2.30). Since the set
$\\{(x,y)\in\Omega(\overset{\circ}{D}):x<x_{d},y<y_{d}\\}$ has an empty
intersection with the sets $\\{(x,y)\in\mathbb{C}^{2}:x=0\\}$ and
$\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\ y=0\\}$, it follows that on
the set $\\{(x,y){\in}\Omega(\overset{\circ}{D}):x{<}x_{d},y{<}y_{d}\\}$, for
any $j\in\mathbb{Z}^{2}_{+}$, the relation

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

holds, or equivalently,

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

Since the function $(x,y)\to 1/(1-P(x,y))$ is analytic in the set
$\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\
(|x|,|y|)\in\overset{\circ}{D}\\}$, this implies that, on the set
$\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\
(|x|,|y|)\in\overset{\circ}{D},\;|x|<x_{d},\;|y|<y_{d}\\}$, we have also the
identity

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

The last identity combined with (9.6) proves (9.1). ∎

Since the functions $(x,y)\to L_{j}(x,y)$ and $(x,y)\to
x^{-k_{1}-1}y^{k_{2}-1}(1-P(x,y))^{-1}$ are analytic in the polycircular set
$\Omega(\overset{\circ}{D})$, and by Theorem 1, the functions
$(x,y)\to(\phi_{1}(x,y)-P(x,y))H_{j}(x,0)$ and
$(x,y)\to(\phi_{2}(x,y)-P(x,y))H_{j}(0,y)$ are analytic respectively on
$\\{(x,y){\in}\Omega(\overset{\circ}{D}):|x|{<}x_{d}\\}$ and
$\\{(x,y)\in\Omega(\overset{\circ}{D}):\,|y|<y_{d}\\}$, as a straightforward
consequence of Lemma 9.1 we obtain

######  Corollary 9.1.

Under the assumptions (A1)-(A4), for any $j{\in}\mathbb{Z}^{2}_{+}$ and
$k{=}(k_{1},k_{2}){\in}\mathbb{Z}^{2}_{+}{\setminus}\\{(0,0)\\}$, relation
(9.2) holds with $I_{0}(j,k)$, $I_{1}(j,k)$ and $I_{2}(j,k)$ defined
respectively by (9.3), (9.4) and (9.5), for any
$(\hat{x}_{0},\hat{y}_{0}){\in}\overset{\circ}{D}$,
$(\hat{x}_{1},\hat{y}_{1}){\in}\\{(x,y){\in}\overset{\circ}{D}:x{<}x_{d}\\}$,
and
$(\hat{x}_{2},\hat{y}_{2}){\in}\\{(x,y)\in\overset{\circ}{D}:y{<}y_{d}\\}$.

###  9.3. Large deviation estimates of the quantities $I_{0}(j,k)$,
$I_{1}(j,k)$ and $I_{2}(j,k)$

######  Lemma 9.2.

Under the assumptions (A1)-(A4), for any $w=(u,v)\in{\mathbb{S}}^{1}_{+}$, and
$j,k\in\mathbb{Z}^{2}_{+}$, as $\|k\|\to\infty$ and $k/\|k\|\to w$,

(9.7)
$\limsup_{k}\|k\|^{-1}\log\left|I_{0}(j,k)\right|\leqslant-\max_{(\hat{x},\hat{y})\in
D}\Bigl{(}u\ln(\hat{x})+v\ln(\hat{y})\Bigr{)},$ (9.8)
$\limsup_{k}\|k\|^{-1}\log\left|I_{1}(j,k)\right|\leqslant-\max_{(\hat{x},\hat{y})\in
D,\,\hat{x}\leqslant x_{d}}\\!\Bigl{(}u\ln(\hat{x})+v\ln(\hat{y})\Bigr{)}.$

and

(9.9)
$\limsup_{k}\|k\|^{-1}\log\left|I_{2}(j,k)\right|\leqslant-\max_{(\hat{x},\hat{y})\in
D,\,\hat{y}\leqslant y_{d}}\\!\Bigl{(}u\ln(\hat{x})+v\ln(\hat{y})\Bigr{)}.$

###### Proof.

Indeed, the definition of $I_{2}(j,k)$ and Cauchy’s inequality give, for any
$j,k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$ and
$(\hat{x},\hat{y})\in\overset{\circ}{D}$ with $\hat{y}<y_{d}$,

$|I_{2}(j,k)|\leqslant M(\hat{x},\hat{y})\hat{x}^{-k_{1}}\hat{y}^{-k_{2}},$

with

$M(\hat{x},\hat{y})=\max\\{|(\phi_{2}(x,y)-P(x,y))H_{j}(0,y)(1-P(x,y))^{-1}|:\leavevmode\nobreak\
(x,y)\in\mathbb{C}^{2},|x|=\hat{x},|y|=\hat{y}\\}.$

This proves that as $\|k\|\to+\infty$ and $k/\|k\|\to w=(u,v)$,

$\limsup_{k}\|k\|^{-1}\log\left|I_{2}(j,k)\right|\leqslant-\sup_{(\hat{x},\hat{y})\in\overset{\circ}{D}:\hat{y}<y_{g}}\bigl{(}u\log(\hat{x})+v\log(\hat{y})\bigr{)}\\\
=-\max_{(\hat{x},\hat{y})\in D,\,\hat{y}\leqslant
y_{d}}\bigl{(}u\log(\hat{x})+v\log(\hat{y})\bigr{)}$

where the second relation holds because the function $(x,y)\to u\ln x+v\ln y$
is continuous on $D$. Relation (9.9) is therefore proved. The proof of (9.7)
and (9.8) is quite similar. ∎

###  9.4. Exact asymptotic behavior of $I_{1}(jk)$ as
$\min\\{k_{1},k_{2}\\}\to\infty$

To get the exact asymptotic of $I_{1}(jk)$ as $\min\\{k_{1},k_{2}\\}\to\infty$
we consider first the following preliminary results.

######  Lemma 9.3.

Under the assumption (A1), for any $w=(u,v)\in{\mathbb{S}}^{1}_{+}$, the
function $x\to u\ln(x)+v\ln(Y_{2}(x))$ is strictly increasing in the line
segment $[x^{*}_{P},x_{D}(w)]$ and strictly decreasing in the line segment
$[x_{D}(w),x^{**}_{P}]$.

###### Proof.

Indeed, recall that under our assumptions, the function
$(\alpha,\beta)\to\tilde{P}(\alpha,\beta)=P(e^{\alpha},e^{\beta})$ is strictly
convex and finite in a neighborhood of the set
$\tilde{D}=\\{(\alpha,\beta)\in\mathbb{R}^{2}:\leavevmode\nobreak\
P(e^{\alpha},e^{\beta})\leqslant 1\\}$, and that for any
$w=(u,v)\in{\mathbb{S}}^{1}$, the point
$(\alpha_{D}(w),\beta_{D}(w)=(\ln(x_{D}(w)),\ln(y_{D}(w)))$ (see Definition
2.6.1) is an only point on the boundary of the set $\tilde{D}$, where the
function $(\alpha,\beta)\to u\alpha+v\beta$ achieves its maximum over
$\tilde{D}$.

Consider the line segment
$[\alpha^{*}_{P},\alpha^{**}_{P}]=\\{\alpha\in\mathbb{R}:\leavevmode\nobreak\
\inf_{\beta\in\mathbb{R}}\tilde{P}(\alpha,\beta)\leqslant 1\\}$. We have, see
the proof of Lemma 4.1, $\alpha^{*}_{P}{=}x\ln(x^{*}_{P})$,
$\alpha^{**}_{P}{=}\ln(x^{**}_{P})$, and that for any
$\alpha\in]\alpha^{*}_{P},\alpha^{**}_{P}[$ and
$\beta_{2}(\alpha)=\ln(Y_{2}(e^{\alpha}))$, we have

$\tilde{P}(\alpha,\beta_{2}(\alpha))=1\quad\text{and}\quad\partial_{\beta}\tilde{P}(\alpha,\beta_{2}(\alpha))>0.$

Since the function $\tilde{P}$ is strictly convex, by the implicit function
theorem, it follows that the function $\beta_{2}$ is also strictly convex on
the line segment $[\alpha^{*}_{P},\alpha^{**}_{P}]$.

Under our assumptions, the definition 2.6.1 of the mapping
$w{\to}(\alpha_{D}(w),\beta_{D}(w)){=}(\ln(x_{D}(w)),\ln(y_{D}(w)))$ gives
that, if $w=(u,v)\in{\mathbb{S}}^{1}_{+}$,

$\partial_{\beta}\tilde{P}(\alpha_{D}(w),\beta_{D}(w))\geqslant 0.$

By the definition of the function
$\alpha\to\beta_{2}(\alpha)=\ln(Y_{2}(e^{\alpha}))$, see the proof of Lemma
4.1, it follows that,

$\beta_{D}(w)=\beta_{2}(\alpha_{D}(w)),$

and that $\alpha_{D}(w)$ is an only point in the line segment
$[\alpha^{*}_{P},\alpha^{**}_{P}]$ where the function

(9.10) $\alpha\to u\alpha+v\beta_{2}(\alpha)=u\alpha+v\ln(Y_{2}(e^{\alpha}))$

achieves its maximum over $[\alpha^{*}_{P},\alpha^{**}_{P}]$. Since the
function $\beta_{2}$ is strictly convex on $[\alpha^{*}_{P},\alpha^{**}_{P}]$,
the function (9.10) is also strictly convex on
$[\alpha^{*}_{P},\alpha^{**}_{P}]$, and consequently, it is strictly
increasing on the segment $[\alpha^{*}_{P},\alpha_{D}(w)]$ and strictly
decreasing on the line segment $[\alpha_{D}(w),\alpha^{**}_{P}]$. Since the
function $x\to\ln(x)$ is strictly increasing on $]0,+\infty[$, this proves
that the function $x\to u\ln(x)+v\ln(Y_{2}(x))$ is strictly increasing on the
line segment $[x^{*}_{P},x_{D}(w)]$ and strictly decreasing on the line
segment $[x_{D}(w),x^{**}_{P}]$. ∎

######  Lemma 9.4.

If condition (A1) is satisfied and let, for some
$x_{0}\in]x^{*}_{P},x^{**}_{P}[$, ${\varepsilon}_{1}>0$ and
${\varepsilon}_{2}>0$, a function $(x,y)\to F(x,y)$ be analytic in the
polycircular set

$\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\
|x_{0}{-}|x||<{\varepsilon}_{1},|Y_{2}(x_{0}){-}|y||<{\varepsilon}_{2}\\}$

and do not vanish at the point $(x_{0},Y_{2}(x_{0}))$. Then for any
$w=(u,v)\in{\mathbb{S}}^{1}_{+}$ such that $x_{D}(w)>x_{0}$ and
$(\hat{x},\hat{y})\in\overset{\circ}{D}$ such that
$x_{0}-{\varepsilon}_{1}<\hat{x}<x_{0}$ and
$|Y_{2}(x_{0})-\hat{y}|<{\varepsilon}_{2}$, as
$\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w$,

(9.11) $\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}}\int_{\|y\|=\hat{y}}\frac{F(x,y)}{x^{k_{1}}y^{k_{2}}(x_{0}-x)(1-P(x,y))}\,dx\,dy\leavevmode\nobreak\
\sim\leavevmode\nobreak\ \frac{C_{1}}{x_{0}^{k_{1}}(Y_{2}(x_{0}))^{k_{2}}}$

with

(9.12)
$C_{1}=\left.\frac{F(x,y)}{\partial_{y}P(x,y)}\right|_{(x,y)=(x_{0},Y_{2}(x_{0}))}\not=0$

and

(9.13) $\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}}\int_{\|y\|=\hat{y}}\frac{F(x,y)}{x^{k_{1}}y^{k_{2}}(x_{0}-x)^{2}(1-P(x,y))}\,dx\,dy\\\
\leavevmode\nobreak\ \sim\leavevmode\nobreak\
\frac{k_{1}C_{1}}{x_{0}^{k_{1}+1}(Y_{2}(x_{0}))^{k_{2}}}-\frac{k_{2}{C}_{2}}{x_{0}^{k_{1}}(Y_{2}(x_{0}))^{k_{2}+1}}$

with $C_{1}$ given by (9.12) and

(9.14)
$C_{2}=\left.\frac{F(x,y)\partial_{x}P(x,y)}{\left(\partial_{y}P(x,y)\right)^{2}}\right|_{(x,y)=(x_{0},Y_{2}(x_{0}))}.$

###### Proof.

Before proving this lemma, remark that under our assumptions, by Lemma 4.1,
$Y_{2}(x_{0})>Y_{1}(x_{0})$ and consequently, without any restriction of
generality, we can assume throughout our proof that

${\varepsilon}_{2}<Y_{2}(x_{0})-Y_{1}(x_{0}).$

Because of Assumption (A1), the function $(x,y)\to 1-P(x,y)$ is analytic in a
neighborhood ${\mathcal{V}}$ of the set $\Omega(D)$ and does not vanishes in
its interior $\Omega(\overset{\circ}{D})$. Hence, for any
$m\in{\mathbb{N}}^{*}$, the function

