# Walsh’s Brownian Motion
and Donsker Scaling Limits
of Perturbed Random Walks

Ilya Pavlyukevich111Institute of Mathematics, Friedrich Schiller University
Jena, Ernst–Abbe–Platz 2, 07743 Jena, Germany<EMAIL_ADDRESS>and Andrey Pilipenko222Institute of Mathematics, National Academy of Sciences
of Ukraine, Tereshchenkivska Str. 3, 01601, Kyiv, Ukraine 333National
Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”,
ave. Pobedy 37, Kyiv 03056, Ukraine<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the Donsker scaling limit of integer-valued random
walks perturbed on a finite subset of $\mathbb{Z}$ called a _membrane_. Under
very mild assumptions about the law of the random walk’s increments inside and
outside of the membrane we show weak convergence of the scaled processes to a
skew Brownian motion and provide the explicit formula for its permeability
parameter in terms of stationary distributions of certain embedded Markov
chains. The proof is based on a representation of the original random walk as
a multidimensional coordinate process and its convergence to a Walsh Brownian
motion.

Keywords: Walsh Brownian motion; skew Brownian motion; oscillating Brownian
motion; skew oscillating Brownian motion; Donsker scaling limit; local time;
perturbed Markov chain

MSC 2020: 60F17, 60G42, 60G50, 60H10, 60J10, 60J55, 60K37

###### Contents

  1. 1 Introduction
  2. 2 Walsh’s and Skew Brownian motions
  3. 3 Convergence to WBM and SMB
  4. 4 Examples
  5. 5 Proof of Theorem 3.3
  6. 6 Proof of Theorem 3.5
  7. 7 Proofs of Theorem 1.1, Corollary 1.3 and Theorem 1.4

##  1 Introduction

Let $S_{\xi}(n):=\xi_{1}+\dots+\xi_{n}$, $n\geq 0$, be a random walk where
$(\xi_{k})$ are independent copies of an integer-valued random variable $\xi$.
The well known Donsker invariance principle (see, e.g., Theorem 14.1 in
Billingsley [2]) states that a properly scaled random walk converges in law to
the standard Brownian motion if ${\mathop{\rm E}}\xi=0$ and
$\sigma^{2}:=\operatorname{Var}\xi\in(0,\infty)$. Namely,

$\displaystyle\frac{S_{\xi}([n\cdot])}{\sigma\sqrt{n}}\Rightarrow W(\cdot),\
n\to\infty,$ (1.1)

in the Skorokhod space $D=D([0,\infty),\mathbb{R})$.

Consider an integer-valued Markov chain $X$ whose transition probabilities
coincide with transition probabilities of $S_{\xi}$ everywhere except for a
finite set $A\subset{\mathbb{Z}}$. We call the set $A$ a _membrane_ and say
that $X$ is a random walk perturbed on $A$. It appears that the Donsker
scaling limit of $X$ is not a Brownian motion any longer but a diffusion with
a singular drift.

The first result on this topic belongs to Harrison and Shepp [8], who
considered a perturbation of a symmetric Bernoulli random walk perturbed on a
one-point membrane $A=\\{0\\}$. They proved that if

$\displaystyle{\mathop{\rm P}}(X(1)=x\pm 1\,|\,X(0)=x)=\frac{1}{2},\ x\neq 0,$
(1.2) $\displaystyle{\mathop{\rm P}}(X(1)=1\,|\,X(0)=0)=p\in[0,1],$
$\displaystyle{\mathop{\rm P}}(X(1)=-1\,|\,X(0)=0)=1-p,$

then the processes $\frac{X([n\cdot])}{\sqrt{n}}$ converge in distribution to
a skew Brownian motion $W^{\text{skew}}_{\gamma}$ with permeability parameter
$\gamma=2p-1$. Recall that the skew Brownian motion $W^{\text{skew}}_{\gamma}$
with permeability parameter $\gamma\in[-1,1]$ is a continuous homogeneous
Markov process with transition probability density function

$\displaystyle p_{t}(x,y)=\varphi_{t}(x-y)+\gamma\mathop{\rm
sgn}(y)\varphi_{t}(|x|+|y|),\quad x,y\in\mathbb{R},\quad t>0,$ (1.3)

where $\varphi_{t}(x)=(2\pi t)^{-1/2}\mathrm{e}^{-\frac{x^{2}}{2t}}$,
$x\in\mathbb{R}$, $t>0$.

The approach by Harrison and Shepp [8] is classical. To show convergence, they
verified tightness and convergence of finite-dimensional distributions, which
can be done directly with the help of the simple structure of transition
probabilities and Andre’s reflection principle.

The case of a general membrane $A$ and arbitrary non-unit jumps outside of $A$
is much harder to deal with. Some partial results on functional limit theorems
that generalize the result by Harrison and Shepp [8] were obtained by Minlos
and Zhizhina [18], Pilipenko et al. [21, 10, 24, 23], Ngo and Peigne [19]. In
these works, the convergence to a skew Brownian motions was established under
various restrictive assumptions, such as boundedness of jumps outside of the
membrane, one-point membrane, etc.

In this paper we obtain a very general convergence results (Theorems 3.3 and
3.5) that use a minimal set of technical assumptions and generalize all the
previous work.

For reader’s convenience, we formulate here an easy-to-use version of the
assumptions and the main result.

Let $(X(k))_{k\geq 0}$ be an integer-valued time-homogeneous Markov chain with
a membrane $\\{-d,\dots,d\\}$, $d\geq 0$.

$1^{\circ}$. We assume that

$\displaystyle{\mathop{\rm P}}(X(1)=x+y|X(0)=x)={\mathop{\rm
P}}(\xi_{+}=y),\quad x>d,$ (1.4) $\displaystyle{\mathop{\rm
P}}(X(1)=x+y|X(0)=x)={\mathop{\rm P}}(\xi_{-}=y),\quad x<-d,$

where the integer valued random variables $\xi_{-}$, $\xi_{+}$ satisfy

$\displaystyle{\mathop{\rm E}}\xi_{\pm}=0,\quad
v_{\pm}^{2}=\operatorname{Var}\xi_{\pm}\in(0,+\infty)$ (1.5)

and generate $1$-arithmetic random walks.

$2^{\circ}$. For any $|x|\leq d$,

$\displaystyle{\mathop{\rm P}}(\text{there is }k\geq 1\text{ such that
}|X(k)|>d\,|\,X(0)=x)=1,$ (1.6)

i.e., the Markov chain $X$ exits from the membrane with probability one.

$3^{\circ}$. The jumps from the membrane are integrable, i.e.,

$\displaystyle\max_{|x|\leq d}{\mathop{\rm
E}}\big{[}|X(1)|\,\big{|}\,X(0)=x\big{]}<\infty.$ (1.7)

$4^{\circ}$. The states ${\mathbb{Z}}\setminus\\{-d,\dots,d\\}$ of Markov
chain $X$ are connected.

Let $\varphi\colon{\mathbb{R}}\to{\mathbb{R}}$ be the following piecewise
linear function:

$\displaystyle\varphi(x):=x\cdot v^{-1}_{\mathop{\rm
sgn}(x)}:=x\big{(}v_{+}^{-1}{1\\!\\!\,{\rm I}_{x\geq
0}}+v_{-}^{-1}{1\\!\\!\,{\rm I}_{x<0}}\big{)}.$ (1.8)

######  Theorem 1.1.

Assume that Assumptions $1^{\circ}$–$4^{\circ}$ hold true. Then for any
initial value $x\in{\mathbb{Z}}$, the Donsker scaling
$\big{\\{}\varphi\big{(}\frac{X([n\cdot])}{\sqrt{n}}\big{)}\big{\\}}_{n\geq
1}$ weakly converges in $D([0,\infty),{\mathbb{R}})$ to a skew Brownian motion
$W^{\text{\emph{skew}}}_{\gamma}(\cdot)$ starting at 0 with some permeability
parameter $\gamma\in(-1,1)$.

The statement of this theorem as well as the explicit value of the
permeability parameter $\gamma$ will follow from Theorem 3.5, see Section 7.
Its formulation requires a consideration of additional construction: entrance
and exit embedded Markov chains, existence of their stationary distributions,
etc. However, formula (3.38) for the permeability parameter is short and
natural.

The case when the membrane $A$ is an arbitrary finite set
$\\{x_{1},\dots,x_{m}\\}\subset{\mathbb{Z}}$ not necessarily equal to
$\\{-d,\dots,d\\}$, is also covered by Theorem 1.1. Indeed, $X$ can be
considered as a random walk perturbed on bigger membrane $\\{-d,\dots,d\\}$,
where $d=\max_{1\leq i\leq m}|x_{i}|$.

Our methods are based on a martingale characterization of the skew and the
Walsh Brownian motions. This approach was initiated by Iksanov and Pilipenko
in [10] and continued in Bogdanskii et al. [3]. In particular, in our proofs
we will utilize a martingale characterization of the Walsh Brownian motion
obtained in [3], see Section 2. Another novelty is the different variance of
the random walk on different sides of the membrane that has not been treated
in previous works except in [19].

######  Remark 1.2.

The case when jumps from the membrane are not integrable and belong to a
domain of attraction of an $\alpha$-stable law with $\alpha\in(0,1)$ are much
harder to deal with even if jumps from the membrane are positive. Some partial
results in this direction were obtained by Pilipenko with co-authors in [25,
12, 22], where a reflected Brownian motion with jump-type exit from 0 appeared
as a limit process. The case when jumps outside of the membrane are not in
Donsker’s setup but belong to a domain of attraction of a stable law with
parameter in $(1,2)$ leads to a skew stable Lévy process as was shown in [11,
4].

The result of Theorem 1.1 allows us to determine the limit of the Donsker
scaling $\big{(}\frac{X([nt])}{\sqrt{n}}\big{)}_{t\geq 0}$, see the proof in
Section 7.

######  Corollary 1.3.

The weak limit of the sequence
$\Big{\\{}\frac{X([n\cdot])}{\sqrt{n}}\Big{\\}}_{n\geq 1}$ is a process
$Y:=\Big{(}\varphi^{-1}(W^{\text{\emph{skew}}}_{\gamma}(t))\Big{)}_{t\geq 0}$,
which is a skew oscillating Brownian motion that satisfies the SDE

$Y(t)=\int_{0}^{t}(v_{-}1\\!\\!\,{\rm I}_{Y(s)<0}+v_{+}1\\!\\!\,{\rm
I}_{Y(s)\geq
0})\,\mathrm{d}W(s)+\frac{\gamma(v_{+}+v_{-})+v_{+}-v_{-}}{\gamma(v_{+}-v_{-})+v_{+}+v_{-}}\,L_{0}^{Y}(t),\quad
t\geq 0,$ (1.9)

where

$\displaystyle L_{0}^{Y}(t)=\lim_{\varepsilon\to
0+}\frac{1}{2\varepsilon}\int_{0}^{t}1\\!\\!\,{\rm
I}_{|Y(s)|<\varepsilon}\,\mathrm{d}\langle Y\rangle_{s}=\lim_{\varepsilon\to
0+}\frac{1}{2\varepsilon}\int_{0}^{t}\Big{(}v_{-}^{2}1\\!\\!\,{\rm
I}_{-\varepsilon<Y(s)<0}+v_{+}^{2}1\\!\\!\,{\rm I}_{0\leq
Y(s)<\varepsilon}\Big{)}\,\mathrm{d}s$ (1.10)

is the symmetric semimartingale local time of $Y$ at 0.

The skew oscillating Brownian motion is a natural generalization of the
oscillating Brownian motion originally introduced by Keilson and Wellner in
[16]. The oscillating Brownian motion is a diffusion that behaves like a
Brownian motion with variance $v_{+}^{2}$ on the positive half-line and like a
Brownian motion with variance $v_{-}^{2}$ on the negative half-line, and
exhibits an infinitesimal push at $0$ in the negative or positive direction.
We mention here the work [7] by Hairer and Manson who obtained the process $Y$
as a limit in the periodic homogenization problem with an interface.

A result similar to Theorem 1.1 is also true for a perturbed random walk on a
graph, where the limit process will be a Walsh Brownian motion. Let us
describe the construction. For some $m\in{\mathbb{N}}$, consider a Markov
chain $\mathcal{X}$ on the state space

$\displaystyle\Big{(}{\mathbb{N}}\times\\{1,\dots,m\\}\Big{)}\cup\\{0\\}=:\mathcal{N}^{m}\cup\\{0\\}.$
(1.11)

The set $\mathcal{N}^{m}\cup\\{0\\}$ consists of $m$ natural-valued rays with
a common end-point 0. If $\mathcal{X}(k)\in{\mathbb{N}}\times\\{i\\}$, we say
that $\mathcal{X}$ belongs to $i$-th ray. We denote the radius of
$\mathcal{X}$ by $R(k)$, i.e.,

$\displaystyle R(k)=\begin{cases}x,\text{ if
}\mathcal{X}(k)=(x,i)\in\mathcal{N}_{m},\\\ 0,\text{ if
}\mathcal{X}(k)=0.\end{cases}$ (1.12)

Let us formulate a list of assumptions concerning the dynamics of
$\mathcal{X}$.

1\. Let $(\xi_{k}^{(1)})_{k\geq 1},\dots,(\xi_{k}^{(m)})_{k\geq 1}$ be $m$
independent sequences of iid integer valued random variables with

$\displaystyle{\mathop{\rm E}}\xi^{(i)}_{k}=0\quad\text{and}\quad
v^{2}_{i}:=\operatorname{Var}\xi^{(i)}_{k}\in(0,\infty),\quad 1\leq i\leq m.$
(1.13)

We assume that for each $k\geq 0$, $\mathcal{X}(k)$ is independent of the
‘future’ $(\xi_{k+j}^{(i)})_{j\geq 1},\dots,(\xi_{k+j}^{(m)})_{j\geq 1}$.

On each ray ${\mathbb{N}}\times\\{i\\}$, $\mathcal{X}$ behaves as a random
walk with increments $\xi^{(i)}$, that is if
$\mathcal{X}(k)=(x,i)\in\mathcal{N}^{m}$ and $x+\xi_{k+1}^{(i)}>0$, then

$\displaystyle\mathcal{X}(k+1):=(x+\xi_{k+1}^{(i)},i).$ (1.14)

If $\mathcal{X}(k)=(x,i)\in\mathcal{N}^{m}$ and $x+\xi_{k+1}^{(i)}=0$, then we
set

$\displaystyle\mathcal{X}(k+1):=0.$ (1.15)

Finally, we assume that if $\mathcal{X}(k)=(x,i)\in\mathcal{N}^{m}$ and
$x+\xi_{k+1}^{(i)}<0$, then the conditional distribution

$\displaystyle\text{Law}\big{(}\mathcal{X}(k+1)\,\big{|}\,\mathcal{X}(k)=(x,i),x+\xi_{k+1}^{(i)}<0\big{)}$
(1.16)

depends only on $i$ and $x+\xi_{k+1}^{(i)}$.

2\. The conditional law $\text{Law}(\mathcal{X}(1)|\mathcal{X}(0)=0)$
satisfies

$\displaystyle{\mathop{\rm
P}}\big{(}R(1)\in{\mathbb{N}}\,\big{|}\,\mathcal{X}(0)=0\big{)}=1,$ (1.17)
$\displaystyle{\mathop{\rm
E}}\big{[}R(1)\,\big{|}\,\mathcal{X}(0)=0\big{]}<\infty.$

3\. There is a constant $C>0$ such that for all $(x,i)\in\mathcal{N}^{m}$

$\displaystyle{\mathop{\rm
E}}\big{[}R(k+1)\,\big{|}\,\mathcal{X}(k)=(x,i),\,x+\xi_{k+1}^{(i)}<0\big{]}\leq
C(1+|x+\xi_{k+1}^{(i)}|).$ (1.18)

4\. All the states $\mathcal{N}^{m}$ of $\mathcal{X}$ are connected.

######  Theorem 1.4.

Assume that Assumptions 1-4 hold true. Then the Donsker scaling of $R$
converges in distribution to a Walsh Brownian motion $W_{\mathbf{p}}(\cdot)$
starting at 0 with some weights $\mathbf{p}=(p_{1},\dots,p_{m})$, i.e.,

$\displaystyle\Big{(}\frac{R([n\cdot])}{v_{1}\sqrt{n}}1\\!\\!\,{\rm
I}_{l(n)=1},\dots,\frac{R([n\cdot])}{v_{m}\sqrt{n}}1\\!\\!\,{\rm
I}_{l(n)=m}\Big{)}\Rightarrow W_{\mathbf{p}}(\cdot),\quad n\to\infty.$ (1.19)

######  Remark 1.5.

The weights $\mathbf{p}=(p_{1},\dots,p_{m})$ of the Walsh Brownian motion will
be specified in Theorem 3.3, where a more general model will be considered.