$(x,y)\to F(x,y)(x_{0}-x)^{-m}x^{-k_{1}}y^{-k_{2}}(1-P(x,y))^{-1}$

is analytic in the polycircular sets
$\\{(x,y)\in\Omega(\overset{\circ}{D}),\,x_{0}-{\varepsilon}_{1}<|x|<x_{0},\,|Y_{2}(x_{0})-y|<{\varepsilon}_{2}\\}$
and
$\\{(x,y)\in\Omega(\overset{\circ}{D}),\,x_{0}<|x|<x_{0}+{\varepsilon}_{1},\,|Y_{2}(x_{0})-y|<{\varepsilon}_{2}\\}$,
and consequently, the function

$(\hat{x},\hat{y})\to J_{m,k}(\hat{x},\hat{y})=\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}}\int_{\|y\|=\hat{y}}\frac{F(x,y)\,dx\,dy}{x^{k_{1}}y^{k_{2}}(x_{0}-x)^{m}(1-P(x,y))}$

is constant on the set
$A_{-}=\\{(x,y)\in\overset{\circ}{D}:\leavevmode\nobreak\
x_{0}-{\varepsilon}_{1}<x<x_{0},\,|Y_{2}(x_{0})-y|<{\varepsilon}_{2}\\}$ and
on the set $A_{+}=\\{(x,y)\in\overset{\circ}{D}:\leavevmode\nobreak\
x_{0}<x<x_{0}+{\varepsilon}_{1},\,|Y_{2}(x_{0})-y|<{\varepsilon}_{2}\\}$. We
denote

$\displaystyle J_{m}(k)$ $\displaystyle=J_{m,k}(\hat{x},\hat{y})\quad\text{
for $(\hat{x},\hat{y})\in A_{-}$},$ $\displaystyle\hat{J}_{m}(k)$
$\displaystyle=J_{m,k}(\hat{x},\hat{y})\quad\text{ for $(\hat{x},\hat{y})\in
A_{+}$.}$

Remark now that for any
$\hat{y}\in]Y_{2}(x_{0})-{\varepsilon}_{2},Y_{2}(x_{0})[$ and $\delta>0$ small
enough, the point $(x_{0}-\delta,\hat{y})$ belongs to the set $A_{-}$, the
point $(x_{0}+\delta,\hat{y})$ belongs to the set $A_{+}$ and by the residue
theorem, for any $y\in\mathbb{C}$ with $|y|=\hat{y}$,

$\int_{|x|=x_{0}+\delta}\\!\frac{F(x,y)\,dx}{x^{k_{1}}(x_{0}-x)(1-P(x,y))}=\int_{|x|=x_{0}-\delta}\,\frac{F(x,y)\,dx}{x^{k_{1}}(x_{0}-x)(1-P(x,y))}-2\pi
i\frac{F(x_{0},y)}{x_{0}^{k_{1}}(1-P(x_{0},y))}$

and

$\int_{|x|=x_{0}+\delta}\frac{F(x,y)\,dx}{x^{k_{1}}(x_{0}-x)^{2}(1-P(x,y))}=\int_{|x|=x_{0}-\delta}\,\frac{F(x,y)\,dx}{x^{k_{1}}(x_{0}-x)^{2}(1-P(x,y))}\\\
+2\pi
i\left(\frac{\partial_{x}F(x_{0},y)}{x_{0}^{k_{1}}(1-P(x_{0},y))}-\frac{k_{1}F(x_{0},y)}{x_{0}^{k_{1}+1}(1-P(x_{0},y))}+\frac{F(x_{0},y)\partial_{x}P(x_{0},y)}{x_{0}^{k_{1}}(1-P(x_{0},y))^{2}}\right)$

Hence, for any $\hat{y}\in]Y_{2}(x_{0})-{\varepsilon}_{2},Y_{2}(x_{0})[$,

(9.15) $J_{1}(k)=\hat{J}_{1}(k)+\frac{1}{2\pi
i\,x_{0}^{k_{1}}}\int_{\|y\|=\hat{y}}\frac{F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy$

and

(9.16) $J_{2}(k)=\hat{J}_{2}(k)-\frac{1}{2\pi
ix_{0}^{k_{1}}}\int_{\|y\|=\hat{y}}\frac{\partial_{x}F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy\\\
+\frac{k_{1}}{(2\pi
i)x_{0}^{k_{1}+1}}\int_{\|y\|=\hat{y}}\frac{F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy-\frac{1}{2\pi
ix_{0}^{k_{1}}}\int_{\|y\|=\hat{y}}\frac{F(x_{0},y)\partial_{x}P(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))^{2}}dy.$

Due to Assumption (A1)(ii), the function $y\to P(x_{0},y)$ is analytic in a
neighborhood of the closed annulus $\\{y\in\mathbb{C}:\leavevmode\nobreak\
Y_{1}(x_{0})\leqslant|y|\leqslant Y_{2}(x_{0})\\}$, satisfies the inequality

$|P(x_{0},y)|\leqslant P(x_{0},|y|)<1$

on the annulus $\\{y\in\mathbb{C}:\leavevmode\nobreak\
Y_{1}(x_{0})<|y|<Y_{2}(x_{0})\\}$, and because of Assumption (A1) i) (we use
here Proposition P7.5 of [24]), for any $y\in\mathbb{C}$ with
$|y|=Y_{2}(x_{0})$,

$|P(x_{0},y)|<P(x_{0},Y_{2}(x_{0}))=1\quad\text{whenever
$y\not=Y_{2}(x_{0})$}.$

Since by Lemma 4.1,

$\left.\partial_{y}P(x,y)\right|_{(x,y)=(x_{0},Y_{2}(x_{0}))}>0,$

this proves that for some $\delta>0$, the point $Y_{2}(x_{0})$ is an only and
simple zero of the function $y\to P(x_{0},y)$ in the annulus
$\\{y\in\mathbb{C}:\leavevmode\nobreak\
Y_{1}(x_{0})<|y|<Y_{2}(x_{0})+\delta)$. Since we assumed that
$Y_{2}(x_{0})-{\varepsilon}_{2}<Y_{1}(x_{0})$, and the functions $y\to
F(x_{0},y)$, $y\to\partial_{x}F(x_{0},y)$, $y\to F(x_{0},y)$ are analytic in
the annulus $\\{y\in\mathbb{C}:\leavevmode\nobreak\
Y_{2}(x_{0})-{\varepsilon}_{2}<|y|<Y_{2}(x_{0})+{\varepsilon}_{2}\\}$, this
implies that for $\delta>0$ small enough, the functions

$y\to\frac{F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}\quad\text{and}\quad
y\to\frac{\partial_{x}F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}$

are analytic in the set $\\{y\in\mathbb{C}:\leavevmode\nobreak\
Y_{2}(x_{0})-{\varepsilon}_{2}<|y|<Y_{2}(x_{0})+\delta,\,y\not=Y_{2}(x_{0})\\}$
and have at the point $Y_{2}(x_{0})$, a simple pole with the residue equal
respectively to

$\frac{-C_{1}}{(Y_{2}(x_{0}))^{k_{2}}}\quad\text{and}\quad\frac{-\tilde{C}_{1}}{(Y_{2}(x_{0}))^{k_{2}}}.$

with $C_{1}$ given by (9.12) and

$\tilde{C}_{1}=\left.\frac{\partial_{x}F(x,y)}{\partial_{y}P(x,y)}\right|_{(x,y)=(x_{0},Y_{2}(x_{0}))}.$

Similarly, the function

$y\to\frac{F(x_{0},y)\partial_{x}P(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))^{2}}$

is analytic in the set $\\{y\in\mathbb{C}:\leavevmode\nobreak\
Y_{2}(x_{0})-{\varepsilon}_{2}<|y|<Y_{2}(x_{0})+\delta,\,y\not=Y_{2}(x_{0})\\}$
and has at the point $Y_{2}(x_{0})$, a pole of the second order with the
residue equal to

$\frac{C}{(Y_{2}(x_{0}))^{k_{2}}}-\frac{k_{2}C_{2}}{(Y_{2}(x_{0}))^{k_{2}+1}}$

with $C_{2}$ given by (9.14) and some constant $C\in\mathbb{C}$ does not
depending on $k_{2}$. By the residue theorem, it follows that for any
$Y_{2}(x_{0})-{\varepsilon}_{2}<\hat{y}<Y_{2}(x_{0})$ and
$Y_{2}(x_{0})<y_{\delta}<Y_{2}(x_{0})+\delta)$

$\int_{\|y\|=\hat{y}}\frac{F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy=\frac{2\pi
iC_{1}}{(Y_{2}(x_{0}))^{k_{2}}}+\int_{\|y\|=y_{\delta}}\frac{F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy,$
$\int_{\|y\|=\hat{y}}\frac{\partial_{x}F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy=\frac{2\pi
i\tilde{C}_{1}}{(Y_{2}(x_{0}))^{k_{2}}}+\int_{\|y\|=y_{\delta}}\frac{\partial_{x}F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy$

and

$\int_{\|y\|=\hat{y}}\frac{F(x_{0},y)\partial_{x}P(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))^{2}}dy\leavevmode\nobreak\
=\leavevmode\nobreak\ -2\pi
i\left(\frac{C}{(Y_{2}(x_{0}))^{k_{2}}}-\frac{k_{2}C_{2}}{(Y_{2}(x_{0}))^{k_{2}+1}}\right)+\int_{\|y\|=y_{\delta}}\frac{F(x_{0},y)\partial_{x}P(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))^{2}}dy$

Using these relations at the right hand side of (9.15) and (9.16) we obtain

(9.17)
$J_{1}(k)=\frac{C_{1}}{x_{0}^{k_{1}}(Y_{2}(x_{0}))^{k_{2}}}+\frac{1}{(2\pi
i)}\int_{\|y\|=y_{\delta}}\frac{F(x_{0},y)}{x_{0}^{k_{1}}y^{k_{2}}(1-P(x_{0},y))}dy+\hat{J}_{1}(k)$

and

(9.18) $\displaystyle J_{2}(k)$
$\displaystyle=\frac{k_{1}C_{1}}{x_{0}^{k_{1}+1}(Y_{2}(x_{0}))^{k_{2}}}-\frac{k_{2}C_{2}}{x_{0}^{k_{1}}(Y_{2}(x_{0}))^{k_{2}+1}}+\frac{\tilde{C}}{x_{0}^{k_{1}}(Y_{2}(x_{0}))^{k_{2}}}~$
$\displaystyle-\frac{1}{2\pi
ix_{0}^{k_{1}}}\int_{\|y\|=y_{\delta}}\frac{\partial_{x}F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy+\frac{k_{1}}{(2\pi
i)x_{0}^{k_{1}+1}}\int_{\|y\|=y_{\delta}}\frac{F(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))}dy$
$\displaystyle-\frac{1}{2\pi
ix_{0}^{k_{1}}}\int_{\|y\|=y_{\delta}}\frac{F(x_{0},y)\partial_{x}P(x_{0},y)}{y^{k_{2}}(1-P(x_{0},y))^{2}}dy+\hat{J}_{2}(k)$

with some constant $\tilde{C}$ does not depending on $k$. Remark now that by
the Cauchy inequality, for $k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, as
$\min\\{k_{1},k_{2}\\}\to\infty$ and $k/\|k\|\to w=(u,v)$,

$\displaystyle\limsup_{k}\|k\|^{-1}\log\left|\frac{1}{(2\pi
i)}\int_{\|y\|=y_{\delta}}\frac{F(x_{0},y)}{x_{0}^{k_{1}}y^{k_{2}}(1-P(x_{0},y))}dy\right|$
$\displaystyle\leqslant-\bigl{(}u\log(x_{0})+v\log(y_{\delta})\bigr{)}$
$\displaystyle\hskip 28.45274pt<-(u\ln(x_{0})+v\ln(Y_{2}(x_{0}))),$

and with the same argument as in the proof of Lemma 9.2,

$\limsup_{k}\|k\|^{-1}\log\bigl{|}\hat{J}_{1}(k)\bigr{|}\leqslant-\sup_{x_{0}\leqslant\hat{x}\leqslant
x_{0}+{\varepsilon}_{1}}\bigl{(}u\ln\hat{x}+v\ln(Y_{2}(\hat{x}))\bigr{)}<-(u\ln(x_{0})+v\ln(Y_{2}(x_{0})))$

where the last relation holds because under our assumptions,
$x^{*}_{P}<x_{0}<x_{D}(w)$ and by Lemma 9.3, the function $x\to u\ln
x+v\ln(Y_{2}(x))$ is strictly increasing on $[x^{*}_{P},x_{D}(w)]$. Since for
$k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, as $\|k\|\to\infty$ and $k/\|k\|\to
w$,

$\lim\|k\|^{-1}\log\left|\frac{F(x_{0},Y_{2}(x_{0}))}{x_{0}^{k_{1}}(Y_{2}(x_{0}))^{k_{2}}\partial_{y}P(x_{0},Y_{2}(x_{0}))}\right|=-\bigl{(}u\log(x_{0})+v\log(Y_{2}(x_{0}))\bigr{)}$

this proves that as $\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w$, the
second and the third terms in the right hand side of (9.17) are negligible
with respect to the first one. Hence (9.11) is verified, and with the same
arguments, from (9.18) one gets (9.13). ∎

As a consequence of Lemma 9.4 and Theorem 2, we obtain the following result.

######  Lemma 9.5.

Under the assumptions (A1)-(A3), for any
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and $w=(u,v)\in{\mathbb{S}}^{1}_{+}$
with $u>u_{D}(x_{d},Y_{2}(x_{d}))$, the following assertions hold

i) If either one of the assertions (B0), (B1), (B3),(B4) holds or (B2) holds
with $x_{d}<x^{**}_{P}$, then for any $k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$,
as $\min\\{k_{1},k_{2}\\}\to\infty$ and $k/\|k\|\to w$,

(9.19) $I_{1}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{1}\,a_{1}\,\varkappa_{1}(j)x_{d}^{-k_{1}-1}(Y_{2}(x_{d}))^{-k_{2}-1}$

where

(9.20)
$c_{1}=\left.\frac{\phi_{1}(x,y)-1}{\partial_{y}P(x,y))}\right|_{(x,y)=(x_{d},Y_{2}(x_{d}))}>0$

and $a_{1}>0$ is given by (2.41).

ii) If (B5) holds with $x_{d}<x^{**}_{P}$, then for
$k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, as $\min\\{k_{1},k_{2}\\}\to\infty$
and $k/\|k\|\to w$,

(9.21) $I_{1}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
\frac{c_{1}\,a_{3}\varkappa_{2}(j)k_{1}}{x_{d}^{k_{1}+2}(Y_{2}(x_{d}))^{k_{2}+1}}+\frac{\tilde{c}_{1}\,a_{3}\varkappa_{2}(j)k_{2}}{x_{d}^{k_{1}+1}(Y_{2}(x_{d}))^{k_{2}+2}}$

where $c_{1}>0$ is given by (9.20),

$\tilde{c}_{1}=\left.\frac{\partial_{x}P(x_{d},Y_{2}(x_{d}))}{\left(\partial_{y}P(x_{d},Y_{2}(x_{d}))\right)^{2}}\right|_{(x,y)=(x_{d},Y_{2}(x_{d}))}$

and $a_{3}>0$ is given by (2.47).

iii) If (B6) holds, then for $k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, as
$\min\\{k_{1},k_{2}\\}\to\infty$ and $k/\|k\|\to w$,

(9.22) $I_{1}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{1}\,a_{5}\,\varkappa_{2}(j)x_{d}^{-k_{1}-1}(Y_{2}(x_{d}))^{-k_{2}-1}$

where $c_{1}>0$ is given by (9.20) and $a_{5}>0$ is given by (2.53).

###### Proof.

Indeed, when either, one of the conditions (B0), (B1), (B3) or (B4) is
satisfied, or (B2) holds with $x_{d}<x^{**}_{P}$, by the first assertion of
Theorem 2 and using Proposition 7.13, one gets that for some
${\varepsilon}>0$, the function $x\to H_{j}(x,0)$ can be extended as an
analytic function to the set $B(0,x_{d}+{\varepsilon}){\setminus}\\{x_{d}\\}$
and has a simple pole at the point $x_{d}$ with the residue

$-a_{1}\varkappa_{1}(j)=-\varkappa_{1}(j)\left(\frac{d}{dx}\phi_{1}(x_{d},Y_{1}(x_{d}))\right)^{-1}<0.$

Since the function $(x,y)\to\phi_{1}(x,y)-P(x,y)$ is analytic in a
neighborhood of the set $\Omega(D)$, this implies that the function $(x,y)\to
F(x,y)=(x_{d}-x)(\phi_{1}(x,y)-P(x,y))H_{j}(x,0)$ can be extended as an
analytic function to a neighborhood of the set
$\\{(x,y)\in\Omega(D):\,|x|<x_{d}+{\varepsilon}\\}$ by letting

$F(x_{d},y)=(\phi(x_{d},y)-P(x_{d},y))a_{1}\varkappa_{1}(j).$

When either, one of the conditions (B0), (B1), (B3) or (B4) is satisfied, or
(B2) holds with $x_{d}<x^{**}_{P}$, one has always
$x^{*}_{p}<x_{d}=x^{**}<x^{**}_{P}$, and consequently, by Lemma 4.1,
$Y_{2}(x_{d})>Y_{1}(x_{d})$ and by Lemma 4.1,
$\phi_{1}(x_{d},Y_{1}(x_{d}))=1$. Since the function $y\to\phi_{1}(x_{d},y)$
is strictly increasing, it follows that $\phi_{1}(x_{d},Y_{2}(x_{d}))>1$.
Since by Theorem 2, $a_{1}>0$, we obtain therefore

$\displaystyle F(x_{d},Y_{2}(x_{d}))$
$\displaystyle=(\phi_{1}(x_{d},Y_{2}(x))-P(x_{d},Y_{2}(x_{d}))a_{1}\varkappa_{1}(j)=(\phi_{1}(x_{d},Y_{2}(x))-1)a_{1}\varkappa_{1}(j)>0.$

By Lemma 9.4, it follows that for any $w=(u,v)\in{\mathbb{S}}^{1}_{+}$ such
that $x_{D}(u)>x_{d}$, and as $\min\\{k_{1},k_{2}\\}\to+\infty$, (9.19) holds
with $c_{1}>0$ given by (9.20). Since for $w=(u,v)\in{\mathbb{S}}^{2}_{+}$ the
inequality $x_{D}(u)>x_{d}$ is equivalent to the inequality
$u>u_{D}(x_{d},Y_{2}(x_{d}))$, the first assertion of our lemma is therefore
proved.

Suppose now that (B5) holds with $x_{d}<x^{**}_{P}$. Then, by Theorem 2 and
using Proposition 7.13, one gets that for some ${\varepsilon}>0$, the function
$x\to H_{j}(x,0)$ can be extended as an analytic function to the set
$B(0,x_{d}+{\varepsilon}){\setminus}\\{x_{d}\\}$ and has a pole of the second
order at the point $x_{d}$ with

$\lim_{x\to x_{0}}(x_{d}-x)^{2}H_{j}(x,0)=a_{3}\varkappa_{2}(j)>0$

where $a_{3}>0$ is given by (2.47). With the same arguments as above, it
follows that the function $(x,y)\to
F(x,y)=(x_{d}-x)^{2}(\phi_{1}(x,y)-1)H_{j}(x,0)$ can be extended as an
analytic function to a neighborhood of the set
$\\{(x,y)\in\Omega(D):\,|x|<x_{d}+{\varepsilon}\\}$ by letting
$F(x_{d},y)=(\phi_{1}(x_{d},y)-1)a_{3}\varkappa_{2}(j)$ and that the extended
function $F$ satisfies the assumptions of Lemma 9.4 with $x_{0}=x_{d}$ and
$F(x_{0},Y_{2}(x_{0}))=a_{3}\varkappa_{2}(j)(\phi_{1}(x_{d},Y_{2}(x_{d}))-1)>0$.
Hence, using (9.13) we obtain (9.21) with $c_{1}>0$ given by (9.20). The
second assertion of our lemma is therefore also proved.

The proof of the third assertion of our lemma is exactly the same as the proof
the first assertion, with an only difference that now, one should use the
fifth assertion of Theorem 2 instead of the first one. ∎

Remark finally that if we exchange the roles of $x$ and $y$, then with the
same arguments as above we obtain the following result.

######  Lemma 9.6.

Under the assumptions (A1)-(A3), for any
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and $w=(u,v)\in{\mathbb{S}}^{1}_{+}$
with $v>v_{D}(X_{2}(y_{d}),y_{d})$, the following assertions hold

  1. i)

If either one of the assertions (B0), (B1), (B5),(B6) holds or (B2) holds with
$y_{d}<y^{**}_{P}$, then for any $k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, as
$\min\\{k_{1},k_{2}\\}\to\infty$ and $k/\|k\|\to w$,

(9.23) $I_{2}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{2}\,b_{2}\varkappa_{2}(j)(X_{2}(y_{d}))^{-k_{1}-1}(y_{d})^{-k_{2}-1}$

where

(9.24)
$c_{2}=\left.(\phi_{2}(x,y)-1)\left(\partial_{x}P(x,y)\right)^{-1}\right|_{(x,y)=(X_{2}(y_{d}),y_{d})}>0$

and

(9.25)
$a_{2}=\left.\left(\frac{d}{dy}\phi_{2}(X_{1}(y),y)\right)^{-1}\right|_{y=y_{d}}$

  2. ii)

If (B3) holds with $y_{d}<y^{**}_{P}$, then for
$k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, as $\min\\{k_{1},k_{2}\\}\to\infty$
and $k/\|k\|\to w$,

(9.26) $I_{2}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
\frac{c_{2}\,b_{3}\varkappa_{1}(j)k_{2}}{(X_{2}(y_{d}))^{k_{1}+1}y_{d}^{k_{2}+2}}+\frac{\tilde{c}_{2}\,b_{3}\varkappa_{1}(j)k_{1}}{(X_{2}(y_{d}))^{k_{1}+2}{y_{d}}^{k_{2}+1}}$

with $c_{2}>0$ given by (9.24),

(9.27)
$a_{3}^{\prime}=\left.(\phi_{1}(x,y)-1)\left(\frac{d}{dy}\phi_{2}(X_{1}(y),y)\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\frac{d}{dy}X_{1}(y)\right)^{-1}\right|_{(x,y)=(X_{2}(y_{d}),y_{d})}>0$

and some $\tilde{c}_{2}\in\mathbb{R}$ do not depending on
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$.

  3. iii)

If (B4) holds, then for $k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, as
$\min\\{k_{1},k_{2}\\}\to\infty$ and $k/\|k\|\to w$,

(9.28) $I_{2}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{2}\,a^{\prime}_{5}\varkappa_{1}(j)(X_{2}(y_{d}))^{-k_{1}-1}(y_{d})^{-k_{2}-1}$

with $c_{2}$ given by (9.24) and

(9.29)
$a^{\prime}_{5}=\left.(\phi_{1}(x,y)-1)\left((1-\phi_{2}(x,y))\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\frac{d}{dy}X_{1}(y)\right)^{-1}\right|_{(x,y)=(X_{2}(y_{d}),y_{d})}>0$

###  9.5. Proof of the assertions i) and ii) of Theorem 4

For $I_{1}(j,k)$ the exact asymptotics as $\min\\{k_{1},k_{2}\\}\to+\infty$
and $k/\|k\|\to w=(u,v)\in{\mathbb{S}}^{1}_{+}$ were obtained only in the case
where $u{>}u_{D}(x_{d},Y_{2}(x_{d}))$. In the following lemma, we show that
this inequality always holds when $w\in{\mathcal{W}}_{1}$. The definitions 3-6
give that the set ${\mathcal{W}}_{1}$ is empty if either, one of the cases
(B5), or (B6) occurs or, (B2) and $x_{d}{=}x^{**}_{P}$ hold, we can assume
that either one of cases (B0), (B1), (B3) or (B4) occurs or (B2) and
$x_{d}<x^{**}_{P}$ hold.

######  Lemma 9.7.

If conditions (A1)-(A3) hold and if either, one of cases (B0), (B1), (B3) or
(B4) holds, or (B2) and $x_{d}{<}x^{**}_{P}$ hold, then for any
$w\in{\mathcal{W}}_{1}$,

(9.30) $u>u_{D}(x_{d},Y_{2}(x_{d})).$

###### Proof.

Suppose first that (B0) holds. Then by Proposition 2.2 and the definition of
the points $x_{d}$ and $y_{d}$ (see (2.33) and (2.34)),

(9.31) $X_{1}(y_{d})<x_{d}=x^{**}<X_{2}(y_{d}),\quad
Y_{1}(x_{d})<y_{d}=y^{**}<Y_{2}(x_{d}).$

In this case ${\mathcal{W}}_{1}=\\{w=(u,v)\in{\mathbb{S}}^{1}_{+}:u>u_{c}\\}$
where $w_{c}=(u_{c},v_{c})$ is the only point in ${\mathbb{S}}^{1}_{+}$ such
that

(9.32)
$u_{c}\ln(x_{d})+v_{c}\ln(Y_{2}(x_{d}))=u_{c}\ln(X_{2}(y_{d}))+v_{c}\ln(y_{d}),$

or equivalently, such that

(9.33)
$(\ln(X_{2}(y_{d}))-\ln(x_{d}))u_{c}=(\ln(Y_{2}(x_{d}))-\ln(y_{d}))v_{c}.$

Since by (9.31) , $X_{2}(y_{d})>x_{d}$ and $Y_{2}(x_{d})>y_{d}$, the last
relation implies that $u_{c}>0$ and $v_{c}>0$, and consequently, if
$u_{D}(x_{d},Y_{2}(x_{d}))\leqslant 0$, one gets

$u_{D}(x_{d},Y_{2}(x_{d}))\leqslant
u_{c}<u,\quad\forall\;w=(u,v)\in{\mathcal{W}}_{1}.$

When (B0) holds and $u_{D}(x_{d},Y_{2}(x_{d}))\leqslant 0$, our lemma is
therefore proved.