Acknowledgments: A.P. acknowledges support by the National Research Foundation
of Ukraine (project 2020.02/0014 “Asymptotic regimes of perturbed random
walks: on the edge of modern and classical probability”) and thanks the
Institute of Mathematics of the FSU Jena for hospitality.

The structure of the paper is as follows. In Section 2 we recall the
definition of the Walsh Brownian motion together with its martingale
characterization and its relation to the skew Brownian motion. We formulate
main general results on limit behavior of perturbed random walks in Section 3.
In Theorem 3.3 we consider perturbed random walks on
${\mathbb{Z}}\times\\{1,\dots,m\\}$, whose Donsker scaling limit is a Walsh
Brownian motion. A result for scaling limits of a perturbed random walk with
values in ${\mathbb{Z}}$ obtained in Theorem 3.5, is a corollary from Theorem
3.3. We determine the weights of the Walsh Brownian motion and the
permeability parameter of the skew Brownian motion explicitly in terms of
stationary distributions of some embedded entrance and exit Markov chains. In
Section 4 contains five illustrative examples. In particular, we show that
formulas obtained in Theorems 3.3 and 3.5 are consistent with already known
results. The proof of Theorem 3.3 is presented in Section 5. Theorem 3.5 is
proven in Sections 6. The last Section 7 contains the proofs of Theorem 1.1,
Corollary 1.3 and Theorem 1.4.

##  2 Walsh’s and Skew Brownian motions

The main result of this paper will be expressed in terms of Walsh’s Brownian
motion $W_{\mathbf{p}}$ (WBM). In this section we recall its definition,
properties and martingale characterizations.

Originally, Walsh’s Brownian motion was introduced in the work [31] as a
diffusion on $m$ rays on a two-dimensional plane with a common origin. Away
from the origin, on each ray, WBM is a standard one-dimensional Brownian
motion that can change the ray upon hitting the origin. In other words, the
rays are characterized by $m$ non-negative weights
$\mathbf{p}=(p_{1},\dots,p_{m})$, $p_{1}+\cdots+p_{m}=1$. The number $p_{i}$
is the probability to leave the origin into the ray number $i$.

In our work [3], we proposed a realization of the WBM as an $m$-dimensional
process (as was also indicated in Walsh [31]) living on positive semi-axes. In
comparison to the 2-dimensional construction with the help of polar
coordinates, in this realization one does not have to treat the origin as a
particular point at which the ray number is not well defined.

Let $m\geq 1$,

$E^{m}=\\{x\in\mathbb{R}^{m}\colon x_{i}\geq 0\text{ and }x_{i}x_{j}=0,\ i\neq
j,\ i,j=1,\dots,m\\},$ (2.1)

and $\mathbf{p}=(p_{1},\dots,p_{m})$ be a vector of weights. Let us work on
the canonical probability space of continuous $\mathbb{R}^{m}$-valued
functions equipped with the filtration $\mathbb{F}$ generated by the
coordinate mappings. We say that WBM is a time-homogeneous continuous Feller
process $W_{\mathbf{p}}=X=(X_{1},\dots,X_{m})$ on $E^{m}$ whose marginal
distributions are given by

$\displaystyle{\mathop{\rm
E}}_{0}\text{e}^{\lambda_{1}X_{1}(t)+\cdots+\lambda_{m}X_{m}(t)}$
$\displaystyle=\sum_{k=1}^{m}p_{k}{\mathop{\rm
E}}_{0}\text{e}^{\lambda_{k}|W(t)|},$ (2.2) $\displaystyle{\mathop{\rm
E}}_{x}\text{e}^{\lambda_{1}X_{1}(t)+\cdots+\lambda_{m}X_{m}(t)}$
$\displaystyle={\mathop{\rm E}}_{x_{j}}\Big{[}1\\!\\!\,{\rm
I}_{t<\tau_{0}}\text{e}^{\lambda_{j}W(t)}\Big{]}+\sum_{k=1}^{m}p_{k}{\mathop{\rm
E}}_{x_{j}}\Big{[}1\\!\\!\,{\rm
I}_{t\geq\tau_{0}}\text{e}^{\lambda_{k}|W(t)|}\Big{]}$
$\displaystyle={\mathop{\rm
E}}_{x_{j}}\Big{[}\sum_{k=1}^{m}p_{k}\text{e}^{\lambda_{k}|W(t)|}\Big{]}+{\mathop{\rm
E}}_{x_{j}}\Big{[}1\\!\\!\,{\rm
I}_{t<\tau_{0}}\Big{(}\text{e}^{\lambda_{j}W(t)}-\sum_{k=1}^{m}p_{k}\text{e}^{\lambda_{k}W(t)}\Big{)}\Big{]},$
$\displaystyle x$ $\displaystyle=(0,\dots,x_{j},\dots,0),\ x_{j}>0,\
\lambda\in\mathbb{C}^{m},$

where $W$ is a one-dimensional standard Brownian motion and
$\tau_{0}=\inf\\{t\geq 0\colon W_{t}=0\\}$. Note that the last expectation is
the expectation of a standard Brownian motion killed upon hitting zero.

In Barlow et al. [1], the authors also provided the martingale
characterization of the WBM realized as a process on the plane. In terms of
the $m$-dimensional realization $X$ of the WBM, their characterization takes
the following form.

######  Theorem 2.1 (Propositon 3.1 and Theorem 3.2 in Barlow et al. [1]).

Let $X=(X_{1}(t),\dots,X_{m}(t))_{t\geq 0}$ be a continuous process. Then $X$
is a WBM with parameters $p_{1},\dots,p_{m}>0$ if and only if it satisfies the
following conditions:

  1. 1.

$X_{i}(t)\geq 0$ and $X_{i}(t)X_{j}(t)=0$ for all $i\neq j$ and $t\geq 0$;

  2. 2.

for each $i=1,\dots,m$ the process

$N_{i}(t):=(1-p_{i})X_{i}(t)-p_{i}\sum_{j\neq i}X_{j}(t),\quad t\geq 0,$ (2.3)

is a continuous martingale;

  3. 3.

for each $i=1,\dots,m$ the process

$(N_{i}(t))^{2}-\int_{0}^{t}\Big{(}(1-p_{i})1\\!\\!\,{\rm
I}_{X_{i}(s)>0}-p_{i}1\\!\\!\,{\rm
I}_{X_{i}(s)=0}\Big{)}^{2}\,\mathrm{d}s,\quad t\geq 0,$ (2.4)

is a continuous martingale.

Note that since the product of the indicator functions in (2.4) is identically
zero, we have

$\displaystyle\langle
N_{i}\rangle_{t}=\int_{0}^{t}\Big{(}(1-p_{i})1\\!\\!\,{\rm
I}_{X_{i}(s)>0}-p_{i}1\\!\\!\,{\rm
I}_{X_{i}(s)=0}\Big{)}^{2}\,\mathrm{d}s=\int_{0}^{t}\Big{(}(1-p_{i})^{2}1\\!\\!\,{\rm
I}_{X_{i}(s)>0}+p_{i}^{2}1\\!\\!\,{\rm I}_{X_{i}(s)=0}\Big{)}\,\mathrm{d}s.$
(2.5)

Furthermore, it is clear, see Lemma 2.2 in Barlow et al. [1], that the radial
process

$R(t):=\sum_{i=1}^{m}X_{i}(t)=\max_{1\leq i\leq m}X_{i}(t)$ (2.6)

is a reflecting Brownian motion, and hence it has a symmetric local time at
$0$ defined by:

$L^{X}_{0}(t):=L^{R}_{0}(t):=\lim_{\varepsilon\to
0+}\frac{1}{2\varepsilon}\int_{0}^{t}1\\!\\!\,{\rm I}_{\max_{1\leq i\leq
m}X_{i}(s)\leq\varepsilon}\,\mathrm{d}s.$ (2.7)

In our paper [3] we gave an equivalent martingale characterization of the WBM
that is more convenient for the proof of the convergence theorem of this
paper.

######  Theorem 2.2 (Theorem 3.2 in [3]).

Let $X=(X_{1}(t),\dots,X_{m}(t))_{t\geq 0}$ and $\nu=(\nu(t))_{t\geq 0}$ be a
continuous processes. Then $X$ is a WBM with parameters $p_{1},\dots,p_{m}>0$,
and $\nu$ is the local time of $X$ at $0$ if and only if they satisfy the
following conditions:

  1. 1.

$X_{i}(t)\geq 0$ and $X_{i}(t)X_{j}(t)=0$ for all $i\neq j$, $t\geq 0$;

  2. 2.

$\nu(0)=0$, $\nu$ is non-decreasing a.s., $\int_{0}^{\infty}1\\!\\!\,{\rm
I}_{X(s)\neq 0}\,\mathrm{d}\nu(s)=0$ a.s.;

  3. 3.

the processes $M_{1},\dots,M_{m}$ defined by

$M_{i}(t):=X_{i}(t)-p_{i}\nu(t),\quad t\geq 0,$ (2.8)

are continuous square integrable martingales with the predictable quadratic
variations

$\langle M_{i}\rangle_{t}=\int_{0}^{t}1\\!\\!\,{\rm
I}_{X_{i}(s)>0}\,\mathrm{d}s.$ (2.9)

  4. 4.

$\int_{0}^{\infty}1\\!\\!\,{\rm I}_{X(s)=0}\,\mathrm{d}s=0$ a.s.

In his paper [31], Walsh addressed the properties of the so-called skew
Brownian motion (SBM) originally constructed by Itô and McKean in [13],
Problem 1, Section 4.2. A SBM with parameter $\gamma\in[-1,1]$ is a diffusion
that behaves like a standard Brownian motion on the half-lines $(-\infty,0)$
and $(0,\infty)$ and upon hitting zero continues to the positive half-line
with probability $p_{+}=(\gamma+1)/2$ and to the negative half line with
probability $p_{-}=(\gamma+1)/2$. In [3], it was shown that a SBM can be
obtained from a WBM by summing together the motions on different rays.

######  Corollary 2.3 ([3], Corollary 3.4).

Let $X=(X_{1}(t),\dots,X_{m}(t))_{t\geq 0}$ be a WBM with parameters
$p_{1},\dots,p_{m}>0$ and let $I\subseteq\\{1,\dots,m\\}$. Set

$\gamma:=\sum_{i\in I}p_{i}-\sum_{j\in I^{c}}p_{j}=2\sum_{i\in I}p_{i}-1.$
(2.10)

Then $\gamma\in[-1,1]$ and the process

$W_{\gamma}^{\emph{\text{skew}}}(t):=\sum_{i\in I}X_{i}(t)-\sum_{j\in
I^{c}}X_{j}(t),\quad t\geq 0,$ (2.11)

is a skew Brownian motion with the parameter $\gamma$.

Recall that due to [8], the skew Brownian motion is the unique strong solution
of the stochastic differential equation

$\mathrm{d}W_{\gamma}^{\text{skew}}(t)=\mathrm{d}W(t)+\gamma\,\mathrm{d}L_{0}^{W^{\text{skew}}_{\gamma}}(t),$
(2.12)

where $L_{0}^{W^{\text{skew}}_{\gamma}}(\cdot)$ is the symmetric
semimartingale local time at 0. Since $\langle
W_{\gamma}^{\text{skew}}\rangle_{t}=t$, the symmetric semimartingale local
time coincides with the symmetric local time.

In the sequel, we will use notation $W_{\mathbf{p}}(\cdot,x)$ and
$W_{\gamma}^{{\text{skew}}}(\cdot,x)$ to denote WBM and SBM starting at a
point $x$.

##  3 Convergence to WBM and SMB

To prove Theorems 1.1 and 1.4 we need to obtain a limit theorem for an
auxiliary Markov chain $\mathcal{X}(k)=(R(k),l(k))$, $k\geq 0$ on the state
space $\mathcal{Z}^{m}:={\mathbb{Z}}\times\\{1,\dots,m\\}$, $m\geq 1$. The
Markov chain $\mathcal{X}$ is defined by the following set of assumptions.

A1. For all $(x,i)\in\mathcal{N}^{m}$

$\displaystyle{\mathop{\rm
P}}\Big{(}\mathcal{X}(1)=(x+y,i)\Big{|}\mathcal{X}(0)=(x,i)\Big{)}={\mathop{\rm
P}}(\xi^{(i)}=y),$ (3.1)

where integer-valued random variables $\xi^{(1)},\dots,\xi^{(m)}$ satisfy

$\displaystyle{\mathop{\rm E}}\xi^{(i)}=0,\quad v_{i}^{2}:={\mathop{\rm
Var}}\,\xi^{(i)}\in(0,\infty),\quad 1\leq i\leq m,$ (3.2)

and each $\xi^{(i)}$, $i=1,\dots,m$, generates a 1-arithmetic random walk,
i.e., the minimal lattice containing $\mathbb{Z}$.

A2. For all $x\leq 0$ and $1\leq i\leq m$

$\displaystyle{\mathop{\rm P}}(R(1)\in{\mathbb{N}}\ |\
\mathcal{X}(0)=(x,i))=1.$ (3.3)

A3. There is $C>0$ such that for all $x\leq 0$ and $1\leq i\leq m$

$\displaystyle{\mathop{\rm
E}}\Big{[}R(1)\Big{|}\mathcal{X}(0)=(x,i)\Big{]}\leq C(1+|x|).$ (3.4)

A4. The states $\mathcal{N}^{m}$ of the Markov chain $\mathcal{X}$ are
connected.

We will call $R(\cdot)$ the radius and $l(\cdot)$ the label of $\mathcal{X}$.
Note that the radius $R$ may attain non-positive values. The qualitative
behaviour of $\mathcal{X}$ is as follows.

Assumption A1 say that for the “positive” initial point
$\mathcal{X}(0)=(R(0),l(0))=(x,i)$, $x>0$, the radius $R$ behaves like a zero
mean square integrable random walk with the increments having distribution
$\xi^{(i)}$, until the first exit time

$\displaystyle\sigma$ $\displaystyle=\inf\\{k\geq 0\colon R(k)\leq 0\\}.$
(3.5)

Note that the label $l$ does not change up to the stopping time $\sigma$,

$\displaystyle l(0)=l(1)=\cdots=l(\sigma)=i.$ (3.6)

Upon reaching a non-positive value at time $\sigma$, the Markov chain
$\mathcal{X}$ for sure jumps to a “positive” value at one step. The label $l$
can change, too:

$\displaystyle R(\sigma+1)>0,\quad l(\sigma+1)\in\\{1,\dots,m\\}.$ (3.7)

The technical Assumption A3 controls the size of the jump of the radius at
time $\sigma+1$.

Let $\\{\sigma_{\iota}\\}_{\iota\geq 1}$ denote the sequence of exit times of
$R$ from ${\mathbb{N}}$, that is

$\displaystyle\sigma=\sigma_{1}$ $\displaystyle=\inf\\{k\geq 0\colon R(k)\leq
0\\},$ (3.8) $\displaystyle\sigma_{\iota+1}$
$\displaystyle=\inf\\{k>\sigma_{\iota}\colon R(k)\leq 0\\}.$

Let also

$\tau_{\iota}:=\sigma_{\iota}+1,\quad\iota\geq 1,$ (3.9)

be the entrance times to ${\mathbb{N}}$.

With the sequences $\\{\sigma_{\iota}\\}_{\iota\geq 1}$ and
$\\{\tau_{\iota}\\}_{\iota\geq 1}$ we associate embedded _exit_ and _entrance_
Markov chains taking values in $\\{\dots,-2,-1,0\\}\times\\{1,\dots,m\\}$ and
$\mathcal{N}^{m}$ respectively and defined as

$\displaystyle\mathcal{X}_{\text{exit}}(\iota)$
$\displaystyle:=\mathcal{X}(\sigma_{\iota}),$ (3.10)
$\displaystyle\mathcal{X}_{\text{entrance}}(\iota)$
$\displaystyle:=\mathcal{X}(\tau_{\iota}),\quad\iota\geq 1,$

Assumption A4 yields the following statement.

######  Lemma 3.1.

The Markov chains $\mathcal{X}_{\emph{exit}}$ and
$\mathcal{X}_{\emph{entrance}}$ have unique stationary distributions
$\pi_{\emph{exit}}$ and $\pi_{\emph{entrance}}$ respectively.