Consider now the case when (B0) holds and $u_{D}(x_{d},Y_{2}(x_{d}))>0$. For
$\hat{w}=(\hat{u},\hat{v})=(u_{D}(x_{d},Y_{2}(x_{d})),v_{D}(x_{d},Y_{2}(x_{d})))$,
by the definition of the mapping $w\to(x_{D}(w),y_{d}(w))$, the point
$(x_{d},Y_{2}(x_{d}))$ is the only point in the set $D$ where the function
$(x,y)\to\hat{u}\ln(x)+\hat{v}\ln(y)$ achieves its maximum over the set $D$.
Since in the case (B0), $(x_{d},Y_{2}(x_{d}))\not=(X_{2}(y_{d}),y_{d})$, it
follows that for
$\hat{w}=(\hat{u},\hat{v})=(u_{D}(x_{d},Y_{2}(x_{d})),v_{D}(x_{d},Y_{2}(x_{d})))$,

$\hat{u}\ln(x_{d})+\hat{v}\ln(Y_{2}(x_{d}))>\hat{u}\ln(X_{2}(y_{d}))+\hat{v}\ln(y_{d})$

or equivalently that

$(\ln(X_{2}(y_{d}))-\ln(x_{d}))\hat{u}<(\ln(Y_{2}(x_{d}))-\ln(y_{d}))\hat{v}.$

Since $X_{2}(y_{d})>x_{d}$ and $Y_{2}(x_{d})>y_{d}$, and since we assumed that
$\hat{u}=u_{D}(x_{d},Y_{2}(x_{d}))>0$, the last inequality shows that
$\hat{v}=v_{D}(x_{d},Y_{2}(x_{d}))>0$ and consequently,
$\hat{v}=\sqrt{1-\hat{u}^{2}}$ and

$(\ln(X_{2}(y_{d}))-\ln(x_{d}))\hat{u}<(\ln(Y_{2}(x_{d}))-\ln(y_{d}))\sqrt{1-\hat{u}^{2}}.$

Finally, we have already proved that $u_{c}>0$ and $v_{c}>0$, and in this
case, relation (9.33) is equivalent to

$(\ln(X_{2}(y_{d}))-\ln(x_{d})){u}_{c}=(\ln(Y_{2}(x_{d}))-\ln(y_{d}))\sqrt{1-{u}_{c}^{2}}.$

Since the function $u\to(\ln(X_{2}(y_{d}))-\ln(x_{d})){u}$ is strictly
increasing and the function
$u\to(\ln(Y_{2}(x_{d}))-\ln(y_{d}))\sqrt{1-{u}_{c}^{2}}$ is decreasing on the
segment $[0,1]$, the last two relations show that
$u_{c}>\hat{u}=u_{D}(x_{d},Y_{2}(x_{d}))$, and consequently using the
definition of ${\mathcal{W}}_{1}$ one gets $u>u_{c}>u_{D}(x_{d},Y_{2}(x_{d}))$
for all $u\in{\mathcal{W}}_{1}$. If (B0) holds, Lemma 9.7 is therefore proved.

When either, (B1) holds, or (B2) and $x_{d}{<}x^{**}_{P}$ hold, relation
(9.30) follows directly from the definition of the set ${\mathcal{W}}_{1}$
(see Definition 4 and Definition 5).

Consider now the case when (B3) holds. With definition 6,
${\mathcal{W}}_{1}=\\{w=(u,v)\in{\mathbb{S}}^{1}_{+}:\leavevmode\nobreak\
u>0\\}$ and by Proposition 2.2 and the definition of the points $x_{d}$ and
$y_{d}$ (see (2.33) and (2.34)), one has
$(x_{d},Y_{2}(x_{d}))\in{\mathcal{S}}_{12}$. By Lemma 4.1 and the definition
of the mapping $(x,y)\to w_{D}(x,y)$, it follows that

$u_{D}(x_{d},Y_{2}(x_{d}))=\frac{x_{d}\partial_{x}P(x_{d},y_{d})}{\sqrt{(x_{d}\partial_{x}P(x_{d},y_{d}))^{2}+(y_{d}\partial_{y}P(x_{d},y_{d}))^{2}}}\leqslant
0.$

and consequently, for any $w=(u,v)\in{\mathcal{W}}_{1}$, (9.30) holds.

Suppose now that (B4) holds. Then, with the Definition 6,
${\mathcal{W}}_{1}={\mathbb{S}}^{1}_{+}$ and by Proposition 2.2 and the
definition of the points $x_{d}$ and $y_{d}$ (see (2.33) and (2.34)) and using
(2.9) one gets

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

By Lemma 4.1 and the definition of the mapping $(x,y)\to w_{D}(x,y)$, it
follows that for any $w=(u,v)\in{\mathcal{W}}_{1}$,

$u_{D}(x_{d},Y_{2}(x_{d})=u_{D}(x_{d},y_{d})=u_{D}(X_{1}(x_{d}),y_{d})=\frac{x_{d}\partial_{x}P(x_{d},y_{d})}{\sqrt{(x_{d}\partial_{x}P(x_{d},y_{d}))^{2}+(y_{d}\partial_{y}P(x_{d},y_{d}))^{2}}}<0\leqslant
u,$

and consequently, (9.30) holds. ∎

As a straightforward consequence of Lemma 9.5 and Lemma 9.7, one gets

######  Corollary 9.2.

Under the assumptions of Theorem 4, for any
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and $w\in{\mathcal{W}}$, as
$\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w$, (9.19) holds.

Now we will show that for any $w=(u,v)\in{\mathcal{W}}$, as
$\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w$, the terms $I_{0}(j,k)$
and $I_{2}(j,k)$ are negligible with respect to $I_{1}(j,k)$. For this we will
use the large deviation estimates for $I_{0}(k,j)$, and

  * –

the large deviation estimates of $I_{2}(j,k)$ when

$\limsup_{k}\frac{1}{\|k\|}\ln I_{2}(j,k)<\lim\frac{1}{\|k\|}\ln I_{1}(j,k),$

  * –

the exact asymptotic of $I_{2}(j,k)$ when

$\limsup_{k}\frac{1}{\|k\|}\ln I_{2}(j,k)=\lim\frac{1}{\|k\|}\ln I_{1}(j,k).$

To compare the limit $\lim_{k}\ln I_{1}(j,k)/\|k\|$ with the large deviation
estimates for $\limsup_{k}\ln I_{0}(j,k)/\|k\|$ and $\limsup_{k}\ln
I_{2}(j,k)/\|k\|$, the following lemma will be useful.

######  Lemma 9.8.

If conditions (A1)-(A3) hold and let $w{=}(u,v){\in}{\mathcal{W}}_{1}$, if one
of the assertions holds,

  * –

one of the conditions (B0), (B1) or (B3) holds,

  * –

(B2) and $x^{**}{<}x^{**}_{P}$ hold,

  * –

(B4) holds and $u>0$,

then

(9.35) $\max_{(x,y)\in D,\,{y}\leqslant
y_{d}}\\!\Bigl{(}u\ln({x})+v\ln({y})\Bigr{)}>u\ln(x_{d})+v\ln(Y_{2}(x_{d})).$

###### Proof.

Consider first the case when (B0) holds. We have, see the proof of Lemma 9.8,
$X_{2}(y_{d})>x_{d}$ and $Y_{2}(x_{d})>y_{d}$, the point
$(w_{c}=(u_{c},v_{c})\in{\mathbb{S}}^{2}_{+}$ satisfies (9.33) and
${\mathcal{W}}_{1}=\\{w=(u,v)\in{\mathbb{S}}^{1}_{+}:u>u_{c}\\}$. For
$w=(u,v)\in{\mathbb{S}}^{1}_{+}$ with $u>u_{c}$ (and consequently also with
$v<v_{c}$), it follows that

$(\ln(X_{2}(y_{d}))-\ln(x_{d}))u>(\ln(Y_{2}(x_{d}))-\ln(y_{d}))v$

or equivalently,

(9.36) $u\ln(X_{2}(y_{d}))+v\ln(y_{d})>u\ln(x_{d})+v\ln(Y_{2}(x_{d})).$

Since the point $(X_{2}(y_{d}),y_{d})$ belongs to the set $\\{(x,y)\in
D:\leavevmode\nobreak\ y\leqslant y_{d}\\}$, this proves that for any
$w=(u,v)\in{\mathbb{S}}^{1}_{+}$ with $u>u_{c}$,

(9.37) $\max_{({x},{y})\in D,\,{y}\leqslant
y_{d}}\\!\Bigl{(}u\ln({x})+v\ln({y})\Bigr{)}\geqslant
u\ln(X_{2}(y_{d}))+v\ln(y_{d})>u\ln(x_{d})+v\ln(Y_{2}(x_{d})),$

and consequently, (9.35) holds.

Consider now the case when (B1) holds, we have the relation

${\mathcal{W}}_{1}=\\{w=(u,v)\in{\mathbb{S}}^{1}_{+}:\leavevmode\nobreak\
u>u_{D}(x_{d},Y_{2}(x_{d}))\\},$

and, by Proposition 2.2 and the definition of the points $x_{d}$ and $y_{d}$
(see (2.33) and (2.34)),

(9.38) $X_{1}(y_{d})<x_{d}=x^{**}=X_{2}(y_{d}),\quad
Y_{1}(x_{d})<y_{d}=y^{**}=Y_{2}(x_{d})$

and $(x_{d},Y_{2}(x_{d}))=(X_{2}(y_{d}),y_{d})\in{\mathcal{S}}_{22}$. Hence,
for any $w=(u,v)\in{\mathcal{W}}_{1}$,

(9.39) $x_{D}(w)>x_{d}=X_{2}(y_{d})\quad\;\text{and}\;\quad
y_{D}(w)<y_{d}=Y_{2}(x_{d}).$

The last relations show that the point $(x_{D}(w),y_{D}(w))$ belongs to the
set $\\{(x,y)\in D:\leavevmode\nobreak\ y\leqslant y_{d}\\}$ and is not equal
to $(x_{d},Y_{2}(x_{d}))$. Since the point $(x_{D}(w),y_{D}(w))$ is an only
point in $D$ where the function $(x,y)\to u\ln(x)+x\ln(y)$ achieves its
maximum over $D$, it follows that

(9.40) $\max_{(\hat{x},\hat{y})\in D,\,\hat{y}\leqslant
y_{d}}\\!\Bigl{(}u\ln(\hat{x})+v\ln(\hat{y})\Bigr{)}=u\ln(x_{D}(w))+v\ln(y_{D}(w))>u\ln(x_{d})+v\ln(Y_{2}(x_{d})),$

and consequently, when the assertion (B1) holds, relation (9.35) is also
proved.

Consider now the case when (B2) and $x_{d}<x^{**}_{P}$ hold. By Proposition
2.2 and the definition of the points $x_{d}$ and $y_{d}$ (see (2.33) and
(2.34)),

(9.41) $x_{d}=x^{**}>X_{2}(y_{d}),\quad
y_{d}=y^{**}>Y_{2}(x_{d}),\quad(x_{d},Y_{2}(x_{d})),\,(X_{2}(y_{d}),y_{d})\in{\mathcal{S}}_{22},$

and with the definition of ${\mathcal{W}}_{1}$,

${\mathcal{W}}_{1}=\\{w=(u,v)\in{\mathbb{S}}^{1}_{+}:\leavevmode\nobreak\
u>u_{D}(x_{d},Y_{2}(x_{d}))\\}.$

Hence, in this case, for any $w=(u,v)\in{\mathcal{W}}_{1}$,

$x_{D}(w)>x_{d}>X_{2}(y_{d})\quad\;\text{and}\;\quad
y_{D}(w)<Y_{2}(x_{d})<y_{d}.$

These relations show that the point $(x_{D}(w),y_{D}(w))$ belongs to the set
$\\{(x,y)\in D:\leavevmode\nobreak\ y\leqslant y_{d}\\}$ and is not equal to
$(x_{d},Y_{2}(x_{d}))$, and consequently, using exactly the same arguments as
in the previous case, one gets (9.35).

Suppose now that (B3) and $y_{d}<y^{**}_{P}$ hold. By Proposition 2.2 and the
definition of the points $x_{d}$ and $y_{d}$ (see (2.33) and (2.34)), we have

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

and

${\mathcal{W}}_{1}=\\{w=(u,v)\in{\mathbb{S}}^{1}_{+}:\leavevmode\nobreak\
u>0\\}.$

Since we assume that $y_{d}<y^{**}_{P}$ and by (2.9), $y^{**}>y^{*}\geqslant
y^{*}_{P}$, by Lemma 4.1 and the definition of the mapping $(x,y)\to
w_{D}(x,y)=(u_{D}(x,y),v_{D}(x,y))$, from (9.42) it follows that

(9.43) $x_{d}=X_{1}(y_{d})<X_{2}(y_{d})\quad\text{and}\quad
u_{D}(x_{d},Y_{2}(x_{d}))=u_{D}(x_{d},y_{d})<0.$

Hence, in this case, for any $w=(u,v)\in{\mathcal{W}}_{1}$,

(9.44)
$u\ln(x_{d})+x\ln(Y_{2}(x_{d}))=u\ln(x_{d})+v\ln(y_{d})<u\ln(X_{2}(y_{d}))+v\ln(y_{d})\\\
\leqslant\max_{({x},{y})\in D,\,{y}\leqslant
y_{d}}\bigl{(}u\ln({x})+v\ln({y})\bigr{)}$

and consequently, (9.35) holds.

Suppose now that (B3) holds and $y_{d}=y^{**}_{P}$. Then
${\mathcal{W}}_{1}=\\{w=(u,v)\in{\mathbb{S}}^{1}_{+}:\leavevmode\nobreak\
u>0\\}$ and (9.42) holds, but now, instead of (9.43) one has

$x_{d}=X_{1}(y_{d})=X_{2}(y_{d}),\quad
y_{d}=y^{**}=y^{**}_{P}\quad\text{and}\quad\quad w_{D}(x_{d},y_{d})=(0,1).$

Hence, in this case, for any $w=(u,v)\in{\mathcal{W}}_{1}$,

(9.45) $x_{D}(w)>x_{d}\quad\;\text{and}\;\quad y_{D}(w)<y_{d}.$

These relations show that the point $(x_{D}(w),y_{D}(w))$ belongs to the set
$\\{(x,y)\in D:\leavevmode\nobreak\ y\leqslant y_{d}\\}$ and is not equal to
$(x_{d},Y_{2}(x_{d}))$ and consequently, using exactly the same arguments as
in the case (B2), one gets (9.35).