###### Proof.

1\. In order to prove the statement for the Markov chain
$\mathcal{X}_{\text{exit}}$ we show that $\\{0\\}\times\\{1,\dots,m\\}$ is its
finite regeneration set with the following property: there is $c>0$ such that
for all $(x,i)\in\\{\dots,-2,-1\\}\times\\{1,\dots,m\\}$

${\mathop{\rm
P}}\Big{(}\mathcal{X}_{\text{exit}}(1)\in\\{0\\}\times\\{1,\dots,m\\}\,\Big{|}\,\mathcal{X}_{\text{exit}}(0)=(x,i)\Big{)}\geq
c.$ (3.11)

To demonstrate this, it is sufficient to verify that there is $c>0$ such that
for all $(x,i)\in\mathcal{N}^{m}$

${\mathop{\rm
P}}\Big{(}\mathcal{X}(\sigma)=(0,i)\,\Big{|}\,\mathcal{X}(0)=(x,i)\Big{)}\geq
c.$ (3.12)

We borrow the reasoning from Vysotsky [30].

Let $i$ be fixed. Consider a random walk with increments $\xi^{(i)}$ and
denote by $H_{-}^{(i)}$ its strictly descending ladder height. From Assumption
A1 we have ${\mathop{\rm E}}|H_{-}^{(i)}|<\infty$, see T1 in Section 18 in
Spitzer [29]. From Section XI.4 in Feller [6] it follows that for any
$y\in{\mathbb{N}}\cup\\{0\\}$

$\displaystyle\lim_{x\to+\infty}{\mathop{\rm
P}}\Big{(}\mathcal{X}(\sigma)=(-y,i)\Big{|}\mathcal{X}(0)=(x,i)\Big{)}=\frac{{\mathop{\rm
P}}(H_{-}^{(i)}\leq-1-y)}{{\mathop{\rm E}}|H_{-}^{(i)}|},$ (3.13)

and in particular for $y=0$

$\displaystyle\lim_{x\to+\infty}{\mathop{\rm
P}}\Big{(}\mathcal{X}(\sigma)=(0,i)\Big{|}\mathcal{X}(0)=(x,i)\Big{)}=\frac{1}{{\mathop{\rm
E}}|H_{-}^{(i)}|}>0.$ (3.14)

Since the probabilities ${\mathop{\rm
P}}(\mathcal{X}(\sigma)=(0,i)|\mathcal{X}(0)=(x,i))$ are obviously positive
for any $x\in{\mathbb{N}}$, equation (3.14) implies that

$\displaystyle\min_{i=1,\dots,m}\inf_{x\in\mathbb{N}}{\mathop{\rm
P}}\Big{(}\mathcal{X}(\sigma)=(0,i)\Big{|}\mathcal{X}(0)=(x,i)\Big{)}>0.$
(3.15)

2\. Existence of the stationary distribution for the Markov chain
$\mathcal{X}_{\text{entrance}}$ follows from the observation that

$\displaystyle\mathcal{X}_{\text{entrance}}(\iota)=\mathcal{X}(\tau_{\iota})=\mathcal{X}(\sigma_{\iota}+1),\quad\mathcal{X}(\sigma_{\iota})=\mathcal{X}_{\text{exit}}(\iota)$
(3.16)

and the existence of the stationary distribution for
$\mathcal{X}_{\text{exit}}$. ∎

Note that the stationary distributions $\pi_{\text{exit}}$ and
$\pi_{\text{entrance}}$ are related as follows:

$\displaystyle{\mathop{\rm P}}_{\pi_{\text{entrance}}}(\mathcal{X}(\sigma)\in
A)$ $\displaystyle={\mathop{\rm P}}_{\pi_{\text{exit}}}(\mathcal{X}(0)\in A),$
(3.17) $\displaystyle{\mathop{\rm P}}_{\pi_{\text{exit}}}(\mathcal{X}(1)\in
A)$ $\displaystyle={\mathop{\rm P}}_{\pi_{\text{entrance}}}(\mathcal{X}(0)\in
A),\quad A\subseteq{\mathbb{Z}}.$

see a detailed investigation of exit and entrance chains constructed for the
general Markov chain in Mijatović and Vysotsky [17].

######  Lemma 3.2.

${{\mathop{\rm E}}}_{\pi_{\emph{\text{exit}}}}|R(0)|=\sum_{x\geq
0}x\,\pi_{\emph{\text{exit}}}(-x)<\infty,$ (3.18) ${{\mathop{\rm
E}}}_{\pi_{\emph{\text{entrance}}}}R(0)=\sum_{x\geq
1}x\,\pi_{\emph{\text{entrance}}}(x)<\infty.$ (3.19)

###### Proof.

It follows from A3 and (3.17) that it suffices to prove (3.18) only.

With the help of formula (18) in Vysotsky [30] we obtain an estimate for the
expectation of an overshoot of a level: for each $\alpha>0$ there is
$K(\alpha)>0$ such that for all $(x,i)\in\mathcal{N}^{m}$

$\displaystyle{\mathop{\rm E}}_{x,i}|R(\sigma_{1})|\leq K(\alpha)+\alpha x.$
(3.20)

Assume that $R(0)=0$. Denote

$\displaystyle x_{n}$ $\displaystyle:={\mathop{\rm
E}}|R(\sigma_{n})|={\mathop{\rm E}}|R_{\text{exit}}(n)|,$ (3.21)
$\displaystyle y_{n}$ $\displaystyle:={\mathop{\rm E}}R(\tau_{n})={\mathop{\rm
E}}R_{\text{entrance}}(n),\ n\geq 1,$

where $R_{\text{exit}}$ and $R_{\text{entrance}}$ are radiuses of the embedded
Markov chains $\mathcal{X}_{\text{exit}}$ and $\mathcal{X}_{\text{entrance}}$
respectively. Then (3.20) and A3 yield inequalities

$\displaystyle y_{n+1}\leq C(1+x_{n}),\quad x_{n+1}\leq K(\alpha)+\alpha
y_{n+1}.$ (3.22)

Take some $\alpha\in(0,C^{-1})$. It can be shown by induction that

$\displaystyle\sup_{n}x_{n}\leq\frac{\alpha C+K(\alpha)}{1-\alpha C}.$ (3.23)

By the ergodic theorem, for any initial distribution we have convergence

$\displaystyle\lim_{n\to\infty}\frac{\sum_{k=1}^{n}|R_{\text{exit}}(k)|}{n}={{\mathop{\rm
E}}}_{\pi_{\text{exit}}}|R(0)|\ \ \text{a.s.}$ (3.24)

Hence, the Fatou lemma yields that

$\displaystyle{{\mathop{\rm E}}}_{\pi_{\text{exit}}}|R(0)|\leq\frac{\alpha
C+K(\alpha)}{1-\alpha C}<\infty,$ (3.25)

and the Lemma is proved. ∎

Now we are ready to formulate our results for scaling limits of $\mathcal{X}$.

Consider a collection of sequences $\\{\mathcal{X}^{n}(k)\\}_{k\geq
0}=(R^{n}(k),l^{n}(k))_{k\geq 0}$ that have the same transition probabilities
as $\\{\mathcal{X}(k)\\}_{k\geq 0}$, but maybe different initial conditions.
The stopping times $\sigma^{n}(k)$ and $\tau^{n}(k)$ are defined respectively.
Set

$\displaystyle X^{n}(t)$
$\displaystyle:=\Big{(}\frac{{R^{n}([nt])}}{v_{1}\sqrt{n}}1\\!\\!\,{\rm
I}_{l^{n}([nt])=1},\dots,\frac{{R^{n}([nt])}}{v_{m}\sqrt{n}}1\\!\\!\,{\rm
I}_{l^{n}([nt])=m}\Big{)},\quad t\geq 0.$ (3.26)

Let $W_{\textbf{p}}(\cdot,x)$ be a Walsh Brownian motion started from $x\in
E^{m}$, see (2.1), with weights $\mathbf{p}=(p_{1},\dots,p_{m})$ and the
symmetric local time $L(\cdot,x)$ at 0, see the definition of the local time
in (2.7).

######  Theorem 3.3.

Assume that conditions A1–A4 are satisfied and

$\displaystyle X^{n}(0)\Rightarrow x\in E^{m}.$ (3.27)

Then we have convergence:

$\displaystyle(X^{n},L^{n})\Rightarrow(W_{\emph{{p}}}(\cdot,x),L(\cdot,x)),\
n\to\infty,$ (3.28)

where

$L^{n}(t)=\frac{1}{\sqrt{n}}\sum_{\tau_{k}^{n}\leq[nt]}\frac{R^{n}(\tau_{k}^{n})-R^{n}(\sigma_{k+1}^{n})}{v_{l^{n}(\tau_{k}^{n})}},\quad
t\geq 0,$ (3.29)

and

$\displaystyle p_{k}=\frac{{{\mathop{\rm E}}}_{\pi_{\text{\rm
entrance}}}R(0)v_{k}^{-1}1\\!\\!\,{\rm I}_{l(0)=k}-{{\mathop{\rm
E}}}_{\pi_{\text{\rm exit}}}R(0)v_{k}^{-1}1\\!\\!\,{\rm
I}_{l(0)=k}}{{{\mathop{\rm E}}}_{\pi_{\text{\rm
entrance}}}R(0)v_{l(0)}^{-1}-{{\mathop{\rm E}}}_{\pi_{\text{\rm
exit}}}R(0)v_{l(0)}^{-1}}$ $\displaystyle=\frac{{{\mathop{\rm
E}}}_{\pi_{\text{\rm entrance}}}(R(0)-R(\sigma))v_{k}^{-1}1\\!\\!\,{\rm
I}_{l(0)=k}}{{{\mathop{\rm E}}}_{\pi_{\text{\rm
entrance}}}(R(0)-R(\sigma))v_{l(0)}^{-1}}$ (3.30)
$\displaystyle=\frac{{{\mathop{\rm E}}}_{\pi_{\text{\rm
exit}}}(R(1)v_{l(1)}^{-1}1\\!\\!\,{\rm
I}_{l(1)=k}-R(0)v_{l(0)}^{-1}1\\!\\!\,{\rm I}_{l(0)=k})}{{{\mathop{\rm
E}}}_{\pi_{\text{\rm exit}}}(R(1)v_{l(1)}^{-1}-R(0)v_{l(0)}^{-1})}.$

This Theorem will be proven in Section 5.

######  Remark 3.4.

Equality of expectations in (3.30) follows from (3.17) and the definition of
the embedded exit and entrance Markov chains.

Let $X$ be a perturbed random walk with a membrane $\\{-d,\dots,d\\}$ from
Theorem 1.1. In order to specify the permeability parameter $\gamma$
analogously to Theorem 3.1 we have to introduce the embedded entrance and exit
Markov chains and entrance and exit stopping times.

We define

$\displaystyle\widetilde{\sigma}_{0}:=0,$ (3.31)
$\displaystyle\widetilde{\tau}_{\iota}:=\inf\\{k\geq\widetilde{\sigma}_{\iota}\colon\
|X(k)|>d\\},\quad\iota\geq 0,$
$\displaystyle\widetilde{\sigma}_{\iota+1}=\inf\\{k>\widetilde{\tau}_{\iota}\colon
X(k-1)>d,\ X(k)\leq d\quad\text{or}\quad X(k-1)<-d,\
X(k)\geq-d\\},\quad\iota\geq 0.$

The difference between definition (LABEL:eq:defn_of_exits_entra) and
definition (3.8), (3.9) is twofold. First, when
$X(\widetilde{\sigma}_{\iota})\in\\{-d,\dots,d\\}$, the Markov chain may stay
on the membrane $\\{-d,\dots,d\\}$ more than one step, and hence in general in
this case we have $\widetilde{\tau}_{\iota}\geq\widetilde{\sigma}_{\iota}+1$.
Second, when $X$ “jumps over” the membrane, i.e., when
$X(\widetilde{\sigma}_{\iota}-1)>d$ and $X(\widetilde{\sigma}_{\iota})<-d$ or
vice versa, we get $\widetilde{\tau}_{\iota}=\widetilde{\sigma}_{\iota}$.

We adapt the set of Assumptions A of Theorem 3.1.

B1.

$\displaystyle{\mathop{\rm
P}}\Big{(}X(1)=x+y\,\Big{|}\,X(0)=x\Big{)}={\mathop{\rm P}}(\xi_{+}=y),\quad
x>d,$ (3.32) $\displaystyle{\mathop{\rm
P}}\Big{(}X(1)=x+y\,\Big{|}\,X(0)=x\Big{)}={\mathop{\rm P}}(\xi_{-}=y),\quad
x<-d,$

where the integer valued random variables $\xi_{-}$, $\xi_{+}$ satisfy

$\displaystyle{\mathop{\rm E}}\xi_{\pm}=0,\quad
v_{\pm}^{2}=\text{Var}\,\xi_{\pm}\in(0,\infty).$ (3.33)

and generate 1-arithmetic random walks.

B2. For all $x\in\\{-d,\dots,d\\}$

$\displaystyle{\mathop{\rm P}}(\widetilde{\tau}_{0}<\infty|X(0)=x)=1.$ (3.34)

B3. For all $x\in\\{-d,\dots,d\\}$

$\displaystyle{\mathop{\rm
E}}\Big{[}|X(\widetilde{\tau}_{0})|\Big{|}X(0)=x\Big{]}<\infty.$ (3.35)

B4. The states ${\mathbb{Z}}\setminus\\{-d,\dots,d\\}$ of the Markov chain $X$
are connected.

As above, we introduce the embedded Markov chains
$X_{\text{exit}}(\iota)=X(\widetilde{\sigma}_{\iota})$, $\iota\geq 1$, and
$X_{\text{entrance}}(\iota)=X(\widetilde{\tau}_{\iota})$, $\iota\geq 0$.
Similarly to Lemmas 3.1 and 3.18 it can be shown that $X_{\text{exit}}$ and
$X_{\text{entrance}}$ have unique stationary distributions $\pi_{\text{exit}}$
and $\pi_{\text{entrance}}$, respectively, which are integrable.

Consider a collection of sequences $\\{X^{n}(k)\\}_{k\geq 0}$ that have the
same transition probabilities as $\\{X(k)\\}_{k\geq 0}.$

######  Theorem 3.5.

Suppose that Assumptions B1—B4 are satisfied and

$\displaystyle\varphi\Big{(}\frac{X^{n}(0)}{\sqrt{n}}\Big{)}\Rightarrow
x\in{\mathbb{R}},$ (3.36)

where $\varphi$ is defined in (1.8). Then we have convergence to a skew
Brownian motion $W^{\text{\emph{skew}}}_{\gamma}(\cdot,x)$ started at $x$:

$\displaystyle\varphi\Big{(}\frac{X^{n}(n\cdot)}{\sqrt{n}}\Big{)}\Rightarrow
W_{\gamma}^{\text{\emph{skew}}}(\cdot,x),\quad n\to\infty,$ (3.37)

where

$\displaystyle\gamma=\frac{{{\mathop{\rm
E}}}_{\pi_{\text{\emph{entrance}}}}\varphi(X(0)-X(\widetilde{\sigma}_{1}))}{{{\mathop{\rm
E}}}_{\pi_{\text{\emph{entrance}}}}|\varphi(X(0)-X(\widetilde{\sigma}_{1}))|}$
$\displaystyle=\frac{{{\mathop{\rm
E}}}_{\pi_{\text{\emph{entrance}}}}(X(0)-X(\widetilde{\sigma}_{1}))v^{-1}_{\mathop{\rm
sgn}(X(0)-X(\widetilde{\sigma}_{1}))}}{{{\mathop{\rm
E}}}_{\pi_{\text{\emph{entrance}}}}|X(0)-X(\widetilde{\sigma}_{1})|v^{-1}_{\mathop{\rm
sgn}(X(0)-X(\widetilde{\sigma}_{1}))}}$ (3.38)
$\displaystyle=\frac{{{\mathop{\rm
E}}}_{\pi_{\text{\emph{entrance}}}}(X(0)-X(\widetilde{\sigma}_{1}))v^{-1}_{\mathop{\rm
sgn}X(0)}}{{{\mathop{\rm
E}}}_{\pi_{\text{\emph{entrance}}}}|X(0)-X(\widetilde{\sigma}_{1})|v^{-1}_{\mathop{\rm
sgn}X(0)}}.$

Theorem 3.5 will be proven in Section 6.