Consider finally the case when (B4) holds. In this case,
${\mathcal{W}}_{1}={\mathbb{S}}^{1}_{+}$ and by Proposition 2.2 and the
definition of the points $x_{d}$ and $y_{d}$ (see (2.33) and (2.34))

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

Hence, in this case, (9.43) holds and consequently, using exactly the same
arguments as above, one gets (9.35) for any $w=(u,v)\in{\mathcal{W}}_{1}$ with
$u>0$. ∎

Now we are ready to complete the proof of the first assertion of Theorem 4:
Since the right hand side of (2.75) does not depend on
$w\in{\mathcal{W}}_{1}$, it is sufficient to show that for any
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and $w=(u,v)\in{\mathcal{W}}_{1}$,
(2.75) holds as $\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w$.

Consider first the case when one of the following assertion holds

  * –

one of the conditions (B0), (B1) or (B3) holds

  * –

(B2) and $x^{**}<x^{**}_{P}$ hold,

  * –

(B4) holds and $u>0$.

By using Lemma 9.2 and Lemma 9.8, as $\|k\|\to+\infty$ and $k/\|k\|\to w$, one
gets

$\displaystyle\limsup_{k}\|k\|^{-1}\log\left|I_{0}(j,k)+I_{2}(j,k)\right|$
$\displaystyle\leqslant-\max_{(\hat{x},\hat{y})\in D,\,\hat{y}\leqslant
y_{d}}\Bigl{(}u\ln(\hat{x})+v\ln(\hat{y})\Bigr{)}$ (9.47)
$\displaystyle<-u\ln(x_{d})+v\ln(Y_{2}(x_{d})).$

By Corollary 9.2, for any $w=(u,v)\in{\mathcal{W}}_{1}$ and
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$, as $\min\\{k_{1},k_{2}\\}\to+\infty$
and $k/\|k\|\to w$,

(9.48) $I_{1}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{1}\,a_{1}\,\varkappa_{1}(j)x_{d}^{-k_{1}-1}(Y_{2}(x_{d}))^{-k_{2}-1}$

where $c_{1}>0$ is given by (9.20) and $a_{1}>0$ is given by (2.41).

Comparison of (9.48) with (9.47) shows that the terms $I_{0}(j,k)$ and
$I_{2}(j,k)$ in (9.2) are negligible with respect to $I_{1}(j,k)$, and
consequently, from the integral representation (9.2) and using (9.48) one gets
(2.75).

The set of directions ${\mathcal{W}}_{1}$ is empty in each of the following
cases:

  * –

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

  * –

(B5) or (B6) holds.

Hence, to complete the proof of the first assertion of Theorem 4 it is
sufficient now to prove (2.75) when (B4) holds and $w{=}(0,1)$. By Proposition
2.2 and the definition of the points $x_{d}$ and $y_{d}$ (see (2.33) and
(2.34)) one has

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

and the definition of the point $y^{**}_{P}$ and the mapping
$w\to(x_{D}(w),y_{D}(w))$ give, for $w=(0,1)$,

(9.50) $(x_{D}(0,1),y_{D}(0,1))=(X_{1}(y^{**}_{P}),y^{**}_{P}).$

Hence, in this case, $y_{D}(0,1)=y^{**}_{P}>y_{d}$, and consequently, also
$1>v_{D}(X_{2}(y_{d}),y_{d})$. By Lemma 9.6 applied with $w=(0,1)$, it follows
that there is $C>0$ such that for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$
and $k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, and as
$\min\\{k_{1},k_{2}\\}\to+\infty$, $k/\|k\|\to(0,1)$

(9.51) $I_{2}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{C}\,\varkappa_{1}(j)(X_{2}(y_{d}))^{-k_{1}-1}(y_{d})^{-k_{2}-1}.$

With Lemma 9.2 applied with $w=(0,1)$, the definition of the point
$y^{**}_{P}$, and using relation (9.49),

(9.52)
$\limsup_{k}\|k\|^{-1}\ln\bigl{|}I_{0}(j,k)\bigr{|}\leqslant-\max_{(x,y)\in
D}\ln(y)=-\ln(y^{**}_{P})<-\ln(y_{d}).$

Remark that by relation (9.49) and Corollary 9.2 applied with
$w=(0,1)\in{\mathcal{W}}_{1}$, for any
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$, as $\min\\{k_{1},k_{2}\\}\to+\infty$
and $k/\|k\|\to(0,1)$,

(9.53) $I_{1}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{1}\,a_{1}\,\varkappa_{1}(j)x_{d}^{-k_{1}-1}(Y_{2}(x_{d}))^{-k_{2}-1}={c}_{1}\,a_{1}\,\varkappa_{1}(j)x_{d}^{-k_{1}-1}y_{d}^{-k_{2}-1}$

where $c_{1}>0$ is given by (9.20) and $a_{1}>0$ is given by (2.41).

A comparison of relation (9.52) with relations (9.51) and (9.53) shows that
for any $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and
$k=(k_{1},k_{2})\in\mathbb{Z}^{2}_{+}$, and as
$\min\\{k_{1},k_{2}\\}\to+\infty$, $k/\|k\|\to(0,1)$, the term $I_{0}(j,k)$ is
negligible with respect to $I_{1}(j,k)+I_{2}(j,k)$ in (9.2), and consequently,
using (9.51) and (9.53) one gets

$g(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
I_{1}(j,k)+I_{2}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{1}\,a_{1}\,\varkappa_{1}(j)x_{d}^{-k_{1}-1}y_{d}^{-k_{2}-1}+{C}\,\varkappa_{1}(j)(X_{2}(y_{d}))^{-k_{1}-1}(y_{d})^{-k_{2}-1}.$

Since by (9.49), $x_{d}=X_{1}(y_{d})<X_{2}(y_{d})$, this proves (2.75) with
$b_{1}=c_{1}a_{1}$.

The first assertion of Theorem 4 is therefore proved. The proof of the second
assertion is the same by exchanging the roles of $x$ and $y$.

###  9.6. Proof of the assertion iii) of Theorem 4

Suppose first that (B0) holds. In this case, see the proof of Lemma 9.7,

(9.54) $u_{c}>u_{D}(x_{d},Y_{2}(x_{d}))\quad\text{and}\quad
v_{c}>v_{D}(X_{2}(y_{d}),y_{d}).$

Consider $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and let
$\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to w_{c}$. Then using the first
relation of (9.54), by Lemma 9.5 applied for $w=w_{c}=(u_{c},v_{c})$, we get

(9.55) $I_{1}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{1}\,a_{1}\varkappa_{1}(j)x_{d}^{-k_{1}-1}(Y_{2}(x_{d}))^{-k_{2}-1},$

where $c_{1}>0$ and $a_{1}>0$ given respectively by (9.20) and (2.41), and
using the second relation of (9.54), by Lemma 9.6 applied for
$w=w_{c}=(u_{c},v_{c})$, we obtain

(9.56) $I_{2}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
c_{2}\,a_{2}\varkappa_{2}(j)(X_{2}(y_{d}))^{-k_{1}-1}y_{d}^{-k_{2}-1}$

where $c_{2}>0$ and $a_{2}>0$ are defined respectively by (9.24) and (9.25).
Comparison of (9.55) and (9.56) with (9.7) shows that the term $I_{0}(j,k)$ in
(9.2) is negligible with respect to $I_{1}(j,k)+I_{1}(j,k)$, and consequently,
using (9.2) together with (9.55) and (9.56) one gets (2.79) with
$b_{1}=c_{1}a_{1}$ and $b_{2}=c_{2}a_{2}$. .

###  9.7. Proof of the assertions iv) and v) of Theorem 4

Suppose that (B3) holds, $y_{d}<y^{**}_{P}$ and let $w=(0,1)$. By Proposition
2.2 and the definition of the points $x_{d}$ and $y_{d}$, one has
$x^{*}_{P}<x_{d}=x^{**}<x^{**}_{P}$,
$(x_{d},y_{d})=(x_{d},Y_{2}(x_{d}))=(X_{1}(y_{d}),y_{d})\in{\mathcal{S}}_{12}$
and $y^{*}_{P}<y_{d}=y^{**}<y^{**}_{P}$. By Lemma 4.1 and the definition of
the mapping $(x,y)\to w_{D}(x,y)=(u_{D}(x,y),v_{D}(x,y))$, it follows that

(9.57) $x_{d}=X_{1}(y_{d})<X_{2}(y_{d}),$ (9.58)
$u_{D}(x_{d},Y_{2}(x_{d}))<0\quad\text{and}\quad v_{D}(X_{2}(y_{d}),y_{d})<1.$

Consider now $j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ and let
$k\in\mathbb{Z}^{2}_{+}$, $\min\\{k_{1},k_{2}\\}\to+\infty$ and $k/\|k\|\to
w=(1,0)$. Then by by Lemma 9.2 and since $y_{d}<y^{**}_{P}$, one gets

(9.59)
$\limsup_{k}\|k\|^{-1}\ln\bigl{|}I_{0}(j,k)\bigr{|}\leqslant-\max_{(x,y)\in
D}\ln(y)=-\ln(y^{**}_{P})<-\ln(y_{d}),$

by Lemma 9.5, from the first relation of (9.58) it follows

(9.60) $I_{1}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
{c}_{1}\,a_{1}\varkappa_{1}(j)x_{d}^{-k_{1}-1}(Y_{2}(x_{d}))^{-k_{2}-1}={c}_{1}\,a_{1}\varkappa_{1}(j)x_{d}^{-k_{1}-1}y_{d}^{-k_{2}-1},$

where $c_{1}>0$ and $a_{1}>0$ are given respectively by (9.20) and (2.41), and
by the second assertion of Lemma 9.6, from the second relation of (9.58) it
follows that there exists $\tilde{c}_{2}\in\mathbb{R}$ do not depending on
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$ such that

(9.61) $I_{2}(j,k)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
\frac{c_{2}\,a^{\prime}_{3}\varkappa_{1}(j)k_{2}}{(X_{2}(y_{d}))^{k_{1}+1}y_{d}^{k_{2}+2}}+\frac{\tilde{c}_{2}\,a^{\prime}_{3}\varkappa_{1}(j)k_{1}}{(X_{2}(y_{d}))^{k_{1}+2}{y_{d}}^{k_{2}+1}}$

where $c_{2}>0$ and $a_{3}^{\prime}>0$ are given respectively by (9.24) and
(9.27).

Comparison of (9.59) with (9.60) shows that the term $I_{0}(j,k)$ is
negligible with respect to $I_{1}(j,k)+I_{2}(j,k)$ in (9.2). Using therefore
(9.2) together with (9.60) and (9.61) we obtain

$\displaystyle g(j,k)$ $\displaystyle\leavevmode\nobreak\
\sim\leavevmode\nobreak\ I_{1}(j,k)+I_{2}(j,k)$ (9.62)
$\displaystyle\leavevmode\nobreak\ \sim\leavevmode\nobreak\
\frac{{c}_{1}\,a_{1}\varkappa_{1}(j)}{x_{d}^{k_{1}+1}y_{d}^{k_{2}+1}}+\frac{c_{2}\,a^{\prime}_{3}\varkappa_{1}(j)k_{2}}{(X_{2}(y_{d}))^{k_{1}+1}y_{d}^{k_{2}+2}}+\frac{\tilde{c}_{2}\,b_{3}\varkappa_{1}(j)k_{1}}{(X_{2}(y_{d}))^{k_{1}+2}{y_{d}}^{k_{2}+1}}$

Finally, by relation (9.57) we have $x_{d}{<}X_{2}(y_{d})$, hence, in relation
(9.62), the last term is negligible with respect to the first one and
consequently, relation (2.80) holds with $b_{1}=c_{1}a_{1}$ and
$b_{3}=c_{2}\,a^{\prime}_{3}$.

The assertion iv) of Theorem 4 is therefore also proved. The proof of the
assertion v) is exactly the same (it is sufficient to exchange the roles of
the first and the second coordinates of the random walk $(Z(n))$).

###  9.8. Proof of the assertion vi) of Theorem 4

To prove the last assertion of Theorem 4 it is sufficient to show that for any
$w\in{\mathcal{W}}_{0}$, (2.84) holds uniformly with respect to
$w_{m}=m/\|m\|$ in some neighborhood of $w$. For this we use Proposition 1 of
[11]. In our setting, this result gives the following lemma.