##  4 Examples

######  Example 4.1.

We work in the setting of Theorem 3.5. Consider perturbations of a simple
symmetric random walk, i.e., assume that
$\xi_{+}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\xi_{-}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\xi$,
where ${\mathop{\rm P}}(\xi=\pm 1)=\frac{1}{2}$. Then $v_{\pm}^{2}=1$ and
$\varphi(x)=x$.

1\. Assume that $d=0$. Then the membrane consists of the single point
$\\{0\\}.$ Notice that $X(\widetilde{\sigma}_{1})=0$, so that the stationary
distribution $\pi_{\text{exit}}$ is trivial, $\pi_{\text{exit}}(0)=1$. Let the
law of the jump from $0$ coincide with the law of a random variable $\eta$,

$\displaystyle{\mathop{\rm P}}(X(1)=j\,|\,X(0)=0)={\mathop{\rm P}}(\eta=j),\
j\in{\mathbb{Z}}.$ (4.1)

Recall that ${\mathop{\rm P}}(\eta=0)<1$ by assumption $\mathbf{B}_{2}$. Hence
the entrance stationary distribution is equal to the distribution of the
random walk at the moment of exit from $0$

$\displaystyle\pi_{\text{entrance}}(j)={\mathop{\rm
P}}(X(\tau_{0})=j\,|\,X(0)=0)=\frac{{\mathop{\rm P}}(\eta=j)}{{\mathop{\rm
P}}(\eta\neq 0)},\quad j\neq 0.$ (4.2)

The permeability parameter $\gamma$ of the limit skew Brownian motion is

$\gamma=\frac{{\mathop{\rm E}}\eta}{{\mathop{\rm E}}|\eta|}.$ (4.3)

In particular, if ${\mathop{\rm P}}(X(1)=1\,|\,X(0)=0)=p$ and ${\mathop{\rm
P}}(X(1)=-1\,|\,X(0)=0)=1-p$ or some $p\in(0,1)$ then

$\gamma=2p-1.$ (4.4)

Formulas (4.4), (4.3) appeared for the first time in the paper by Harrison and
Shepp [8] (formula (4.3) was only announced there).

2\. Assume that $d\in{\mathbb{N}}$, but the random walk can exit from
$\\{-d,\dots,d\\}$ only to the neighboring points $d+1$ or $-d-1$. Then the
distribution $\pi_{\text{entrance}}$ is supported on the two point set
$\\{-d-1,d+1\\}$. Denote

$\displaystyle\alpha:={\mathop{\rm
P}}(X(\widetilde{\tau}_{0})=d+1\,|\,X(0)=-d),\quad\beta:={\mathop{\rm
P}}(X(\widetilde{\tau}_{0})=-d-1\,|\,X(0)=d).$ (4.5)

The stationary distribution $\pi_{\text{entrance}}$ is calculated explicitly:

$\displaystyle\pi_{\text{entrance}}(d+1)=\frac{\alpha}{\alpha+\beta},\quad\pi_{\text{entrance}}(-d-1)=\frac{\beta}{\alpha+\beta}.$
(4.6)

Notice that $X(\widetilde{\sigma}_{1})=d$ if $X(0)=d+1$ and
$X(\widetilde{\sigma}_{1})=-d$ if $X(0)=-d-1$. Hence the permeability
parameter $\gamma$ has the form

$\displaystyle\gamma=\pi_{\text{entrance}}(d+1)-\pi_{\text{entrance}}(-d-1)=\frac{\alpha-\beta}{{\alpha+\beta}}.$
(4.7)

This result was already obtained in Pilipenko and Pryhod’ko [21].

######  Example 4.2.

We work in the setting of Theorem 3.5. Assume that a perturbed random walk on
${\mathbb{Z}}$ is such that ${\mathop{\rm P}}(\xi_{+}\geq-1)={\mathop{\rm
P}}(\xi_{-}\leq 1)=1$ and assume that $d\in{\mathbb{N}}$. Then
$\pi_{\text{exit}}$ is concentrated in two points $\\{-d\\}$ and $\\{d\\}$.
Denote

$\displaystyle\widetilde{\tau}_{0}:=\inf\\{k\geq 0\colon|X(k)|>d\\}.$ (4.8)

and

$\displaystyle\alpha:={\mathop{\rm
P}}(X(\widetilde{\tau}_{0})>0\,|\,X(0)=-d),\quad\beta:={\mathop{\rm
P}}(X(\widetilde{\tau}_{0})<0\,|\,X(0)=d).$ (4.9)

Then the stationary exit distribution $\pi_{\text{exit}}$ is

$\displaystyle\pi_{+}:=\pi_{\text{exit}}(d)=\frac{\alpha}{\alpha+\beta},\quad\pi_{-}:=\pi_{\text{exit}}(-d)=\frac{\beta}{\alpha+\beta},$
(4.10)

and the stationary entrance distribution $\pi_{\text{entrance}}$ is

$\displaystyle\pi_{\text{entrance}}(x)=\pi_{-}{\mathop{\rm
P}}(X(\widetilde{\tau}_{0})=x\,|\,X(0)=-d)+\pi_{+}{\mathop{\rm
P}}(X(\widetilde{\tau}_{0})=x\,|\,X(0)=d),\ |x|>d.$ (4.11)

Notice that $X(\widetilde{\sigma}_{1})=d\,\mathop{\rm sgn}X(0)$ and
$X(0)-X(\widetilde{\sigma}_{1})=X(0)-d\,\mathop{\rm sgn}X(0)$ if $|X(0)|>d.$
Hence the parameter $\gamma$ from Theorem 3.5 equals

$\displaystyle\gamma=\frac{{{\mathop{\rm
E}}}_{\pi_{\text{entrance}}}(X(0)-d\mathop{\rm sgn}X(0))v^{-1}_{\mathop{\rm
sgn}X(0)}}{{{\mathop{\rm E}}}_{\pi_{\text{entrance}}}|X(0)-d\mathop{\rm
sgn}X(0)|v^{-1}_{\mathop{\rm sgn}X(0)}}.$ (4.12)

In particular, if $v_{+}=v_{-},$ then

$\displaystyle\gamma=\frac{{{\mathop{\rm
E}}}_{\pi_{\text{entrance}}}(X(0)-d\mathop{\rm sgn}X(0))}{{{\mathop{\rm
E}}}_{\pi_{\text{entrance}}}|X(0)-d\mathop{\rm sgn}X(0)|}.$ (4.13)

If additionally (4.5) holds true, then

$\displaystyle\gamma=\frac{\alpha-\beta}{{\alpha+\beta}}.$ (4.14)

according to formula (4.7).

######  Example 4.3.

We work in the setting of Theorem 3.5. Assume that the random walk
$\mathcal{X}$ on $\mathcal{Z}^{m}$, $m\geq 1$, is such that ${\mathop{\rm
P}}(\xi^{(i)}=\pm 1)=1/2$, ${\mathop{\rm P}}(\mathcal{X}(0)=(1,j)|\
\mathcal{X}(0)=(0,i))=q_{ij}$, where $Q=\|q_{ij}\|$ is an irreducible
stochastic matrix. By $\pi=(\pi_{i})_{1\leq i\leq m}$ denote its stationary
distribution. Then the stationary distribution $\pi_{\text{entrance}}$ is
supported on the set $\\{(1,i)\\}_{1\leq i\leq m}$, and

$\displaystyle\pi_{\text{entrance}}(1,i)=\pi_{i},\quad 1\leq i\leq m.$ (4.15)

Moreover the weights $\mathbf{p}=(p_{1},\dots,p_{m})$ of the limit Walsh
Brownian motion $W_{\textbf{p}}$ coincide with $\pi$, i.e., $\mathbf{p}=\pi$.

If $q_{ij}=q_{j}$, i.e., if the choice of the next label is independent on the
previous one, then $\pi_{i}=q_{i}$. The case when $q_{ij}$ depends on
$(j-i)\operatorname{mod}m$ was considered by Enriquez and Kifer [5]; in this
case the stationary distribution $\pi$ is the uniform one.

######  Example 4.4.

In the work [19], Ngo and Paigné considered a time homogeneous Markov chain
$\\{Y(k)\\}_{k\geq 0}$ defined by

$\displaystyle Y(k+1)=\begin{cases}Y(k)+\xi_{k+1}^{+},&\ \text{if}\ Y(k)>0\
\text{and}\ Y(k)+\xi_{k+1}^{+}>0,\\\ Y(k)+\xi_{k+1}^{-},&\ \text{if}\ Y(k)<0\
\text{and}\ Y(k)+\xi_{k+1}^{-}<0,\\\ \eta_{k+1},&\ \text{if}\ Y(k)=0,\\\ 0,&\
\text{otherwise},\end{cases}$ (4.16)

where $\xi_{k}^{+}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\xi_{+}$,
$\xi_{k}^{-}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\xi_{-}$, $\eta_{k}$,
$k\geq 1$, are mutually independent random variables that are independent of
$Y(0)$. It is assumed that $\xi_{-}$ and $\xi_{+}$ are zero-mean integer-
valued random variables with variances
$v^{2}_{\pm}=\operatorname{Var}\xi_{\pm}\in(0,\infty)$, and ${\mathop{\rm
E}}|\eta|<\infty$.

Since $Y$ is forced to hit 0 at any attempt to jump over the origin, it does
not satisfy the setting of Theorem 3.5. However it can be fit into our theory
with the help of a Markov chain $\mathcal{X}(k)=(R(k),l(k))$, $k\geq 0$, on
the state space ${\mathbb{Z}}\times\\{-,+\\}$ defined as follows.

We set

$\displaystyle R(k+1)$ $\displaystyle=\begin{cases}R(k)+\xi_{k+1}^{+}&\text{
if }l(k)=+\text{ and }R(k)>0,\\\ R(k)-\xi_{k+1}^{-}&\text{ if }l(k)=-\text{
and }R(k)>0,\\\ |\eta_{k}|&\text{ if }R(k)\leq 0,\end{cases}$ (4.17)
$\displaystyle l(k+1)$ $\displaystyle=\begin{cases}l(k)&\text{ if }R(k)>0,\\\
+&\text{ if }R(k)\leq 0\text{ and }\eta_{k}\geq 0,\\\ -&\text{ if }R(k)\leq
0\text{ and }\eta_{k}<0,\end{cases}$ $\displaystyle R(0)$
$\displaystyle=|Y(0)|,$ $\displaystyle l(0)$ $\displaystyle=\begin{cases}+,\
\text{ if }Y(0)\geq 0,\\\ -,\ \text{ if }Y(0)<0.\end{cases}$

We have that $Y(k)=l(k)R(k)$ in each of the following three cases:

$\displaystyle\Big{\\{}Y(k)>0\ \text{and}\ Y(k)+\xi_{k+1}^{+}>0\Big{\\}}\text{
or }\Big{\\{}Y(k)<0\ \text{and}\ Y(k)+\xi_{k+1}^{-}<0\Big{\\}}\text{ or
}\Big{\\{}Y(k)=0\Big{\\}}.$ (4.18)

Otherwise,

$\displaystyle|Y(k)-l(k)R(k)|\leq|\xi^{-}_{k+1}|\vee|\xi^{+}_{k+1}|.$ (4.19)

Since $\xi^{\pm}_{k}$ are square integrable we have

$\displaystyle\frac{1}{\sqrt{n}}\max_{k\leq[nt]}\Big{(}|\xi^{-}_{k+1}|\vee|\xi^{+}_{k+1}|\Big{)}\stackrel{{\scriptstyle{\mathop{\rm
P}}}}{{\to}}0,\quad n\to\infty,$ (4.20)

for any $t\geq 0$.

We apply Theorem 3.3 to study the limit behaviour of $\mathcal{X}$. The
stationary distribution $\pi_{\text{entrance}}$ of the Markov chain
$\mathcal{X}_{\text{entrance}}$ can be determined explicitly as

$\displaystyle\pi_{\text{entrance}}(x,+)$ $\displaystyle={\mathop{\rm
P}}(\eta=x\,|\,\eta\neq 0),$ (4.21) $\displaystyle\pi_{\text{entrance}}(x,-)$
$\displaystyle={\mathop{\rm P}}(\eta=-x\,|\,\eta\neq 0),\quad
x\in{\mathbb{N}}.$

Let $\sigma=\inf\\{k\geq 0\colon R(k)\leq 0\\}$ be the time instant, when the
radius becomes non-positive for the first time. Denote by

$\displaystyle f(x,\pm):={\mathop{\rm
E}}[R(\sigma)\,|\,(R(0),l(0))=(x,\pm)],\quad x\in{\mathbb{N}}.$ (4.22)

the expected value of the overshoot over 0. Hence by the formula (3.30)

$\displaystyle p_{+}$ $\displaystyle=\frac{{{\mathop{\rm E}}}_{\pi_{\text{\rm
entrance}}}(R(0)-R(\sigma))v_{+}^{-1}1\\!\\!\,{\rm I}_{l(0)=+}}{{{\mathop{\rm
E}}}_{\pi_{\text{\rm
entrance}}}(R(0)-R(\sigma))v_{l(0)}^{-1}}=\frac{{\mathop{\rm
E}}(\eta-f(\eta,+))v_{+}^{-1}1\\!\\!\,{\rm I}_{\eta>0}}{{\mathop{\rm
E}}(\eta-f(\eta,+))v_{+}^{-1}1\\!\\!\,{\rm I}_{\eta>0}+{\mathop{\rm
E}}(-\eta-f(-\eta,-))v_{-}^{-1}1\\!\\!\,{\rm I}_{\eta<0}},$ (4.23)
$\displaystyle p_{-}$ $\displaystyle=1-p_{+}.$

Eventually, the Donsker scaling limit of
$\varphi\Big{(}Y([n\cdot])/\sqrt{n}\Big{)}$, where $\varphi$ is defined in
(1.8), is the skew Brownian motion with $\gamma=p_{+}-p_{-},$ see Corollary
2.3.

Our formula (4.23) coincides with the result obtained in Ngo and Paigné [19]
where the formula for $\gamma$ was characterized in terms of descending ladder
epochs and descending renewal functions of the random walks with steps
$\\{\xi^{+}_{k}\\}$ and $\\{\xi^{-}_{k}\\}$, respectively.

######  Example 4.5.

We work in the setting of Theorem 1.1 and assume that all the jumps starting
from the membrane have mean value zero:

$\displaystyle{\mathop{\rm E}}[X(1)\,|\,X(0)=x]=0,\quad|x|\leq d.$ (4.24)

According to Corollary 1.3 the weak limit of
$\big{(}\frac{X([nt])}{\sqrt{n}}\big{)}_{t\geq 0}$ is a skew oscillating
Brownian motion satisfying (1.9). On the other hand, it is easy to see that
the Markov chain $X=(X(k))_{k\geq 0}$ is a martingale, and that the limit skew
oscillating Brownian motion is a martingale, too. Therefore the local time
term in (1.9) vanishes. This means that the limit of
$\big{(}\frac{X([nt])}{\sqrt{n}}\big{)}_{t\geq 0}$ is the oscillating Brownian
motion that solves the SDE

$\mathrm{d}Y(t)=(v_{-}1\\!\\!\,{\rm I}_{Y(t)<0}+v_{+}1\\!\\!\,{\rm
I}_{Y(t)\geq 0})\,\mathrm{d}W(t).$ (4.25)

This convergence was established by Helland in the case of a one-point
membrane, $d=0$, see Corollary 8.4. in [9].

##  5 Proof of Theorem 3.3

The proof of Theorem 3.3 is divided into seven steps.

Step 1. First we introduce several objects related to the random walk
$\mathcal{X}(k)=(R(k),l(k))$, $k\geq 0$.

We say that the radial part $R$ is in the ‘normal’ mode at time $k$ if
$R(k)>0$ and is in the ‘critical’ mode otherwise. Hence, for $k\geq 0$, we
denote by

$\displaystyle T_{\text{normal}}(k):=\sum_{i=0}^{k-1}1\\!\\!\,{\rm
I}_{R(i)\in{\mathbb{N}}},$ (5.1)

the number of jumps in the normal mode up to time $k$, and

$\displaystyle T_{\text{critical}}(k):=\sum_{i=0}^{k-1}1\\!\\!\,{\rm
I}_{R(i)\leq 0}=k-T_{\text{normal}}(k),$ (5.2)

the number of jumps in the critical mode, where $\sum_{i=0}^{-1}:=0$. We also
consider the inverses

$\displaystyle T_{\text{normal}}^{-1}(\iota)$ $\displaystyle:=\min\\{k\geq
0\colon T_{\text{normal}}(k)\geq\iota\\},$ (5.3) $\displaystyle
T_{\text{critical}}^{-1}(\iota)$ $\displaystyle:=\min\\{k\geq 0\colon
T_{\text{critical}}(k)\geq\iota\\},\quad\iota\geq 1.$

Also note that

$\displaystyle\sigma_{\iota}=T_{\text{critical}}^{-1}(\iota)-1,\quad\iota\geq
1,$ (5.4)

is a moment of $k$-th exit of $R$ from ${\mathbb{N}}$, and

$\displaystyle\tau_{\iota}=T_{\text{critical}}^{-1}(\iota),\quad\iota\geq 1,$
(5.5)

see (3.8) and (3.9) for the definition of $\tau_{\iota}$, $\sigma_{\iota}$.

Define random sequences $V$ and $U$ as follows:

$\displaystyle V(0)$ $\displaystyle:=0,\quad U(0):=0,$ (5.6) $\displaystyle
V(\iota)-V(\iota-1)$
$\displaystyle:=\frac{R(T_{\text{normal}}^{-1}(\iota))}{v_{l(T_{\text{normal}}^{-1}(\iota))}}-\frac{R(T_{\text{normal}}^{-1}(\iota)-1)}{v_{l(T_{\text{normal}}^{-1}(\iota)-1)}}$
(5.7)
$\displaystyle=\frac{R(T_{\text{normal}}^{-1}(\iota))-R(T_{\text{normal}}^{-1}(\iota)-1)}{v_{l(T_{\text{normal}}^{-1}(\iota)-1)}}$
(5.8)
$\displaystyle=\frac{R(T_{\text{normal}}^{-1}(\iota))-R(T_{\text{normal}}^{-1}(\iota)-1)}{v_{l(T_{\text{normal}}^{-1}(\iota))}},\
\iota\geq 1,$ (5.9) $\displaystyle U(\iota)-U(\iota-1)$
$\displaystyle:=\frac{R(T_{\text{critical}}^{-1}(\iota))}{v_{l(T_{\text{critical}}^{-1}(\iota))}}-\frac{R(T_{\text{critical}}^{-1}(\iota)-1)}{v_{l(T_{\text{critical}}^{-1}(\iota)-1)}}=\frac{R(\tau_{\iota})}{v_{l(\tau_{\iota})}}-\frac{R(\sigma_{\iota})}{v_{l(\sigma_{\iota})}},\
\iota\geq 1.$ (5.10)

In (5.8) we used that the label $l$ does not change as long as $R$ is in the
normal mode.

Notice that

$\displaystyle\frac{R(k)}{v_{l(k)}}-\frac{R(k-1)}{v_{l(k-1)}}$
$\displaystyle=\Big{(}V(T_{\text{normal}}(k))-V(T_{\text{normal}}(k)-1)\Big{)}1\\!\\!\,{\rm
I}_{R(k-1)>0}$ (5.11)
$\displaystyle+\Big{(}U(T_{\text{critical}}(k))-U(T_{\text{critical}}(k)-1)\Big{)}1\\!\\!\,{\rm
I}_{R(k-1)\leq 0}.$

Hence, for $k\geq 0$

$\displaystyle\frac{R(k)}{v_{l(k)}}$
$\displaystyle=\frac{R(0)}{v_{l(0)}}+\sum_{i=1}^{k}\frac{R(i)-R(i-1)}{v_{l(i)}}1\\!\\!\,{\rm
I}_{R(i-1)>0}+\sum_{i=1}^{k}\Big{(}\frac{R(i)}{v_{l(i)}}-\frac{R(i-1)}{v_{l(i-1)}}\Big{)}1\\!\\!\,{\rm
I}_{R(i-1)\leq 0}$ (5.12)
$\displaystyle=\frac{R(0)}{v_{l(0)}}+\sum_{\iota=1}^{T_{\text{normal}}(k)}(V(\iota)-V(\iota-1))+\sum_{\iota=1}^{T_{\text{critical}}(k)}(U(\iota)-U(\iota-1))$
$\displaystyle=\frac{R(0)}{v_{l(0)}}+V(T_{\text{normal}}(k))+U(T_{\text{critical}}(k)).$

Since $T_{\text{normal}}^{-1}(\iota)$ and $T_{\text{critical}}^{-1}(\iota)$,
$\iota\geq 1$, are stopping times w.r.t. the filtration generated by
$\mathcal{X}$, the sequence $(V(k))_{k\geq 0}$ is a martingale. The sequences

$\displaystyle\Big{(}R(T_{\text{critical}}^{-1}(\iota)),v_{l(T_{\text{normal}}^{-1}(\iota))}\Big{)}_{\iota\geq
0}\quad\text{and}\quad\Big{(}R(T_{\text{critical}}^{-1}(\iota)-1),v_{l(T_{\text{critical}}^{-1}(\iota)-1}\Big{)}_{\iota\geq
1}$ (5.13)

are the embedded entrance and exit Markov chains introduced in (3.10).

For $t\in[0,\infty)$ we define

$\displaystyle R(t)$ $\displaystyle:=R([t]),\quad V(t):=V([t]),\quad
U(t):=U([t]),\quad l(t):=l([t]),$ (5.14) $\displaystyle T_{\text{normal}}(t)$
$\displaystyle:=T_{\text{normal}}([t])+(t-[t]),\quad
T_{\text{critical}}(t):=t-T_{\text{normal}}([t])$

so that

$\displaystyle\frac{R(t)}{v_{l(t)}}=\frac{R(0)}{v_{l(0)}}+{V(T_{\text{normal}}(t))}+U(T_{\text{critical}}(t)).$
(5.15)

Introducing similar notation for processes $\mathcal{X}^{n}$ we get from
(5.12) that

$\frac{R^{n}(nt)}{v_{l^{n}(nt)}\sqrt{n}}=\frac{R^{n}(0)}{v_{l^{n}(0)}\sqrt{n}}+\frac{V^{n}(T^{n}_{\text{normal}}(nt))}{\sqrt{n}}+\frac{U^{n}(T^{n}_{\text{critical}}(nt))}{\sqrt{n}}.$
(5.16)

######  Remark 5.1.

We have the equality

$\displaystyle\frac{U^{n}(T^{n}_{\text{critical}}(nt))}{\sqrt{n}}=L^{n}(t),$
(5.17)

where $L^{n}(\cdot)$ is defined in (3.29).

Step 2. We prove the following Lemma.

######  Lemma 5.2.

The following weak convergence

$\Big{(}\frac{V^{n}(nt)}{\sqrt{n}},\frac{U^{n}(\sqrt{n}t)}{\sqrt{n}}\Big{)}_{t\geq
0}\Rightarrow\Big{(}W(t),\mu t\Big{)}_{t\geq 0},\quad n\to\infty,$ (5.18)

holds true, where $W$ is a standard Brownian motion,

$\displaystyle\mu$ $\displaystyle={\mathop{\rm E}}_{\pi_{\text{\rm
entrance}}}\frac{R(0)}{v_{l(0)}}-{\mathop{\rm E}}_{\pi_{\text{\rm
exit}}}\frac{R(0)}{v_{l(0)}}$ (5.19) $\displaystyle={\mathop{\rm
E}}_{\pi_{\text{\rm
exit}}}\left(\frac{R(1)}{v_{l(1)}}-\frac{R(0)}{v_{l(0)}}\right)$
$\displaystyle={\mathop{\rm E}}_{\pi_{\text{\rm
entrance}}}\left(\frac{R(0)}{v_{l(0)}}-\frac{R(\sigma)}{v_{l(\sigma)}}\right)$
$\displaystyle={\mathop{\rm E}}_{\pi_{\text{\rm
entrance}}}\left(\frac{R(0)-R(\sigma)}{v_{l(0)}}\right)\in(0,\infty),$

and $\sigma$ is defined in (3.5).

###### Proof of Lemma 5.2.

Since the second coordinate in of the limit in (5.18) is non-random, it is
sufficient to prove the coordinate-wise convergence, see the generalization of
Slutsky’s theorem in Ressel [26].

Convergence $\frac{V^{n}(nt)}{\sqrt{n}}\Rightarrow W(t)$, $n\to\infty$ follows
from the functional central limit theorem for martingale differences, see
Theorem 18.2 in Billingsley [2]. Indeed, due to Assumption A1 since
$\operatorname{Var}\xi^{(j)}=v_{j}^{2}\in(0,\infty)$, for every $t\geq 0$ and
$\varepsilon>0$ the Lindeberg condition holds true, namely

$\displaystyle\frac{1}{n}\sum_{k=1}^{[nt]}$ $\displaystyle{\mathop{\rm
E}}\Big{[}(V^{n}(k)-V^{n}(k-1))^{2}1\\!\\!\,{\rm
I}_{|V^{n}(k)-V^{n}(k-1)|\geq\sqrt{n}\varepsilon}\Big{]}\leq
t\sum_{j=1}^{m}{\mathop{\rm
E}}\Big{[}\frac{(\xi^{(j)})^{2}}{v_{j}^{2}}1\\!\\!\,{\rm
I}_{|\frac{|\xi^{(j)}|}{v_{j}}|\geq\sqrt{n}\varepsilon}\Big{]}\to
0,n\to\infty.$ (5.20)

Furthermore, for each $t\geq 0$

$\displaystyle\frac{1}{n}\sum_{k=1}^{[nt]}{\mathop{\rm
E}}(V^{n}(k)-V^{n}(k-1))^{2}=\frac{[nt]}{n}\to t,\quad n\to\infty,$ (5.21)

what yields the result.

Now we prove the convergence of the second coordinate.

1\. First assume that the initial condition $\mathcal{X}(0)$ does not depend
on $n$. Let the initial condition $\mathcal{X}(0)=(R(0),l(0))$ be fixed.
Recall the entrance and exit embedded Markov chains
$\mathcal{X}_{\text{entrance}}$ and $\mathcal{X}_{\text{exit}}$ defined in
(3.10). Representation (5.10) together with the ergodic theorem for Markov
chains immediately imply that for each $t\geq 0$ the limit

$\displaystyle\frac{U(\sqrt{n}t)}{\sqrt{n}}=\frac{1}{\sqrt{n}}\sum_{k=1}^{[\sqrt{n}t]}\Big{(}U(k)-U(k-1)\Big{)}=\frac{[\sqrt{n}t]}{\sqrt{n}}\frac{1}{[\sqrt{n}t]}\sum_{k=1}^{[\sqrt{n}t]}\Big{(}\frac{R(\tau_{k})}{v_{l(\tau_{k})}}-\frac{R(\sigma_{k})}{v_{l(\sigma_{k})}}\Big{)}\to\mu
t,\quad n\to\infty.$ (5.22)

holds almost surely. The functional convergence follows from Theorem 2.15 c)
(i) in Chapter VI of Jacod and Shiryaev [14] because all the functions are
non-decreasing and the limit is continuous.

2\. In the situation when the distribution of the initial value
$\mathcal{X}^{n}(0)$ depends on $n$, we deal with weak convergence in the the
scheme of series. Hence, certain modifications of the above argument have to
be made.

Consider the set $A:=\\{0\\}\times\\{1,\dots,m\\}$ and let
$\theta^{n}:=\inf\\{k\geq 1\colon R^{n}(\sigma^{n}_{k})=0\\}$. We write

$\displaystyle{\mathop{\rm
P}}\Big{(}\Big{|}\frac{1}{\sqrt{n}}\sum_{k=1}^{[\sqrt{n}t]}$
$\displaystyle\Big{(}\frac{R^{n}(\tau^{n}_{k})}{v_{l(\tau^{n}_{k})}}-\frac{R^{n}(\sigma^{n}_{k})}{v_{l(\sigma^{n}_{k})}}\Big{)}-\mu
t\Big{|}>\varepsilon\Big{)}$ (5.23) $\displaystyle\leq{\mathop{\rm
P}}\Big{(}\Big{|}\frac{1}{\sqrt{n}}\sum_{k=1}^{\theta^{n}}\Big{(}\frac{R^{n}(\tau^{n}_{k})}{v_{l(\tau^{n}_{k})}}-\frac{R^{n}(\sigma^{n}_{k})}{v_{l(\sigma^{n}_{k})}}\Big{)}\Big{|}>\frac{\varepsilon}{3}\Big{)}$
$\displaystyle+{\mathop{\rm
P}}\Big{(}\Big{|}\frac{1}{\sqrt{n}}\sum_{k=\theta^{n}}^{[\sqrt{n}t]+\theta^{n}}\Big{(}\frac{R^{n}(\tau^{n}_{k})}{v_{l(\tau^{n}_{k})}}-\frac{R^{n}(\sigma^{n}_{k})}{v_{l(\sigma^{n}_{k})}}\Big{)}-\mu
t\Big{|}>\frac{\varepsilon}{3}\Big{)}$ $\displaystyle+{\mathop{\rm
P}}\Big{(}\Big{|}\frac{1}{\sqrt{n}}\sum_{k=[\sqrt{n}t]+1}^{[\sqrt{n}t]+\theta^{n}}\Big{(}\frac{R^{n}(\tau^{n}_{k})}{v_{l(\tau^{n}_{k})}}-\frac{R^{n}(\sigma^{n}_{k})}{v_{l(\sigma^{n}_{k})}}\Big{)}\Big{|}>\frac{\varepsilon}{3}\Big{)}$
$\displaystyle=S_{1}^{n}+S_{2}^{n}+S^{n}_{3}.$

It is easy to see that for any $\varepsilon>0$ and $t>0$

$\displaystyle S_{2}^{n}\leq\sup_{\text{Law}(\mathcal{X}(0))\in
M(A)}\lim_{n\to\infty}{\mathop{\rm
P}}\Big{(}\Big{|}\frac{1}{\sqrt{n}}\sum_{k=1}^{[\sqrt{n}t]}\Big{(}\frac{R(\tau_{k})}{v_{l(\tau_{k})}}-\frac{R(\sigma_{k})}{v_{l(\sigma_{k})}}\Big{)}-\mu
t\Big{|}>\frac{\varepsilon}{3}\Big{)}=0.$ (5.24)

It follows from (3.12) that there is $N=N_{\delta}\in{\mathbb{N}}$ such that
for all $n\geq 1$

$\displaystyle{\mathop{\rm P}}(\theta^{n}>N)<\frac{\delta}{2}.$ (5.25)

Hence we can estimate

$\displaystyle S_{1}^{n}+S_{3}^{n}\leq\delta$ $\displaystyle+{\mathop{\rm
P}}\Big{(}\frac{1}{\sqrt{n}}\sum_{k=1}^{N}\Big{(}\frac{|R^{n}(\tau^{n}_{k})|}{v_{l(\tau^{n}_{k})}}+\frac{|R^{n}(\sigma^{n}_{k})|}{v_{l(\sigma^{n}_{k})}}\Big{)}>\frac{\varepsilon}{3}\Big{)}$
(5.26) $\displaystyle+{\mathop{\rm
P}}\Big{(}\frac{1}{\sqrt{n}}\sum_{k=[\sqrt{n}t]+1}^{[\sqrt{n}t]+N}\Big{(}\frac{R^{n}(\tau^{n}_{k})}{v_{l(\tau^{n}_{k})}}+\frac{|R^{n}(\sigma^{n}_{k})|}{v_{l(\sigma^{n}_{k})}}\Big{)}>\frac{\varepsilon}{3}\Big{)}$
$\displaystyle\leq\delta+\sum_{k=1}^{N}{\mathop{\rm
P}}\Big{(}\frac{1}{\sqrt{n}}\frac{R^{n}(\tau^{n}_{k})}{v_{l(\tau^{n}_{k})}}>\frac{\varepsilon}{6N}\Big{)}+\sum_{k=1}^{N}{\mathop{\rm
P}}\Big{(}\frac{1}{\sqrt{n}}\frac{|R^{n}(\sigma^{n}_{k})|}{v_{l(\sigma^{n}_{k})}}>\frac{\varepsilon}{6N}\Big{)}$
$\displaystyle+\sum_{k=[\sqrt{n}t]+1}^{[\sqrt{n}t]+N}{\mathop{\rm
P}}\Big{(}\frac{1}{\sqrt{n}}\frac{|R^{n}(\tau^{n}_{k})|}{v_{l(\tau^{n}_{k})}}>\frac{\varepsilon}{6N}\Big{)}+\sum_{k=[\sqrt{n}t]+1}^{[\sqrt{n}t]+N}{\mathop{\rm
P}}\Big{(}\frac{1}{\sqrt{n}}\frac{|R^{n}(\sigma^{n}_{k})|}{v_{l(\sigma^{n}_{k})}}>\frac{\varepsilon}{6N}\Big{)}$
$\displaystyle\leq\delta+2N\sup_{k\geq 1}{\mathop{\rm
P}}\Big{(}|R^{n}(\tau^{n}_{k})|>\frac{\varepsilon\sqrt{n}\min
v_{i}}{6N}\Big{)}+2N\sup_{k\geq 1}{\mathop{\rm
P}}\Big{(}|R^{n}(\sigma^{n}_{k})|>\frac{\varepsilon\sqrt{n}\min
v_{i}}{6N}\Big{)}.$

Therefore it is enough to prove that for any $c_{1}$, $c_{2}>0$

$\displaystyle\sup_{k\geq 1}{\mathop{\rm P}}(R^{n}(\sigma^{n}_{k})\leq-
c_{1}\sqrt{n})<\frac{\delta}{8N},$ (5.27) $\displaystyle\sup_{k\geq
1}{\mathop{\rm P}}(R^{n}(\tau^{n}_{k})>c_{2}\sqrt{n})<\frac{\delta}{4N}$
(5.28)

for $n$ large enough.