######  Lemma 9.9.

Suppose that the Assumption (A1) is satisfied and let
$w=(u,v){\in}{\mathbb{S}}^{1}_{+}$ and ${\varepsilon}{>}0$ and a function
$(x,y){\to}F(x,y)$ be analytic in the polycircular set

(9.63) $\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\
\bigl{|}|x|-x_{D}(w)\bigr{|}<{\varepsilon},\;\bigl{|}|y|-y_{D}(w)\bigr{|}<{\varepsilon}\\},$

and not vanishing at the point $(x_{D}(w),y_{D}(w))$. Then the integrals

(9.64) $I(m)=\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}}\int_{|y|=\hat{y}}\frac{F(x,y)}{x^{m_{1}+1}y^{m_{2}+1}(1-P(x,y))}\,dxdy,\quad
m=(m_{1},m_{2})\in\mathbb{Z}^{2}_{+},$

are well defined and does not depend on the point $(\hat{x},\hat{y})$ on the
set

(9.65) $\\{(\hat{x},\hat{y})\in\overset{\circ}{D}:\leavevmode\nobreak\
|\hat{x}-x_{D}(w)|<{\varepsilon},\quad|\hat{y}-y_{D}(w)|<{\varepsilon}\\},$

and as $\|m\|\to+\infty$, uniformly with respect to $w_{m}=m/\|m\|$ in some
neighborhood of $w$,

(9.66) $I(m)\leavevmode\nobreak\ \sim\leavevmode\nobreak\
\frac{F(x_{D}(w_{k}),y_{D}(w_{k}))\sqrt{w_{k}^{\perp}\cdot{\mathcal{Q}}(w_{k})w_{k}^{\perp}}}{\bigl{(}2\pi\|k\|)^{1/2}\|{\mathfrak{m}}(w_{k})\|^{-1}(x_{D}(w_{k}))^{k_{1}}(y_{D}(w_{k}))^{k_{2}}}$

By Corollary 9.1, for any $(\hat{x},\hat{y})\in\overset{\circ}{D}$, with
$\hat{x}<x_{d}$ and $\hat{y}<y_{d}$, we have

(9.67) $g(j,k)=\frac{1}{(2\pi
i)^{2}}\int_{|x|=\hat{x}}\int_{|y|=\hat{y}}\frac{F_{j}(x,y)}{x^{k_{1}+1}y^{k_{2}+1}(1-P(x,y))}\,dxdy.$

with

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

Remark moreover that the set of directions ${\mathcal{W}}_{0}$ is non-empty if
and only if (B2) holds, and that in this case, for any
$w=(u,v)\in{\mathcal{W}}_{0}$, one has

(9.68) $x_{D}(w)<x_{d},\quad\quad y_{D}(w)<y_{d}.$

By Theorem 1, it follows that for some neighborhood $V(x_{D}(w),y_{D}(w))$ of
the point $(x_{D}(w),y_{D}(w))$ in $\mathbb{R}^{2}$, the function $(x,y)\to
F_{j}(x,y)=L_{j}(x,y)+(\phi_{1}(x,y)-1)H_{j}(x,0)+(\phi_{2}(x,y)-1)H_{j}(0,y)$
is analytic in the polycircular set
$\\{(x,y)\in\mathbb{C}^{2}:\leavevmode\nobreak\ (|x|,|y|)\in
V(x_{D}(w),y_{D}(w))\\}$, and by Proposition 7.13,

$F_{j}(x_{D}(w),y_{D}(w))=\varkappa_{(x_{D}(w),y_{D}(w))}(j)>0,\quad\forall
j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}.$

For any $w{\in}{\mathcal{W}}_{0}$ and
$j{\in}\mathbb{Z}^{2}_{+}{{\setminus}}E_{0}$, the conditions of Lemma 9.9 are
therefore satisfied with $F{=}F_{j}$, and consequently relation (2.84) holds
uniformly with respect to $w_{k}{=}k/\|k\|$ in some neighborhood of $w$.

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##  Appendix A Relations with Positive Recurrence and Transience Conditions

This section discusses the conditions (B0)-(B7) defining our classification
with transience and positive recurrence conditions of of these random walks.
As we will see, the regions of our classification are in fact not defined in
terms of positive recurrence or transience. Transience and positive recurrence
are possible in several regions.

Throughout this section, $\mu_{0}$, $\mu_{1}$ or $\mu_{2}$ are assumed to be
stochastic, i.e. probability distributions. Before getting these results, note
that Assumption (A1)‘(ii) is equivalent to the usual Cramer’s condition for
the distribution of the jumps of $(S(n))$.

$(\alpha,\beta)\to\tilde{P}(\alpha,\beta)\stackrel{{\scriptstyle\text{\rm
def.}}}{{=}}P(e^{\alpha},e^{\beta})=\sum_{k=(k_{1},k_{2})}\mu(k)e^{\alpha
k_{1}+\beta k_{2}}.$

It is satisfied if and only if the function $\tilde{P}$ is finite in a
neighborhood of the set $\tilde{D}\stackrel{{\scriptstyle\text{\rm
def.}}}{{=}}\\{(\alpha,\beta){\in}\mathbb{R}^{2}:\leavevmode\nobreak\
P(e^{\alpha},e^{\beta}){\leqslant}1\\}$. Similarly, Assumption (A3) (ii) is
satisfied if and only if the generating functions

$\tilde{\phi}_{i}(\alpha,\beta)\stackrel{{\scriptstyle\text{\rm
def.}}}{{=}}\phi_{1}(e^{\alpha},e^{\beta})=\sum_{k=(k_{1},k_{2})}\mu_{i}(k)e^{\alpha
k_{1}+\beta k_{2}},\quad i{\in}\\{0,1,2\\},$

are finite in a neighborhood of $\tilde{D}$. Since $(0,0){\in}\tilde{D}$, all
jumps of the random walk $(Z(n))$ are in particular integrable.

Remark furthermore that because of Assumptions (A1) (ii) and (A3) (iv), the
mean jumps

$(M_{1},M_{2})=\sum_{j\in\mathbb{Z}^{2}}j\mu(j),\;(M^{1}_{1},M^{1}_{2})=\sum_{j\in\mathbb{Z}^{2}}j\mu_{1}(j)\quad
and\quad(M^{2}_{1},M^{2}_{2})=\sum_{j\in\mathbb{Z}^{2}}j\mu_{1}(j)$

are non zero, $M^{1}_{2}{>}0$ and $M^{2}_{1}{>}0$. Moreover, since with
Assumption (A1) (iii), the sets $D{\cap}D_{1}$ and $D{\cap}D_{2}$ have a non-
empty interior, one has also $M_{1}M^{1}_{2}{\not=}M_{2}M^{1}_{1}$ and
$M_{2}M^{2}_{1}{\not=}M_{1}M^{2}_{2}$.

Under our assumptions, the necessary and sufficient conditions of positive
recurrence and transience for the Markov chain $(Z(n))$, are given in the next
proposition. See Theorem 3.3.1 of [7] for example.

######  Proposition A.1.

Under the assumptions (A1)-(A3) then, the following assertions hold for the
Markov chain $(Z(n))$ on $\mathbb{Z}_{+}^{2}$.

  1. (1)

Positive recurrence. If and only if one of the following conditions is
satisfied:

    1. (R0)

$M_{1}{<}0$, $M_{2}{<}0$, $M_{1}M^{1}_{2}{<}M_{2}M^{1}_{1}$ and
$M_{2}M^{2}_{1}{<}M_{1}M^{2}_{2}$;

    2. (R1)

$M_{2}{<}0$, $M_{1}{\geqslant}0$ and $M_{1}M^{1}_{2}{<}M_{2}M^{1}_{1}$;

    3. (R2)

$M_{1}{<}0$, $M_{2}{\geqslant}0$ and $M_{2}M^{2}_{1}{<}M_{1}M^{2}_{2}$.

  2. (2)

Transience. If one of the following conditions is satisfied:

    1. (T0)

$M_{1}{>}0$ and $M_{2}{>}0$;

    2. (T1)

$M_{2}{<}0$ and $M_{1}M^{1}_{2}{>}M_{2}M^{1}_{1}$;

    3. (T2)

$M_{1}{<}0$ and $M_{2}M^{2}_{1}{>}M_{1}M^{2}_{2}$;

The proposition below studies the relation between conditions of transience
and positive recurrence for the Markov chain $(Z(n))$ and the location of the
points defining the regions (B0)-(B7). We formulate the conditions (R0)-(R2)
and (T0)-(T3) in terms of the location of the points $(x^{*},Y_{i}(x^{*}))$,
$(X_{i}(y^{*}),y^{*})$, and $(x^{**},Y_{i}(x^{**}))$,
$(X_{i}(y^{**}),y^{**})$, $i{\in}\\{1,2\\}$. In Proposition A.4, the relation
with the location of the dominant singularities $x_{d}$ and $y_{d}$ is
analyzed.

We first establish a technical lemma.

######  Lemma A.1.

Under the assumptions (A1)-(A3), then

  1. (i)

$M_{2}{<}0$ if and only if
$(1,1){\in}{\mathcal{S}}_{11}{\cup}{\mathcal{S}}_{21}$ and
$x^{*}_{P}{<}1{<}x^{**}_{P}$.
In this case, $Y_{1}(1){=}1$ and

(A.1) $1=\begin{cases}x^{*}&\text{ if }M_{1}M^{1}_{2}<M_{2}M^{1}_{1},\\\
x^{**}&\text{ if }M_{1}M^{1}_{2}>M_{2}M^{1}_{1}.\end{cases}$

  2. (ii)

$M_{1}{<}0$ if and only if
$(1,1){\in}{\mathcal{S}}_{11}{\cup}{\mathcal{S}}_{12}$ and
$y^{*}_{P}{<}1{<}y^{**}_{P}$.
In this case, $X_{1}(1){=}1$ and

(A.2) $1=\begin{cases}y^{*}&\text{ if }M_{2}M^{2}_{1}<M_{1}M^{2}_{2},\\\
y^{**}&\text{ if }M_{2}M^{2}_{1}>M_{1}M^{2}_{2}.\end{cases}$

###### Proof.

To prove (i), it is sufficient to remark that $M_{2}{=}\partial_{y}P(1,1)$ and
that, according to the definition of the curves $S_{ij}$, $i$
,$j{\in}\\{1,2\\}$,

${\mathcal{S}}_{11}\cup{\mathcal{S}}_{21}=\\{(x,y)\in\partial
D:\leavevmode\nobreak\ \partial_{y}P(x,y)\leqslant 0\\}$

and that by Lemma 4.1, the points
$(x^{*}_{P},Y_{1}(x^{*}_{P})){=}(x^{*}_{P},Y_{2}(x^{*}_{P}))$ and
$(x^{**}_{P},Y_{1}(x^{**}_{P})){=}(x^{**}_{P},Y_{1}(x^{**}_{P}))$ are the only
points on the boundary $\partial D$ of the set $D$ satisfying the relation
$\partial_{y}P(x,y){=}0$.

Moreover, by relations (2.15) and (2.17), we have
${\mathcal{S}}_{11}{\cup}{\mathcal{S}}_{21}{=}\\{(x,y)\in\partial
D:\leavevmode\nobreak\ y{=}Y_{1}(x)\\}$. Hence, when $M_{2}{<}0$, one has
$Y_{1}(1){=}1$ and therefore $\phi_{1}(1,Y_{1}(1)){=}\phi_{1}(1,1)=1$. By
Corollary 4.1, it follows that the point $1$ is an end point of the line
segment $[x^{*},x^{**}]$, and, consequently, with relations (2.9) and (2.10),
we conclude that one and only one of the following cases occurs:

  * –

$x^{*}_{P}<x^{*}=1<x^{**}\leqslant x^{**}_{P}$;

  * –

$x^{*}_{P}\leqslant x^{*}<1=x^{**}<x^{**}_{P}$.

By the implicit function theorem we have, for any
$x{\in}]x^{*}_{P},x^{**}_{P}[$,

$\frac{d}{dx}\phi_{1}(x,Y_{1}(x))=\left.\partial_{x}\phi_{1}(x,y)+\partial_{y}\phi_{1}(x,y)\partial_{x}P(x,y)/\partial_{y}P(x,y)\right|_{(y=Y_{1}(x))},$

hence, when $M_{2}{<}0$, the relation

$\displaystyle\left.\frac{d}{dx}\phi_{1}(x,Y_{1}(x))\right|_{x=1}$
$\displaystyle=\partial_{x}\phi_{1}(1,1)+\partial_{y}\phi_{1}(1,1)\partial_{x}P(1,1)/\partial_{y}P(1,1)\leavevmode\nobreak\
=\leavevmode\nobreak\ M^{1}_{1}-M^{1}_{2}M_{1}/M_{2},$

holds, and, consequently, in a neighborhood of $x{=}1$, the function
$x{\to}\phi_{1}(x,Y_{1}x))$ is

  * –

non-decreasing if $M^{1}_{1}{-}M^{1}_{2}M_{1}/M_{2}>0$,

  * –

non-increasing if $M^{1}_{1}{-}M^{1}_{2}M_{1}/M_{2}<0$.

By Corollary 4.1, we have the relation $\phi_{1}(x,Y_{1}(x)){<}1$, for all
$x{\in}]x^{*},x^{**}[$, and $\phi_{1}(x,Y_{1}(x)){>}1$, for
$x{\in}[x^{*}_{P},x^{*}[\,\cup\,]x^{**},x^{**}_{P}]$. Hence, when
$x^{*}_{P}{<}x^{*}{=}1{<}x^{**}{\leqslant}x^{**}_{P}$, the function
$x{\to}\phi_{1}(x,Y_{1}(x))$ is non-decreasing in a neighborhood of $x{=}1$
and $M^{1}_{1}M_{2}{-}M^{1}_{2}M_{1}{>}0$, and, when
$x^{*}_{P}{\leqslant}x^{*}{<}1{=}x^{**}{<}x^{**}_{P}$, the function
$x{\to}\phi_{1}(x,Y_{1}(x))$ is non-decreasing in a neighborhood of $x{=}1$
and $M^{1}_{1}M_{2}{-}M^{1}_{2}M_{1}{<}0$.