Let us show that for any $j=1,\dots,m$

$\displaystyle\lim_{a\to\infty}\sup_{i\in\mathbb{N}}{\mathop{\rm
P}}\Big{(}|R(\sigma)|\geq a\,\Big{|}\,R(0)=i,l(0)=j\Big{)}=0.$ (5.29)

Let $\varepsilon>0$. From (3.13) it follows that there are $i_{0}>0$ and
$a_{0}>0$ such that

$\displaystyle\sup_{i>i_{0}}{\mathop{\rm P}}\Big{(}|R(\sigma)|\,\geq
a_{0}\,\Big{|}\,R(0)=i,l(0)=j\Big{)}\leq\varepsilon.$ (5.30)

Since $\sigma<\infty$ a.s. for any starting value $(i,j)$, there is $a_{1}>0$
such that

$\displaystyle\max_{i=1,\dots,i_{0}}{\mathop{\rm P}}\Big{(}|R(\sigma)|\geq
a_{1}\,\Big{|}\,R(0)=i,l(0)=j\Big{)}\leq\varepsilon$ (5.31)

and (5.27) follows.

To demonstrate (5.28) we note that

$\displaystyle{\mathop{\rm P}}(R^{n}(\tau^{n}_{k})>c_{2}\sqrt{n})$
$\displaystyle=\sum_{x\leq 0,j=1,\dots,m}{\mathop{\rm
P}}\Big{(}R^{n}(\sigma^{n}_{k}+1)>c_{2}\sqrt{n}\,\Big{|}\,R^{n}(\sigma^{n}_{k})=x,l^{n}(0)=j\Big{)}{\mathop{\rm
P}}(R^{n}(\sigma^{n}_{k})=x,l^{n}(\sigma_{k}^{n})=j)$ (5.32)
$\displaystyle\leq\sum_{x\leq 0,j=1,\dots,m}{\mathop{\rm
P}}\Big{(}R^{n}(\sigma^{n}_{k}+1)>c_{2}\sqrt{n}\,\Big{|}\,R^{n}(\sigma^{n}_{k}\Big{)}=x,l^{n}(0)=j){\mathop{\rm
P}}(R^{n}(\sigma^{n}_{k})=x).$

Let $C>0$ be the constant from A3. Choose $c_{1}>0$ such that
$Cc_{1}/c_{2}<\delta/(8N)$. Then we obtain

$\displaystyle{\mathop{\rm P}}(R^{n}(\tau^{n}_{k})>c_{2}\sqrt{n})$
$\displaystyle\leq\sum_{x\leq
0,j=1,\dots,m}\Big{(}\frac{C(1+|x|)}{c_{2}\sqrt{n}}\wedge 1\Big{)}{\mathop{\rm
P}}(R^{n}(\sigma^{n}_{k})=x)$ (5.33) $\displaystyle\leq\sum_{x\leq-
c_{1}\sqrt{n},j=1,\dots,m}\Big{(}\frac{C(1+|x|)}{c_{2}\sqrt{n}}\wedge
1\Big{)}{\mathop{\rm P}}(R^{n}(\sigma^{n}_{k})=x)$
$\displaystyle+\sum_{-c_{1}\sqrt{n}<x\leq
0,j=1,\dots,m}\Big{(}\frac{C(1+|x|)}{c_{2}\sqrt{n}}\wedge 1\Big{)}{\mathop{\rm
P}}(R^{n}(\sigma^{n}_{k})=x)$ $\displaystyle\leq{\mathop{\rm
P}}(R^{n}(\sigma^{n}_{k})\leq-
c_{1}\sqrt{n})+\frac{C(1+c_{1}\sqrt{n})}{c_{2}\sqrt{n}}\leq\frac{\delta}{4N}$

for $n\to\infty$ by (5.27). The Lemma is proven. ∎

Step 3. The main result of this Step is Theorem 5.4 about convergence of the
scaled radius process to the reflected Brownian motion.

We will use the following result on the deterministic convergence in the
Skorokhod space which is a particular case of Theorem 1 from Pilipenko and
Sarantsev [22].

######  Theorem 5.3.

Assume that sequence of càdlàg functions
$(f_{n},g_{n},h_{n},T_{A_{n}},T_{B_{n}})$ is such that

  1. 1.

$f_{n}(t)=g_{n}(T_{A_{n}}(t))+h_{n}(T_{B_{n}}(t))$, $t\geq 0$, where

$T_{A_{n}}(t)=\int_{0}^{t}1\\!\\!\,{\rm I}_{f_{n}(s)>0}\,\mathrm{d}s,\quad
T_{B_{n}}(t)=\int_{0}^{t}1\\!\\!\,{\rm I}_{f_{n}(s)\leq 0}\,\mathrm{d}s$
(5.34)

and the set $\\{s\in[0,t]\colon f_{n}(s)>0\\}$ consists of finitely many
intervals for any $t\geq 0$;

  2. 2.

$g_{n}(\cdot)\to g_{0}(\cdot)$ in $D([0,\infty),{\mathbb{R}})$;

  3. 3.

there is a sequence $(\rho_{n})$ of positive numbers such that $\rho_{n}\to 0$
and $h_{n}(\rho_{n}\cdot)\to h_{0}(\cdot)$ in $D([0,\infty),{\mathbb{R}})$;

  4. 4.

$g_{0}$, $h_{0}$ are continuous functions and $h_{0}$ is strictly increasing;

  5. 5.

$g_{0}(0)\geq 0$, $h_{n}(0)=0$.

Then $(f_{n})$ converges in $D([0,\infty),{\mathbb{R}})$ to the solution of
the Skorokhod reflection problem at zero for the function $g_{0}$, i.e.,

$\displaystyle f_{n}(t)\to g_{0}(t)+\max_{s\in[0,t]}(-g_{0}(s))\vee 0.$ (5.35)

Moreover,

$\displaystyle h_{n}(T_{B_{n}}(t))\to\max_{s\in[0,t]}(-g_{0}(s))\vee 0$ (5.36)

in $D([0,\infty),{\mathbb{R}})$.

Let the processes $R^{n}$ and $L^{n}$ be from Theorem 3.3.

######  Theorem 5.4.

We have convergence in distribution in $D([0,\infty),{\mathbb{R}}^{2})$:

$\displaystyle\Big{(}\frac{{R^{n}(nt)}}{v_{l^{n}(nt)}\sqrt{n}},L^{n}(t)\Big{)}_{t\geq
0}\Rightarrow\Big{(}|\|x\|+W(t)|,L_{t}^{0}(|\|x\|+W|)\Big{)}_{t\geq 0},\quad
n\to\infty,$ (5.37)

where $L^{0}(|\|x\|+W|)$ is the symmetric local time at 0 of the reflected
Brownian motion $(|\|x\|+W(t)|)_{t\geq 0}$ and $x\in E^{m}$ is from (3.27).

###### Proof.

It follows from Lemma 5.2 and the Skorokhod representation theorem that there
are copies
$(\widetilde{V}^{n}(\cdot),\widetilde{U}^{n}(\cdot))\overset{\mathrm{d}}{=}(V^{n}(\cdot),U^{n}(\cdot))$
and $\widetilde{W}\overset{\mathrm{d}}{=}W$ defined on the same probability
space such that we have a.s. locally uniform convergence

$\displaystyle\Big{(}\frac{\widetilde{V}^{n}(nt)}{\sqrt{n}},\frac{\widetilde{U}^{n}(\sqrt{n}t)}{\sqrt{n}}\Big{)}_{t\geq
0}\to(\widetilde{W}(t),\mu t)_{t\geq 0},\quad n\to\infty,$ (5.38)

and

$\displaystyle\frac{\widetilde{R}^{n}(0)}{v_{\widetilde{l}^{n}(0)}\sqrt{n}}\to\|x\|,\quad
n\to\infty\text{ a.s.},$ (5.39)

where $x\in E^{m}$ is from (3.27).

The functions $t\mapsto\widetilde{T}^{n}_{\text{normal}}(nt)$ and
$t\mapsto\widetilde{T}^{n}_{\text{critical}}(nt)$ increase if
$\frac{\widetilde{R}^{n}(nt)}{v_{\widetilde{l}^{n}(nt)}\sqrt{n}}$ belongs to
$(0,\infty)$ and $(-\infty,0]$, respectively. Hence Theorem 5.3 applied to the
functions

$\displaystyle f_{n}(t)$
$\displaystyle=\frac{\widetilde{R}^{n}(nt)}{\sqrt{n}v_{\widetilde{l}^{n}(nt)}},\quad
g_{n}(t)=\frac{\widetilde{R}^{n}(0)}{v_{\widetilde{l}^{n}(0)}}+\frac{\widetilde{V}^{n}(nt)}{\sqrt{n}},\quad
h_{n}(t)=\frac{\widetilde{U}^{n}(nt)}{\sqrt{n}},$ (5.40) $\displaystyle
T_{A_{n}}(t)$
$\displaystyle=\frac{\widetilde{T}^{n}_{\text{normal}}(nt)}{n},\quad
T_{B_{n}}(t)=\frac{\widetilde{T}^{n}_{\text{critical}}(nt)}{n},\quad\rho_{n}=\frac{1}{\sqrt{n}},$

yields that the term

$\displaystyle\frac{\widetilde{R}^{n}(nt)}{\sqrt{n}v_{\widetilde{l}^{n}(nt)}}=\frac{\widetilde{R}^{n}(0)}{v_{\widetilde{l}^{n}(0)}\sqrt{n}}+\frac{\widetilde{V}^{n}(\widetilde{T}^{n}_{\text{normal}}(nt))}{\sqrt{n}}+\frac{\widetilde{U}^{n}(\widetilde{T}^{n}_{\text{critical}}(nt))}{\sqrt{n}}$
(5.41)

converges with probability one in $D([0,\infty),{\mathbb{R}})$ to the solution
of the Skorokhod reflection problem for the process $\|x\|+\widetilde{W}(t)$,
$t\geq 0,$ i.e., to

$\displaystyle\|x\|+\widetilde{W}(t)+\max_{s\in[0,t]}(-\|x\|-\widetilde{W}(s))\vee
0.$ (5.42)

Moreover, we have convergence

$\displaystyle\frac{\widetilde{U}^{n}(\widetilde{T}^{n}_{\text{critical}}(nt))}{\sqrt{n}}\to\max_{s\in[0,t]}(-\|x\|-\widetilde{W}(s))\vee
0,\ n\to\infty.$ (5.43)

In particular we have that

$\displaystyle\frac{\widetilde{V}^{n}(\widetilde{T}^{n}_{\text{normal}}(nt))}{\sqrt{n}}\to\widetilde{W}(t).$
(5.44)

It is well known that the law of the solution of the Skorokhod reflection
problem coincides with the law of the reflected Brownian motion, see, e.g.,
§1.3 in Pilipenko [20]. This means that

$\displaystyle\Big{(}\|x\|+\widetilde{W}(t)+\max_{s\in[0,t]}(-\|x\|-\widetilde{W}(s))\vee
0,\max_{s\in[0,t]}(-\|x\|-\widetilde{W}(s))\vee 0\Big{)}_{t\geq
0}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\Big{(}|\|x\|+W(t)|,L_{t}^{0}(|\|x\|+W|)\Big{)}_{t\geq
0},$ (5.45)

and Theorem 5.4 is proven. ∎

Step 4. We continue proving Theorem 3.3. First assume for simplicity that the
initial value $X(0)$ does not depend on $n$. Consider projections of $R$ on
the line ${\mathbb{Z}}\times\\{j\\}$, $j=1,\dots,m$:

$\displaystyle R_{j}(k)$ $\displaystyle:=R(k)1\\!\\!\,{\rm I}_{l(k)=j},$
(5.46) $\displaystyle R(k)$ $\displaystyle:=R_{1}(k)+\dots+R_{m}(k),$

Similarly to (5.1) we define

$\displaystyle T_{\text{normal},j}(k):=\sum_{i=0}^{k-1}1\\!\\!\,{\rm
I}_{R_{j}(i)\in\mathbb{N}},$ (5.47)

the number of jumps made from the positive side of $j$-th ray, as well as the
inverse mapping

$\displaystyle T_{\text{normal},j}^{-1}(\iota):=\inf\\{k\geq 0\colon
T_{\text{normal},j}(k)\geq\iota\\}.$ (5.48) $\displaystyle V_{j}(0)$
$\displaystyle=0,\quad U_{j}(0):=0,$ (5.49) $\displaystyle
V_{j}(\iota)-V_{j}(\iota-1)$
$\displaystyle:=\frac{R_{j}(T_{\text{normal},j}^{-1}(\iota))}{v_{l(T_{\text{normal},j}^{-1}(\iota))}}-\frac{R_{j}(T_{\text{normal},j}^{-1}(\iota)-1)}{v_{l(T_{\text{normal},j}^{-1}(\iota)-1)}}$
(5.50)
$\displaystyle=\frac{R_{j}(T_{\text{normal},j}^{-1}(\iota))-R_{j}(T_{\text{normal},j}^{-1}(\iota)-1)}{v_{j}},\quad\iota\geq
1.$ (5.51)

It follows from the assumptions of the Theorem that $(V_{j}(\iota))_{\iota\geq
0}$, $1\leq j\leq m,$ are independent random walks, which increments have
distributions of $\frac{\xi^{(j)}}{v_{j}}$ respectively. Observe that

$\displaystyle\frac{R_{j}(k)}{v_{j}}$
$\displaystyle=\frac{R_{j}(k)1\\!\\!\,{\rm I}_{l(k)=j}}{v_{l(k)}}$ (5.52)
$\displaystyle=\frac{R(0)1\\!\\!\,{\rm
I}_{l(0)=j}}{v_{j}}+\sum_{i=1}^{k}\Big{(}\frac{R(i)1\\!\\!\,{\rm
I}_{l(i)=j}}{v_{j}}-\frac{R(i-1)1\\!\\!\,{\rm
I}_{l(i-1)=j}}{v_{j}}\Big{)}1\\!\\!\,{\rm I}_{R(i-1)>0}$
$\displaystyle+\sum_{i=1}^{k}\Big{(}\frac{R(i)1\\!\\!\,{\rm
I}_{l(i)=j}}{v_{j}}-\frac{R(i-1)1\\!\\!\,{\rm
I}_{l(i-1)=j}}{v_{j}}\Big{)}1\\!\\!\,{\rm I}_{R(i-1)\leq 0}$
$\displaystyle=\frac{R_{j}(0)}{v_{j}}+\sum_{\iota=1}^{T_{\text{normal},j}(k)}(V_{j}(\iota)-V_{j}(\iota-1))$
$\displaystyle+\sum_{i=1}^{k}\Big{(}\frac{R(i)1\\!\\!\,{\rm
I}_{l(i)=j}}{v_{j}}-\frac{R(i-1)1\\!\\!\,{\rm
I}_{l(i-1)=j}}{v_{j}}\Big{)}1\\!\\!\,{\rm I}_{R(i-1)\leq 0}$

Note that

$\displaystyle R(i-1)\leq 0\quad\Leftrightarrow\quad
i-1=\sigma_{\iota}\quad\text{for some $\iota$}\quad\Leftrightarrow\quad
i=\tau_{\iota}\quad\text{for some $\iota$}$ (5.53)

and

$\displaystyle\tau_{\iota}\leq k\quad\Leftrightarrow\quad\sigma_{\iota}\leq
k\quad\Leftrightarrow\quad\iota\leq T_{\text{critical}}(k).$ (5.54)

Therefore,

$\displaystyle\frac{R_{j}(k)}{v_{j}}$
$\displaystyle=\frac{R_{j}(0)}{v_{j}}+V_{j}(T_{\text{normal},j}(k))+U_{j}(T_{\text{critical}}(k)),$
(5.55)

where

$\displaystyle
U_{j}(l)=\sum_{\iota=1}^{l}\Big{(}\frac{R(\tau_{\iota})}{v_{j}}1\\!\\!\,{\rm
I}_{l(\tau_{\iota})=j}-\frac{R(\sigma_{\iota})}{v_{j}}1\\!\\!\,{\rm
I}_{l(\sigma_{\iota})=j}\Big{)}.$ (5.56)

The sequences $k\mapsto U_{j}(T_{\text{critical}}(k))$ are non-decreasing for
all $j=1,\dots,m$.

Notice that $V(\iota)$ and $U(\iota)$, $\iota\geq 1$, defined in (5.8) and
(5.10) satisfy the property

$\displaystyle
V(T_{\text{normal}}(k))=V_{1}(T_{\text{normal},1}(k))+\dots+V_{m}(T_{\text{normal},m}(k)),$
(5.57)
$U(T_{\text{critical}}(k))=U_{1}(T_{\text{critical}}(k))+\dots+U_{m}(T_{\text{critical}}(k)).$
(5.58)

When the initial value $X(0)$ depends on $n$, similarly to notations above we
define the sequences $R^{n}_{j}$, $V^{n}_{j}$, $U^{n}_{j}$, etc.

Step 5. To prove convergence, we will utilize the martingale characterization
of the WBM. First, we show that the sequence

$\displaystyle\Big{\\{}\Big{(}\frac{R^{n}_{j}(n\cdot)}{\sqrt{n}},\frac{V_{j}^{n}(T^{n}_{\text{normal},j}(n\cdot))}{\sqrt{n}},\frac{U^{n}_{j}(T^{n}_{\text{critical}}(n\cdot))}{\sqrt{n}}\Big{)}_{1\leq
j\leq m}\Big{\\}}_{n\geq 1}$ (5.59)

is weakly relatively compact in $D([0,\infty),{\mathbb{R}}^{3m})$ and its
limit is continuous.

Due (5.43), the sequence of processes
$\\{\frac{U^{n}(T^{n}_{\text{critical}}(n\cdot))}{\sqrt{n}}\\}_{n\geq 1}$
converges in distribution to a continuous non-decreasing process. Equation
(5.58) implies that the modulus of continuity of each non-decreasing process
$t\mapsto U^{n}_{j}(T^{n}_{\text{critical}}(nt))/\sqrt{n}$ is dominated by the
modulus of continuity of $t\mapsto
U^{n}(T^{n}_{\text{critical}}(nt))/\sqrt{n}$. Hence this sequence is weakly
relatively compact, and any limit process
$\widehat{U}^{\infty}=(\widehat{U}^{\infty}_{1}(t),\dots,\widehat{U}^{\infty}_{m}(t))_{t\geq
0}$, is continuous in $t$ and each coordinate
$t\mapsto\widehat{U}^{\infty}_{j}(t)$ is non-decreasing.

By Donsker’s theorem,

$\displaystyle\Big{(}\frac{V^{n}_{j}(n\cdot)}{\sqrt{n}},\dots,\frac{V^{n}_{m}(n\cdot)}{\sqrt{n}}\Big{)}\Rightarrow\Big{(}B_{1}(t),\dots,B_{m}(\cdot)\Big{)},\
n\to\infty,$ (5.60)

where $B_{1},\dots,B_{m}$ are independent Brownian motions. Since
$T^{n}_{\text{normal},j}(t)-T^{n}_{\text{normal},j}(s)\leq t-s$, for $0\leq
s\leq t$, the sequence of processes
$\\{(\frac{V^{n}_{j}(T^{n}_{\text{normal},j}(n\cdot))}{\sqrt{n}})_{1\leq j\leq
m}\\}_{n\geq 1}$ is weakly relatively compact, too, and any limit point
$M^{\infty}=(M^{\infty}_{1}(t),\dots,M^{\infty}_{m}(t))_{t\geq 0}$ is a
continuous process. Weak relative compactness of
$\\{\frac{R^{n}_{j}(n\cdot)}{\sqrt{n}}\\}_{n\geq 1}$ follows from (5.55).
Since all limits $M_{j}^{\infty}$ and $\widehat{U}_{j}^{\infty}$ are
continuous, the limits $R_{j}^{\infty}$ are also continuous, and the sequence
(5.59) is tight in $D([0,\infty),{\mathbb{R}}^{3m})$.

We have

$\displaystyle\frac{{R^{n}(nt)}}{v_{l^{n}(nt)}\sqrt{n}}=\sum_{j=1}^{m}\frac{{R^{n}_{j}(nt)}}{v_{j}\sqrt{n}}$
$\displaystyle=\sum_{j=1}^{m}\frac{{R^{n}_{j}(0)}}{v_{j}\sqrt{n}}+\sum_{j=1}^{m}\frac{V_{j}^{n}(T^{n}_{\text{normal},j}(nt))}{\sqrt{n}}+\sum_{j=1}^{m}\frac{U^{n}_{j}(T^{n}_{\text{critical}}(nt))}{\sqrt{n}}$
(5.61)
$\displaystyle=\frac{{R^{n}(0)}}{v_{l^{n}(0)}\sqrt{n}}+\frac{V^{n}(T^{n}_{\text{normal}}(nt))}{\sqrt{n}}+\frac{U^{n}(T^{n}_{\text{critical}}(nt))}{\sqrt{n}}.$

Recall that by Theorem 5.4, see (5.43) and (5.44),
$\frac{V^{n}(T^{n}_{\text{normal}}(n\cdot))}{\sqrt{n}}$ converges to a
Brownian motion $W(\cdot)$, $\frac{{R^{n}(nt)}}{v_{l^{n}(nt)}\sqrt{n}}$
converges to a reflected Brownian motion

$\displaystyle\Big{(}\|x\|+W(t)+\max_{s\in[0,t]}(-\|x\|-W(s))\vee
0\Big{)}_{t\geq 0},$ (5.62)

and $\frac{U^{n}(T^{n}_{\text{critical}}(n\cdot))}{\sqrt{n}}$ converges to its
local time at $0$ that will be denoted by $\nu(\cdot)$.

Hence, any condensation point

$\displaystyle(R^{\infty}_{j}(\cdot),M^{\infty}_{j}(\cdot),\widehat{U}^{\infty}_{j}(\cdot))_{1\leq
j\leq m}$ (5.63)

of (5.59) is a continuous process such that

  1. 1.

$\displaystyle\sum_{j=1}^{m}M^{\infty}_{j}(t)$ $\displaystyle=W(t)$ (5.64)
$\displaystyle\sum_{j=1}^{m}\widehat{U}^{\infty}_{j}(t)$
$\displaystyle=\nu(t),\quad t\geq 0;$ (5.65)

  2. 2.

for each $1\leq j\leq m$

$\displaystyle
X^{\infty}_{j}(t):=\frac{R^{\infty}_{j}(t)}{v_{j}}=x_{j}+M^{\infty}_{j}(t)+\widehat{U}^{\infty}_{j}(t),\quad
t\geq 0;$ (5.66)

  3. 3.

$t\mapsto\widehat{U}^{\infty}_{j}(t)$ are non-decreasing for $t\geq 0$,
$\widehat{U}^{\infty}_{j}(0)=0,$ for each $j=1,\dots,m$.

Since the process

$\displaystyle\sum_{j=1}^{m}X^{\infty}_{j}(t),\quad t\geq 0,$ (5.67)

is a reflected Brownian motion whose local time at 0 is $\nu(\cdot)$ we have

$\displaystyle\int_{0}^{\infty}1\\!\\!\,{\rm
I}_{\sum_{j=1}^{m}X^{\infty}_{j}(t)>0}\,\mathrm{d}\nu(t)=0\mbox{ and
}\int_{0}^{\infty}1\\!\\!\,{\rm
I}_{\sum_{j=1}^{m}X^{\infty}_{j}(t)=0}\,\mathrm{d}t=0$ (5.68)

with probability 1

Also note that $X^{\infty}_{i}(t)X^{\infty}_{j}(t)=0$ for every $i\neq j$ and
$t\geq 0$. Since $\sum_{j=1}^{m}X^{\infty}_{j}(t)\geq 0$ we conclude that all
$X^{\infty}_{j}(t)\geq 0$, $j=1,\dots,m$.

Step 6. It is left to demonstrate that any condensation point
$(X^{\infty}_{j}(\cdot))_{1\leq j\leq m}$ is a WBM with weights
$p_{1},\dots,p_{m}$ and the starting point $x$. Without loss of generality, we
assume that the sequence (5.59) weakly converges to the condensation point
itself.

Let us check conditions of the martingale characterization of the Walsh
Brownian motion, see Theorem 2.2. It remains to verify the following:

  1. 1.

for any $j=1,\dots,m$ and $t\geq 0$

$\frac{\widehat{U}^{\infty}_{j}(t)}{\nu(t)}=\frac{\widehat{U}^{\infty}_{j}(t)}{\sum_{l=1}^{m}\widehat{U}^{\infty}_{l}(t)}=p_{j}\
\mbox{a.s.},$ (5.69)

where $p_{j}$ are specified in (3.30);

  2. 2.

for any $j=1,\dots,m$ the process $M^{\infty}_{j}$ is a continuous square
integrable martingale with respect to the filtration generated by
$(X^{\infty},M^{\infty},\widehat{U}^{\infty})$ with the predictable quadratic
variation

$\displaystyle\langle M^{\infty}_{j}\rangle_{t}=\int_{0}^{t}1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(s)>0}\,\mathrm{d}s.$ (5.70)

By the Skorokhod representation theorem, see Theorem 6.7 in Billingsley [2],
we may assume without loss of generality that the sequence (5.59) is defined
on the same probability space and that with probability 1 convergence

$\displaystyle\Big{(}\frac{R^{n}_{j}(nt)}{v_{j}\sqrt{n}},\frac{V^{n}_{j}(T^{n}_{\text{normal},j}(nt))}{\sqrt{n}},\frac{U^{n}_{j}(T^{n}_{\text{critical}}(nt))}{\sqrt{n}}\Big{)}_{1\leq
j\leq
m}\to(X^{\infty}_{j}(t),M^{\infty}_{j}(t),\widehat{U}^{\infty}_{j}(t))_{1\leq
j\leq m},\quad n\to\infty,$ (5.71)

holds true locally uniformly in $t$.

1\. We demonstrate (5.69). Recall that (5.18) in Lemma 5.2 imply that

$\displaystyle\sum_{l=1}^{m}\frac{U^{n}_{l}(\sqrt{n}t)}{\sqrt{n}}\to\mu
t,\quad n\to\infty,$ (5.72)

where $\mu$ is defined in (5.19). By (5.43) and (5.65) we have:

$\displaystyle\sum_{l=1}^{m}\frac{U^{n}_{l}\Big{(}\sqrt{n}\frac{T^{n}_{\text{critical}}(nt)}{\sqrt{n}}\Big{)}}{\sqrt{n}}=\sum_{l=1}^{m}\frac{U^{n}_{l}(T^{n}_{\text{critical}}(nt))}{\sqrt{n}}\to\sum_{l=1}^{m}\widehat{U}^{\infty}_{l}(t)=\nu(t),\quad
n\to\infty,$ (5.73)

with probability 1 locally uniformly in $t$. Therefore for any $t>0$

$\displaystyle\frac{T^{n}_{\text{critical}}(nt)}{\sqrt{n}}\to\frac{\nu(t)}{\mu},\quad
n\to\infty.$ (5.74)

Analogously to the proof of (5.18) in Lemma 5.2 we obtain that

$\displaystyle\frac{U^{n}_{j}(\sqrt{n}t)}{\sqrt{n}}\to p_{j}\mu t,\quad
n\to\infty,$ (5.75)

in probability locally uniformly in $t$, where $p_{j}$ is defined in (3.30).
With the help of (5.74) and (5.75) we conclude that

$\displaystyle\frac{U^{n}_{j}(T^{n}_{\text{critical}}(nt))}{\sqrt{n}}=\frac{U^{n}_{j}\Big{(}\sqrt{n}\frac{T^{n}_{\text{critical}}(nt)}{\sqrt{n}}\Big{)}}{\sqrt{n}}\to\widehat{U}_{j}^{\infty}(t)=p_{j}\nu(t).$
(5.76)

The limit (5.69) follows from (5.73) and (5.76).

2\. To study martingale properties of $M^{\infty}$ we need an auxiliary result
about limits of local martingales.

Let $({\mathcal{F}}_{t}^{\infty})_{t\geq 0}$ be the natural filtration
generated by
$(X^{\infty}_{j}(t),M^{\infty}_{j}(t),\widehat{U}^{\infty}_{j}(t))_{1\leq
j\leq m}$, $t\geq 0$. It is easy to see that $W$ defined in (5.64) is a
$({\mathcal{F}}_{t}^{\infty})$-Brownian motion. To complete the proof of the
Theorem it suffices to show that

$M^{\infty}_{j}(t)=\int_{0}^{t}1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(s)>0}\,\mathrm{d}W(s)\ \ \text{a.s.},$ (5.77)

for any $t\geq 0$. Note that

$\displaystyle\frac{V^{n}_{j}(T^{n}_{\text{normal},j}(nt))}{\sqrt{n}}=\int_{0}^{t}1\\!\\!\,{\rm
I}_{\frac{R^{n}_{j}(ns-)}{v_{j}\sqrt{n}}>0}\,\mathrm{d}\frac{V^{n}(T^{n}_{\text{normal}}(ns))}{\sqrt{n}},$
(5.78)

where this integral is understood as a stochastic integral w.r.t. to the
square integrable martingal
$(\frac{V^{n}(T^{n}_{\text{normal}}(nt))}{\sqrt{n}},{\mathcal{F}}_{t}^{n})_{t\geq
0}$, where
${\mathcal{F}}^{n}_{t}=\sigma\\{R^{n}_{j}(ns),V^{n}_{j}(T^{n}_{\text{normal},j}(ns)),\widehat{U}^{n}_{j}(ns),\,j=1,\dots,m,\,s\leq
t\\}$.

Step 7. The proof of the Theorem will follow from the following Lemma.

######  Lemma 5.5.

For all $j=1,\dots,m$ and $t\geq 0$ we have

$\displaystyle\lim_{n\to\infty}{\mathop{\rm
E}}\Big{|}\int_{0}^{t}1\\!\\!\,{\rm
I}_{\frac{R^{n}_{j}(ns-)}{v_{j}\sqrt{n}}>0}\,\mathrm{d}\frac{V^{n}(T^{n}_{\text{\rm
normal}}(ns))}{\sqrt{n}}-\int_{0}^{t}1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(s)>0}\,\mathrm{d}W(s)\Big{|}^{2}=0.$ (5.79)

###### Proof.

We note that for any $t_{0}>0$

$\displaystyle{\mathop{\rm E}}\Big{|}\int_{0}^{t_{0}}1\\!\\!\,{\rm
I}_{\frac{R^{n}_{j}(ns-)}{v_{j}\sqrt{n}}>0}\,\mathrm{d}\frac{V^{n}(T^{n}_{\text{\rm
normal}}(ns))}{\sqrt{n}}\Big{|}^{2}\leq t_{0},$ (5.80)
$\displaystyle{\mathop{\rm E}}\Big{|}\int_{0}^{t_{0}}1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(s)>0}\,\mathrm{d}W(s)\Big{|}^{2}\leq t_{0}.$

Therefore to prove (5.79) it is sufficient to check that for any
$t_{0}\in(0,t]$

$\displaystyle\lim_{n\to\infty}{\mathop{\rm
E}}\Big{|}\int_{t_{0}}^{t}1\\!\\!\,{\rm
I}_{\frac{R^{n}_{j}(ns-)}{v_{j}\sqrt{n}}>0}\,\mathrm{d}\frac{V^{n}(T^{n}_{\text{\rm
normal}}(ns))}{\sqrt{n}}-\int_{t_{0}}^{t}1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(s)>0}\,\mathrm{d}W(s)\Big{|}^{2}=0.$ (5.81)

This will follow from Skorokhod’s Theorem from §3 of Chapter II in [28].
Recall the notation

$\displaystyle X^{n}_{j}(t)=\frac{R^{n}_{j}(nt)}{v_{j}\sqrt{n}},\quad
j=1,\dots,m,$ (5.82)

and let

$\displaystyle M^{n}(t)=\frac{V^{n}(T^{n}_{\text{\rm
normal}}(nt))}{\sqrt{n}},\quad j=1,\dots,m,$ (5.83)

To apply Skorokhod’s Theorem we have to verify the following conditions:

i) for any fixed $s\in[t_{0},t]$

$\displaystyle M_{n}(s)\to W(s)\quad\text{a.s.},\quad n\to\infty;$ (5.84)

ii) ${\mathop{\rm E}}|M_{n}(t)|^{2}\leq t$, $n\geq 1$;

iii) for any fixed $s\in[t_{0},t]$

$\displaystyle 1\\!\\!\,{\rm I}_{X^{n}_{j}(s-)>0}\to 1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(s)>0}\quad\text{a.s.},\quad n\to\infty,$ (5.85)

iv) for any $\varepsilon>0$

$\displaystyle\lim_{h\to 0}\lim_{n\to\infty}\sup_{|t_{2}-t_{1}|<h}{\mathop{\rm
P}}\Big{(}|1\\!\\!\,{\rm I}_{X^{n}_{j}(t_{2}-)>0}-1\\!\\!\,{\rm
I}_{X^{n}_{j}(t_{1}-)>0}|>\varepsilon\Big{)}=0.$ (5.86)

Indeed, convergence i) was established in (5.44). Condition ii) holds true by
the construction of the sequence $V^{n}$. We emphasize that in [28], it was
demanded that ${\mathop{\rm E}}|M_{n}(s)|^{2}\to s$, $n\to\infty$, for all
$s\in[t_{0},t]$. However the inspection of Skorokhod’s argument shows that
only boundedness was used in the proof.