The first assertion of our lemma is proved. The second assertion is
symmetrical by exchanging the roles of $x$ and $y$. ∎

######  Proposition A.2.

Under Assumptions (A1)-(A3), then

  1. (i)

(R0) holds if and only if
$(x^{*},Y_{1}(x^{*})){=}(X_{1}(y^{*}),y^{*}){=}(1,1)$, $x^{*}_{P}{<}1$ and
$y^{*}_{P}{<}1$;

  2. (ii)

(R1) holds if"and only if $(x^{*},Y_{1}(x^{*})){=}(1,1)$ and $x^{*}_{P}{<}1$;

  3. (iii)

(R2) holds if"and only if $(X_{1}(y^{*}),y^{*}){=}(1,1)$ and $y^{*}_{P}{<}1$.

  4. (iv)

(T0) holds if"and only if $(1,1){\in}{\mathcal{S}}_{22}$, $1{<}x^{**}$ and
$1{<}y^{**}$.
In this case, $Y_{2}(x^{**}){<}1$ and $X_{2}(y^{**}){<}1$.

  5. (v)

(T1) holds if and only if $(1,1){=}(x^{**},Y_{1}(x^{**}))$ and
$1{<}x^{**}_{P}$.

  6. (vi)

(T2) holds if"and only if $(1,1){=}(X_{1}(y^{**}),y^{**})$ and
$1{<}y^{**}_{P}$.

###### Proof.

The assertions (i)-(iii) and (v)-(vi) follow directly from Lemma A.1.

We have only to establish (iv). By using again the relations
$M_{1}{=}\partial_{x}P(1,1)$ and $M_{2}{=}\partial_{y}P(1,1)$, when
$(1,1){\in}{\mathcal{S}}_{22}$, by relations (2.9), (2.10) and (2.18), and
with Lemma 4.1, we get the identities $(1,1){=}(1,Y_{2}(1)){=}(X_{2}(1),1)$
and the relations

(A.3) $x^{*}_{P}<X_{1}(y^{**}_{P})\leqslant 1\leqslant x^{**}\leqslant
x^{**}_{P}\quad\text{and}\quad y^{*}_{P}<Y_{1}(x^{**}_{P})\leqslant 1\leqslant
y^{**}\leqslant y^{**}_{P},$

and

(A.4) $\displaystyle M_{1}{>}0$
$\displaystyle\;\Leftrightarrow\;(1,1)\not=(X_{1}(y^{**}_{P}),y^{**}_{P})\;\Leftrightarrow\;1<y^{**}_{P},$
(A.5) $\displaystyle M_{2}{>}0$
$\displaystyle\;\Leftrightarrow\;(1,1){\not=}(x^{**}_{P},Y_{1}(x^{**}_{P}))\quad\Leftrightarrow\;1<x^{**}_{P}.$

Hence, when $(1,1){\in}{\mathcal{S}}_{22}$, we have $1{<}x^{**}$ and
$1{<}y^{**}$, the condition (T0) holds.

Conversely, if (T0) holds, then, according to the definition of the curve
${\mathcal{S}}_{22}$,

(A.6) $(1,1)=(1,Y_{2}(1))=(X_{2}(1),1)\in{\mathcal{S}}_{22}$

and, by using relations (A.4) and (A.5), we get

$1<x^{**}_{P},\quad\text{and}\quad 1<y^{**}_{P}.$

By relation (A.3), we have always $1{\leqslant}x^{**}$ and, if we assume that
$x^{**}{=}1$, then, by relations (A.6) and (A.3), we obtain
$x^{*}_{P}{<}1{=}x^{**}{<}x^{**}_{P}$. Consequently, by Lemma 4.1 and
Corollary 4.1, we have $Y_{1}(1){<}Y_{2}(1)$ and $\phi_{1}(1,Y_{1}(1)){=}1$.

Since under our assumptions (A3) (ii) and (A3) (v), the function
$y{\to}\phi_{1}(1,y)$ is strictly increasing on the line segment
$[0,Y_{1}(1)]$, these relations imply that $\phi_{2}(1,Y_{2}(1)){>}1$ and,
consequently, $(1,Y_{1}(1)){\not=}(1,1)$. Since this last relation contradicts
relation (A.6), we conclude that $x^{**}{>}1$.

By exchanging the roles of $x$ and $y$, the same arguments prove also that
when (T0) holds, the relation $y^{**}{>}1$ holds. By relation (2.18), we have

${\mathcal{S}}_{22}{=}\\{(x,Y_{2}(x)):\leavevmode\nobreak\
x{\in}[X_{1}(y^{**}_{P}),x^{**}_{P}]\\}{=}\\{(X_{2}(y),y):\leavevmode\nobreak\
y{\in}[Y_{1}(x^{**}_{P}),y^{**}_{P}]\\},$

and, by Lemma 4.1, the functions

$X_{2}:[Y_{1}(x^{**}_{P}),y^{**}_{P}]{\to}[X_{1}(y^{**}_{P}),x^{**}_{P}]\text{
and }Y_{2}:[X_{1}(y^{**}_{P}),x^{**}_{P}]{\to}[Y_{1}(x^{**}_{P}),y^{**}_{P}]$

are strictly decreasing. It follows that $Y_{2}(x^{**}){<}Y_{2}(1){=}1$ holds
and, similarly, $X_{2}(y^{**}){<}X_{2}(1){=}1$. The assertion (iv) of
Proposition A.2 is proved. ∎

The following statement give relations between the conditions (B0)-(B7) and
the conditions (R0)-(R2) and (T0)-(T2).

######  Proposition A.3.

Under the assumptions (A1)-(A3), then

  1. (1)

both conditions (T1) and (T2) hold if and only if (B7) holds;

  2. (2)

(B2) holds if (T0) holds.

  3. (3)

If either (B3) or (B4) holds, then

     * –

(T2) is not possible;

     * –

(T1) holds if and only if $x^{**}{=}1$, and in this case, $Y_{1}(1){=}1$;

     * –

either (R0) or (R1) or (R2) holds whenever $x^{**}{>}1$.

  4. (4)

If either (B5) or (B6) holds, then

     * –

(T1) is not possible;

     * –

(T2) holds if and only if $y^{**}{=}1$, and in this case, $X_{1}(1){=}1$;

     * –

either (R0) or (R1) or (R2) holds whenever $y^{**}{>}1$.

###### Proof.

The first and the second assertions of this statement follow from directly
Proposition 2.2 and Proposition A.2.

If either (B3) or (B4) holds, by Proposition 2.2,
$1{<}Y_{2}(x^{**}){\leqslant}y^{**}$ holds, and, consequently, by Proposition
A.2, the case (T2) is not possible.

Still under the condition that either (B3) or (B4) holds. By Proposition 2.2
and with relations (2.9) and (2.10), we have
$1{\leqslant}x^{**}{<}x^{**}_{P}$. Therefore, by Proposition A.2, (T1) holds
if and only if $(1,1){=}(x^{**},Y_{1}(x^{**}))$ holds. Since $P(1,1){=}1$, by
Lemma 4.1, either $Y_{1}(1){=}1$ or $Y_{2}(1){=}1$ holds, since, in our case,
we have $1{<}Y_{2}(x^{**})$ by Proposition 2.2, it follows that when either
(B3) or (B4) holds, the condition (T1) is satisfied if and only if
$x^{**}{=}1$.

Under the condition that either (B3) or (B4) holds and that $x^{**}{>}1$
holds. We have that either $M_{2}{\leqslant}0$ or $M_{1}{\leqslant}0$, because
otherwise (T0) and, consequently, (B2) holds. By Proposition 2.2 and with
relation (2.9), we get the relations
$1{<}Y_{2}(x^{**}){\leqslant}y^{**}{\leqslant}y^{**}_{P}$ and
$1{<}x^{**}{\leqslant}x^{**}_{P}$, and, consequently,
$(1,1){\not=}(x^{**}_{P},Y_{1}(x^{**}_{P}))$ and
$(1,1){\not=}(X_{1}(y^{**}_{P}),y^{**}_{P})$. Since the point
$(x,y)=(x^{**}_{P},Y_{1}(x^{**}_{P}))$, resp.
$(x,y){=}(X_{1}(y^{**}_{P}),y^{**}_{P})$ ) is the only point on the boundary
$\partial D$ of $D$ for which $\partial_{x}P(x,y)>0$ and
$\partial_{y}P(x,y)=0$, resp. $\partial_{x}P(x,y){=}0$ and
$\partial_{y}P(x,y){>}0$) and, since $M_{1}{=}\partial_{x}P(1,1)$ and
$M_{2}{=}\partial_{y}P(1,1)$, we conclude therefore that, either $M_{1}{<}0$
or $M_{2}{<}0$. If $M_{2}{<}0$, then, by Lemma A.1, and the assumption
$x^{**}{>}1$, we get $x^{*}{=}1$ and $M_{1}M^{1}_{2}{<}M_{2}M^{1}_{1}$.
Similarly, if $M_{1}{<}0$, by Lemma A.1 and since, by Proposition 2.2,
$1{<}Y_{2}(x^{**}){\leqslant}y^{**}$ holds, we obtain $y^{*}{=}1$ and
$M_{2}M^{2}_{1}{<}M_{1}M^{2}_{2}$. We conclude that in the case when either
(B3) or (B4) holds and $x^{**}{>}1$, one of the conditions (R0), (R1) or (R2)
is satisfied.

The third assertion of Proposition A.3 is therefore also proved. The last
assertion follows by exchanging the roles of $x$ and $y$. ∎

We can now establish the relations with the locations of the dominant
singularities $x_{d}$ and $y_{d}$.

######  Proposition A.4.

Under the assumptions (A1)-(A3) the following assertions hold:

  1. (i)

If the Markov chain $(Z(n))$ is positive recurrent then $x_{d}{>}1$ and
$y_{d}{>}1$.

  2. (ii)

If (T0) is satisfied then (B2), $x_{d}{=}x^{**}{>}1$ and $y_{d}{=}y^{**}{>}1$
hold.

  3. (iii)

If (T1) is satisfied then

     * –

in the cases (B0)-(B2), relations $x_{d}{=}x^{**}{=}1{<}x^{**}_{P}$ and
$y_{d}{=}y^{**}{>}Y_{1}(1){=}1$ hold;

     * –

in the cases (B3) and (B4), relations $x_{d}{=}x^{**}{=}1{<}x^{**}_{P}$ and
$y_{d}{=}Y_{2}(1){>}1$ holds;

     * –

(B5) and (B6) do not hold.

  4. (iv)

If (T2) is satisfied then

     * –

in the cases (B0)-(B2), relations $y_{d}{=}y^{**}{=}1{<}y^{**}_{P}$ and
$x_{d}{=}x^{**}{>}X_{1}(1){=}1$ hold;

     * –

and (B3) and (B4) do not hold;

     * –

in the cases (B5) and (B6), relations $y_{d}{=}y^{**}{=}1{<}y^{**}_{P}$ and
$x_{d}{=}X_{2}(1){>}1$ hold.

###### Proof.

Consider first the case when one of the cases (B0)-(B2) occurs. Then by
Proposition 2.2 and according to the definition of $x_{d}$ and $y_{d}$,

(A.7) $x_{d}=x^{**}>X_{1}(y^{**})\quad\text{and}\quad
y_{d}=y^{**}>Y_{1}(x^{**}).$

Hence, if we suppose moreover that (R0) holds, by Lemma A.1 and using (2.9),
we obtain

$1=x^{*}<x^{**}=x_{d}\quad\text{and}\quad 1=y^{*}<y^{**}=y_{d}.$

If we suppose that (R1) holds, then by Proposition A.2 and using (2.9), and
(A.7),

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

And similarly, if we suppose that (R2) holds, then by Proposition A.2 and
using (2.9), and (A.7),

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

In the case when one of the cases (B0)-(B2) holds, the first assertion of out
statement is therefore proved.

Suppose now that either (B3) or (B4) holds. In this case, by Proposition 2.2
and according to the definition of $x_{d}$ and $y_{d}$, we have

(A.8) $x_{d}=x^{**}<x^{**}_{P}\quad\text{and}\quad y_{d}=Y_{2}(x^{**})>1.$

Moreover, by relation (2.10), the inequality $1\leqslant x^{**}$ always holds,
and consequently, in this case, by Proposition A.3, one of the conditions
(R0)-(R2) is satisfied if and only if $1<x^{**}=x_{d}$. Hence, when either
(B3) or (B4) holds, the first assertion of Proposition A.4 is also proved. The
same arguments (it is sufficient to exchange the roles of $x$ and $y$) prove
the first assertion of our proposition in the case when either (B5) or (B6)
holds.

Furthermore, if the condition (T0) is satisfied, then by Proposition A.3, (B2)
holds and consequently, by Proposition A.2 and using (A.7) one gets
$x_{d}=x^{**}>1$ and $y_{d}=y^{**}>1$. The second assertion of Proposition A.4
is therefore also proved.

Suppose now that the condition (T1) is satisfied. Then by Proposition A.3, the
cases (B5) and (B6) are impossible, and by Proposition A.2,

(A.9) $x^{**}=1\quad\text{ and }\quad Y_{1}(x^{**})=1.$

Moreover, if one of the cases (B0)-(B2) holds, then by (A.7), one gets
$y_{d}=y^{**}>Y_{1}(x^{**})=1$, and if either (B3) or (B4) holds, then by
(A.8) and using the second relation of (A.9), we obtain
$y_{d}=Y_{2}(x^{**})>Y_{1}(x^{**})=1$. The third assertion of Proposition A.4
is therefore also proved. The proof of fourth assertion is the same, it is
sufficient to exchange the roles of $x$ and $y$. ∎

##  Appendix B Irreducibility Properties of the killed Markov Chain

### Proof of Lemma 2.1

Since by the assumption (A1), the homogeneous random walk $(S(n))$ with
transition probabilities ${\mathbb{P}}_{j}(S(1)=k)=\mu(k-j)$ is irreducible on
$\mathbb{Z}^{2}$, for any $k\in\mathbb{Z}^{2}$, there is a sequence of points
$\ell^{k}(1)=(\ell_{1}^{k}(1),\ell_{2}^{k}(1)),\ldots\ell^{k}(N_{k})=(\ell_{1}^{k}(N_{k}),\ell_{2}^{k}(N_{k}))\in\mathbb{Z}^{2}$
such that $\ell^{k}(1)+\cdots+\ell^{k}(N_{k})=k$ and
${\mathbb{P}}_{0}(S(N_{k})=k)\geqslant\mu(\ell_{1}^{k})\times\cdots\times\mu(\ell^{k}(N_{k}))>0$.

Let $N_{1}=1+\max\\{N_{(-1,0)},N_{(1,0)},N_{0,-1)},N_{(0,1)}\\}$. Then because
of Assumption (A2), for any $j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with
$j_{1},j_{2}\geqslant N_{1}$ and $k\in\\{(-1,0),(1,0),(0,-1),(0,1)\\}$ the
sequence of points
$m^{k}(1)=j+\ell^{k}(1),\ldots,m^{k}(N_{k})=j+\ell^{k}(1)+\cdots+\ell^{k}(N_{k})$
does not exit from the set $({\mathbb{N}}^{*})^{2}$, and consequently

${\mathbb{P}}_{j}\bigl{(}Z(n)=j+k,\;\text{for some
$n<\tau$}\bigr{)}\geqslant{\mathbb{P}}_{j}\bigl{(}Z(n)=m^{k}(n),\quad\forall
n\in\\{1,\ldots,N_{k}\\}\bigr{)}>0.$

This proves that

(B.1) $g(j,k)\geqslant{\mathbb{P}}_{j}(Z(n)=k,\;\text{for some
$n<\tau$})>0\quad\text{whenever \; $j_{1},j_{2},k_{1},k_{2}\geqslant N_{1}$}.$

Consider now a random walk $(\hat{S}(n))$ on $\mathbb{Z}\times{\mathbb{N}}$
with transition probabilities

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

Remark that under our hypotheses (see Assumptions (A1) and (A3) (v) and (vi))
such a random walk is irreducible on $\mathbb{Z}\times{\mathbb{N}}$, and
consequently, for any $j_{2}\in{\mathbb{N}}$ and
$k=(k_{1},k_{2})\in\mathbb{Z}^{2}$ such that
$(k_{1},k_{2}+j_{2})\in\mathbb{Z}\times{\mathbb{N}}$, there is a sequence of
points
$\ell^{j_{2},k}(1)=(\ell_{1}^{j_{2},k}(1),\ell_{2}^{j_{2},k}(1)),\ldots\ell^{j_{2},k}(N_{j_{2},k})=(\ell_{1}^{j_{2},k}(N_{j_{2},k}),\ell_{2}^{j_{2},k}(N_{j_{2},k}))\in\mathbb{Z}^{2}$
such that for any $n\in\\{1,\ldots,N_{j_{2},k}\\}$,

$(0,j_{2})+\ell^{j_{2},k}(1)+\cdots+\ell^{j_{2},k}(n)\in\mathbb{Z}\times{\mathbb{N}},$

and

${\mathbb{P}}_{(0,j_{2})}\bigl{(}\hat{S}(N_{j_{2},k})=(0,j_{2})+k\bigr{)}={\mathbb{P}}_{(0,j_{2})}\bigl{(}\hat{S}(n)-\hat{S}(n-1)=\ell^{j_{2},k}(n),\;\forall
n\in\\{1,\ldots,N_{j_{2},k}\\}\bigr{)}>0.$

Letting

$N_{2}=1+\max\\{N_{j_{2},k}:\leavevmode\nobreak\ \;j_{2}\leqslant
N_{1},\,(k_{1},k_{2})\in\\{(-1,0),(1,0),(0,-1),(0,1)\\},\;(k_{1},k_{2}+j_{2})\in\mathbb{Z}\times{\mathbb{N}}\\}$

and using Assumptions (A2) and (A3) (iv), we obtain that for any
$j=(j_{1},j_{2})\in\mathbb{Z}^{2}_{+}$ with $j_{1}\geqslant N_{2}$, and any
$k=(k_{1},k_{2})\in\\{(-1,0),(1,0),(0,-1),(0,1)\\}$ such that
$j+k\in\mathbb{Z}\times{\mathbb{N}}$, the sequence of points
$m^{j,k}(1)=j+\ell^{j_{2},k}(1),\ldots,m^{j,k}(N_{j_{2},k})=j+\ell^{j_{2},k}(1)+\cdots+\ell^{j_{2},k}(N_{j_{2},k})=(\ell_{1}^{j_{2},k}(N_{j_{2},k}),\ell_{2}^{j_{2},k}(N_{j_{2},k}))$
does not exist from the set ${\mathbb{N}}^{*}\times{\mathbb{N}}$, and
consequently,

${\mathbb{P}}_{j}\bigl{(}Z(n)=j+k\;\text{for some
$n<\tau$}\bigr{)}\geqslant{\mathbb{P}}_{j}\bigl{(}\hat{Z}(n)=m^{j_{2},k}(n)\;\forall
n\in\\{1,\ldots,N_{j_{2},k}\\}\bigr{)}>0.$

This proves that

(B.2) $g(j,k)>0\quad\text{whenever \, $j_{2}\leqslant N_{1},k_{2}\leqslant
N_{1}+1$ and $j_{1},k_{1}\geqslant N_{2}$},$

and with exactly the same arguments (it is sufficient to exchange the roles of
the first and the second coordinates of the points in $\mathbb{Z}^{2}_{+}$),
one gets that for some $N_{3}>0$,

(B.3) $g(j,k)>0\quad\text{whenever \, $j_{1}\leqslant N_{1},k_{1}\leqslant
N_{1}+1$ and $j_{2},k_{2}\geqslant N_{3}$}.$

When combined together, relations (B.1), (B.2) and (B.3) show that for some
$N_{0}\geqslant 0$,

(B.4) $g(j,k)>0\quad\text{for all \; $j,k\in\mathbb{Z}^{2}_{+}$ with
$\|j\|\geqslant N_{0}$ and $\|k\|\geqslant N_{0}$}.$

Remark now that if $\ell\in\mathbb{Z}^{2}_{+}$ is such that $g(\ell,j_{0})>0$
for some $j_{0}\in\mathbb{Z}^{2}_{+}$ with $\|j_{0}\|\geqslant N_{0}$, then
also ${\mathbb{P}}_{\ell}(Z(n)=j_{0},\;\tau>n)>0$ for some $n\in{\mathbb{N}}$,
and consequently, by (B.4) and using the inequality

(B.5)
$g(\ell,k)\geqslant\sum_{j\in\mathbb{Z}^{2}_{+}\setminus\\{0\\}}{\mathbb{P}}_{\ell}(Z(n)=j,\;\tau>n)g(j,k)\leavevmode\nobreak\
\geqslant\leavevmode\nobreak\
{\mathbb{P}}_{\ell}(Z(n)=j_{0},\;\tau>n)g(j_{0},k)$

we obtain

$g(\ell,k)\geqslant{\mathbb{P}}_{\ell}(Z(n)=j,\;\tau>n)g(j,k)>0\quad\text{for
all $k\in\mathbb{Z}^{2}_{+}$ with $\|k\|\geqslant N_{0}$}.$

Hence, for any $\ell\in\mathbb{Z}^{2}_{+}$ we have either $g(\ell,k)>0$ for
all $k\in\mathbb{Z}^{2}_{+}$ with $\|k\|\geqslant N_{0}$, or $g(\ell,k)=0$
also for all $k\in\mathbb{Z}^{2}_{+}$ with $\|k\|\geqslant N_{0}$, and in the
last case, because of (B.4), $\|\ell\|<N_{0}$. Letting therefore
$E_{0}=\\{\ell\in\mathbb{Z}^{2}_{+}:\leavevmode\nobreak\
g(\ell,k)=0\;\text{for all $k\in\mathbb{Z}^{2}_{+}$ with $\|k\|\geqslant
N_{0}$}\\}$ we obtain a finite subset of $\mathbb{Z}^{2}_{+}$ satisfying
(2.32). Remark moreover that this set satisfies also (2.31) because if suppose
that $g(\ell,j)>0$ for some $\ell\in E_{0}$ and
$j\in\mathbb{Z}^{2}_{+}{\setminus}E_{0}$, then using again the inequality
(B.5) and the same arguments as above we would get $g(\ell,k)>0$ for all
$k\in\mathbb{Z}^{2}_{+}$ with $\|k\|\geqslant N_{0}$. Now, to complete the
proof of our lemma, it is sufficient to notice that $g(0,k)>0$ for any
$k\in\mathbb{Z}^{2}_{+}{\setminus}\\{0\\}$, because under our assumptions, the
random walk $(Z(n))$ is irreducible on $\mathbb{Z}^{2}_{+}$, and consequently,
the point $0=(0,0)$ does not belong to the set $E_{0}$.

##  Appendix C Figures

$D\cap D_{1}$$D$$(x^{**},0)$$(x^{**}_{P},0)$$(x^{*},0)$$(x^{*}_{P},0)$

Figure 1.

$D\cap D_{1}$$D$$(x^{**},0)=(x^{**}_{P},0)$$(x^{*},0)$

Figure 2.

$D$$(x,Y_{1}(x))$$(x,Y_{2}(x))$$(x,0)$

Figure 3.

${\mathcal{S}}_{22}$${\mathcal{S}}_{21}$$D$${\mathcal{S}}_{11}$${\mathcal{S}}_{12}$

Figure 4.

(B0)$a$$b$$d$$c$(B1)$a$$b=d$$c$(B2)$a$$b$$c$$d$(B3)$a=d$$b$$c$(B4)$a$$b$$d$$c$(B5)$d$$a$$b=c$(B6)$c$$a$$b$$d$(B7)$d$$a=c$$b$

Figure 5. Possible cases for the location of the line segments
$[a,b]=[(X_{1}(y^{**}),y^{**}),(X_{2}(y^{**}),y^{**})]$ and
$[cd]=[(x^{**},Y_{1}(x^{**})),(x^{**},Y_{2}(x^{**}))]$

(B0)$a$$b$$d$$c$(B1)$a$$b=d$$c$(B2)$a$$b$$c$$d$(B3)$a=d$$b$$c$(B4)$a$$b$$d$$c$(B5)$d$$a$$b=c$(B6)$c$$a$$b$$d$

Figure 6. All possible configurations for the line segments
$[a,b]=[(X_{1}(y^{**}),y^{**}),(X_{2}(y^{**}),y^{**})]$ and
$[cd]=[(x^{**},Y_{1}(x^{**})),(x^{**},Y_{2}(x^{**}))]$ and the trace of the
set $\\{(x,y)\in{\mathcal{D}}:\;|x|<x_{d},\;|y|<y_{d}\\}$. Cases (B0)-(B6).

$D\cap D_{2}$$(1,1)$$D\cap
D_{1}$$D$$(X_{1}(y_{d}),y_{d})$$(X_{2}(y_{d}),y_{d})$$(0,0)$$(x_{d},0)$$(x_{d},Y_{1}(x_{d}))$

Figure 7. Case (B2) with $x_{d}=x^{**}=x^{**}_{P}$. In this case, the line
segment $[(x^{**},Y_{1}(x^{**})),(x^{**},Y_{2}(x^{**}))]$ is a single point
and $y_{d}=y^{**}\leqslant y^{**}_{P}$.

$(0,y^{**}_{P})$$D$$D\cap D_{2}$$D\cap
D_{1}$$(X_{1}(y_{d}),y_{d})$$(0,0)$$(x_{d},0)$$(x_{d},y_{d})$

Figure 8. Case (B5) with $x_{d}=x^{**}=x^{**}_{P}$. In this case, the line
segment $[(x^{**},Y_{1}(x^{**})),(x^{**},Y_{2}(x^{**}))]$ is a single point
and $y_{d}=y^{**}=Y_{1}(x_{d})<y^{**}_{P}$.

$D\cap D_{2}$$(1,1)$$D\cap D_{1}$$(0,0)$$(0,y_{d})$$(x_{d},0)$

Figure 9. Case (B2) with $x_{d}=x^{**}=x^{**}_{P}$ and
$y_{d}=y^{**}=y^{**}_{P}$. Here, each of the line segments
$[(x^{**},Y_{1}(x^{**})),(x^{**},Y_{2}(x^{**}))]$ and
$[(X_{1}(y^{**}),y^{**}),(X_{2}(y^{**}),y^{**})]$ is a single point.

$D$$\Theta$$(0,0)$$(0,Y_{1}(x^{**}))$$(X_{1}(y^{**}),0)$$(0,y^{**})$$(x^{**},0)$$(x^{**},Y_{1}(x^{**}))$

Figure 10. Set $\Theta$, cases (B0)-(B2).