To demonstrate iii), we recall that $X^{\infty}=\sum_{l=1}^{m}X_{l}^{\infty}$
is a reflected Brownian motion. Hence with probability 1, $X^{\infty}(s)>0$
for any fixed $s>0$. Therefore, for a.a. $\omega$ there exists $k=k(s,\omega)$
such that $X^{\infty}_{k}(s,\omega)>0$. Let $\kappa=\kappa(s,\omega)>0$ be
such that

$\displaystyle\inf_{r\in[s-\kappa,s+\kappa]}X^{\infty}_{k}(r)>0.$ (5.87)

Correspondingly, for $i\neq k$, we have

$\displaystyle X^{\infty}_{i}(r)=0\quad\text{on}\quad
r\in[s-\kappa,s+\kappa].$ (5.88)

Recall that $X^{n}_{j}(\cdot)\to X^{\infty}_{j}(\cdot)$ locally uniformly for
all $j$ a.s. and in particular on the interval $[s-\kappa,s+\kappa]$.
Therefore, for $n$ large enough

$\displaystyle X^{n}_{k}(r)>0,\quad r\in[s-\kappa,s+\kappa]$ (5.89)

and for all $i\neq k$

$\displaystyle X^{n}_{i}(r)=0,\quad r\in[s-\kappa,s+\kappa].$ (5.90)

This yields iii).

Eventually we establish iv). First we note that by (5.87) and (5.88) the
mapping $s\mapsto 1\\!\\!\,{\rm I}_{X^{\infty}(s)>0}$ is continuous in
probability on $[t_{0},t]$, and thus is uniformly continuous in probability on
$[t_{0},t]$, i.e., for each $\varepsilon>0$

$\displaystyle\lim_{h\to 0}\sup_{|t_{2}-t_{1}|<h}{\mathop{\rm
P}}\Big{(}|1\\!\\!\,{\rm I}_{X^{\infty}(t_{2})>0}-1\\!\\!\,{\rm
I}_{X^{\infty}(t_{1})>0}|>\varepsilon\Big{)}=0$ (5.91)

that is equivalent to

$\displaystyle\lim_{h\to 0}\sup_{|t_{2}-t_{1}|<h}{\mathop{\rm
P}}\Big{(}1\\!\\!\,{\rm I}_{X^{\infty}(t_{2})>0}\neq 1\\!\\!\,{\rm
I}_{X^{\infty}(t_{1})>0}\Big{)}=0.$ (5.92)

From (5.87), (5.88), (5.89) and (5.90) we get that for any $\delta>0$ and
$s\in[t_{0},t]$ there is $\kappa=\kappa(s)$ and $n_{0}=n_{0}(s)$ such that for
all $j=1,\dots,m$

$\displaystyle\sup_{|z-s|<\kappa(s)}{\mathop{\rm P}}\Big{(}1\\!\\!\,{\rm
I}_{X^{n}_{j}(z-)>0}\neq 1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(z)>0}\Big{)}<\delta,\quad n\geq n_{0}(s).$ (5.93)

Since $[t_{0},t]$ is a compact, from its infinite covering by the sets
$(s-\frac{1}{2}\kappa(s),s+\frac{1}{2}\kappa(s))$, $s\in[t_{0},t]$, we may
choose a finite subcovering
$(s_{p}-\frac{1}{2}\kappa(s_{p}),s_{p}+\frac{1}{2}\kappa(s_{p}))$,
$p=1,\dots,N$. Hence, for $|t_{2}-t_{1}|<h$ with

$\displaystyle h:=\frac{1}{2}\min_{p=1,\dots,N}\kappa(s_{p}),$ (5.94)

there is $p=p(t_{1},t_{2})$ such that

$\displaystyle t_{1},t_{2}\in(s_{p}-\kappa(s_{p}),s_{p}+\kappa(s_{p})).$
(5.95)

Hence for all $n\geq\max\\{n_{0}(s_{1}),\dots,n_{0}(s_{N})\\}$ we get

$\displaystyle{\mathop{\rm P}}\Big{(}1\\!\\!\,{\rm
I}_{X^{n}_{j}(t_{1}-)>0}\neq 1\\!\\!\,{\rm I}_{X^{n}_{j}(t_{2}-)>0}\Big{)}$
$\displaystyle\leq{\mathop{\rm P}}\Big{(}1\\!\\!\,{\rm
I}_{X^{n}_{j}(t_{1}-)>0}\neq 1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(t_{1})>0}\Big{)}$ (5.96) $\displaystyle+{\mathop{\rm
P}}\Big{(}1\\!\\!\,{\rm I}_{X^{\infty}_{j}(t_{1})>0}\neq 1\\!\\!\,{\rm
I}_{X^{\infty}_{j}(t_{2})>0}\Big{)}$ $\displaystyle+{\mathop{\rm
P}}\Big{(}1\\!\\!\,{\rm I}_{X^{\infty}_{j}(t_{2})>0}\neq 1\\!\\!\,{\rm
I}_{X^{n}_{j}(t_{2}-)>0}\Big{)}$ $\displaystyle\leq
2\delta+\sup_{|s_{2}-s_{1}|<h}{\mathop{\rm P}}\Big{(}1\\!\\!\,{\rm
I}_{X^{\infty}(s_{2})>0}\neq 1\\!\\!\,{\rm I}_{X^{\infty}(s_{1})>0}\Big{)}.$

Therefore,

$\displaystyle\limsup_{h\to
0}\lim_{n\to\infty}\sup_{|t_{2}-t_{1}|<h}{\mathop{\rm P}}\Big{(}|1\\!\\!\,{\rm
I}_{X^{n}_{j}(t_{2}-)>0}-1\\!\\!\,{\rm
I}_{X^{n}_{j}(t_{1}-)>0}|>\varepsilon\Big{)}\leq 2\delta,$ (5.97)

what finishes the proof of (5.86) and (5.81). The Lemma is proven. ∎

##  6 Proof of Theorem 3.5

Before we start proofs let us formulate two results on convergence of
compositions of functions and processes.

######  Lemma 6.1.

Let $(f_{n},\lambda_{n})_{n\geq 0}$ be a sequence of càdlàg functions such
that

  1. 1.

$f_{n}\to f_{0},n\to\infty$ in $D([0,+\infty),{\mathbb{R}}^{m})$;

  2. 2.

$(\lambda_{n})_{n\geq 1}$ are non-negative and non-decreasing functions;

  3. 3.

$\lambda_{n}\to\lambda_{0},n\to\infty,$ in $D([0,+\infty),{\mathbb{R}})$;

  4. 4.

$\lambda_{0}$ is strictly increasing and continuous function.

Then

$f_{n}\circ\lambda_{n}\to f_{0}\circ\lambda_{0},\ n\to\infty,$ (6.1)

in $D([0,+\infty),{\mathbb{R}}^{m})$.

###### Proof.

See Theorem 13.2.2, p. 430, in Whitt [32]. ∎

######  Corollary 6.2.

Let $(\xi_{n})_{n\geq 0}$ and $(\eta_{n})_{n\geq 1}$ be sequences of
stochastic càdlàg processes with values in ${\mathbb{R}}^{m}$ and
${\mathbb{R}}$ respectively. Assume that

  1. 1.

$(\xi_{n}(t))_{t\geq 0}\Rightarrow(\xi_{0}(t))_{t\geq 0}$, $n\to\infty$ in
$D([0,+\infty),{\mathbb{R}}^{m})$;

  2. 2.

$(\eta_{n})_{n\geq 1}$ are non-negative and non-decreasing processes;

  3. 3.

$(\eta_{n}(t))_{t\geq 0}\Rightarrow(ct)_{t\geq 0},n\to\infty$ in
$D([0,+\infty),{\mathbb{R}})$ where $c>0$ is a constant.

Then

$(\xi_{n}(\eta_{n}(t))_{t\geq 0}\Rightarrow(\xi_{0}(ct))_{t\geq
0},n\to\infty\text{ in }D([0,+\infty),{\mathbb{R}}^{m}).$ (6.2)

###### Proof.

Since the process $(ct)_{t\geq 0}$ is non-random, assumptions of the corollary
are equivalent to the convergence of pairs

$(\xi_{n}(t),\eta_{n}(t))_{t\geq 0}\Rightarrow(\xi(t),ct)_{t\geq 0},$ (6.3)

see Ressel [26].

The statement follows from Lemma 6.1 and the Skorokhod representation theorem,
see for example Kallenberg [15, Theorem 3.30, p. 56]. ∎

######  Remark 6.3.

Since all $\eta_{n}$ are non-decreasing and the limit is non-random, condition
3 of the corollary is equivalent to weak convergence (or convergence in
probability) $\eta_{n}(t)\to ct,n\to\infty$ for any fixed $t\geq 0$.

Next lemma gives a reverse result: convergence of compositions yields
convergence of functions. We formulate the result for functions and then a
corollary for processes.

######  Lemma 6.4.

[12, Lemma 3.5] Let $(f_{n})_{n\geq 0}\subset D([0,\infty),{\mathbb{R}}^{m})$
and $(\mu_{n})_{n\geq 0}\subset D([0,\infty),{\mathbb{R}})$, $\mu_{n}$ be non
negative and non decreasing. Assume that, for all $T>0$,

$\lim_{n\to\infty}\sup_{t\in[0,\,T]}|\mu_{n}(t)-t|=0$ (6.4)

and

$\lim_{n\to\infty}f_{n}\circ\mu_{n}=f_{0}$ (6.5)

in the $J_{1}$-topology in $D([0,\infty),{\mathbb{R}}^{m})$. For
$n\in{\mathbb{N}}$, denote by $(t_{k}^{(n)})_{k}$ the points of discontinuity
of $\mu_{n}$ and put $u_{k}^{(n)}:=\mu_{n}(t_{k}^{(n)}-)$ and
$v_{k}^{(n)}:=\mu_{n}(t_{k}^{(n)})$. If, in addition to (6.4) and (6.5), for
all $T>0$,

$\lim_{n\to\infty}\sup_{k}\sup_{s\in[u_{k}^{(n)},v_{k}^{(n)})\cap[0,\,T]}|f_{n}(s)-f_{n}(u_{k}^{(n)}-)|=0,$
(6.6)

then

$\lim_{n\to\infty}f_{n}=f_{0}$

in the $J_{1}$-topology in $D([0,\infty),{\mathbb{R}}^{m})$.

######  Remark 6.5.

If $\mu_{n}$ are continuous, $\mu_{n}(0)=0$, and $\mu_{n}(\infty)=\infty,$
then the result of the lemma follows from Lemma 6.1. Indeed, let
$\lambda_{n}(t)=\mu_{n}^{-1}(t):=\inf\\{s\geq 0\colon\mu_{n}(s)\geq t\\}.$
Then $\mu_{n}\circ\lambda_{n}=\text{id}$. So
$f_{n}=(f_{n}\circ\mu_{n})\circ\lambda_{n}$. It can be easily seen that
$\lambda_{n}$ locally uniformly converges to $t$, so all conditions of Lemma
6.1 are satisfied. New features appear if $\mu_{n}$ may be discontinuous, so
the compositions $f_{n}\circ\mu_{n}$ do not use parts of $f_{n}$ paths
corresponding to jumps of $\mu_{n}.$

The proof of the next corollary is obtained with the help of Skorokhod’s
representation theorem similarly to the proof of Corollary 6.2.

######  Corollary 6.6.

Let $(\xi_{n})_{n\geq 0}$ and $(\eta_{n})_{n\geq 1}$ be sequences of
stochastic càdlàg processes with values in ${\mathbb{R}}^{m}$ and
${\mathbb{R}}$ respectively such that $\eta_{n}$, $n\geq 1$, are non negative
and non decreasing. Assume that for all $T>0$, we have locally uniform
convergence in probability

$\sup_{t\in[0,T]}|\eta_{n}(t)-t|\overset{{\mathop{\rm
P}}}{\to}0,{n\to\infty},$ (6.7)

and convergence in distribution

$\xi_{n}\circ\eta_{n}\Rightarrow\xi_{0},\ {n\to\infty},\ \text{ in
}D([0,\infty),{\mathbb{R}}^{m}).$ (6.8)

For $n\in{\mathbb{N}}$, denote by $(t_{k}^{(n)})_{k}$ the points of
discontinuity of $\eta_{n}$ and put $u_{k}^{(n)}:=\eta_{n}(t_{k}^{(n)}-)$ and
$v_{k}^{(n)}:=\eta_{n}(t_{k}^{(n)})$. If, in addition to (6.7) and (6.8), for
all $T>0$, we have the uniform convergence in probability

$\sup_{k}\sup_{s\in[u_{k}^{(n)},v_{k}^{(n)})\cap[0,\,T]}|\xi_{n}(s)-\xi_{n}(u_{k}^{(n)}-)|\overset{{\mathop{\rm
P}}}{\to}0,\ {n\to\infty},$ (6.9)

then

$\displaystyle\xi_{n}\Rightarrow\xi_{0},\ {n\to\infty},$ (6.10)

in $D([0,\infty),{\mathbb{R}}^{m})$.

###### Proof of Theorem 3.5.

The proof of the Theorem will be divided into four steps.

Step 1. We will employ Theorem 3.1 for the convergence to the WBM and
construct Markov chains $\mathcal{X}^{n}=(R^{n},l^{n})$, $n\geq 1$, on
$\mathbb{Z}\times\\{-,+\\}$. The original Markov chain $X^{n}$ on
${\mathbb{Z}}$ will be obtained from $\mathcal{X}^{n}$ by two transformations.

The transition probabilities of $\mathcal{X}^{n}$ do not depend on $n$ and are
defined as follows. If $R^{n}(0)=x\in{\mathbb{N}}$, i.e., if $R^{n}(0)$ is on
the positive side, then we set

$\displaystyle{\mathop{\rm P}}((R^{n}(1),l^{n}(1))=(y,+)\ |\
(R^{n}(0),l^{n}(0))=(x,+)):={\mathop{\rm P}}(\xi_{+}=y-x),$ (6.11)
$\displaystyle{\mathop{\rm P}}((R^{n}(1),l^{n}(1))=(y,-)\ |\
(R^{n}(0),l^{n}(0))=(x,-)):={\mathop{\rm P}}(-\xi_{-}=y-x),$

that is the distribution of a jump is equal to the distribution of $\xi_{+}$
if the label is $+$ and $(-\xi_{-})$ if the label is $-$.

Assume that $R^{n}(0)=x\in{\mathbb{Z}}\setminus\mathbb{N}$, i.e., the Markov
chain $\mathcal{X}^{n}$ starts on the membrane.

If $R^{n}(0)\leq-2d-1$, then we set

$\displaystyle R^{n}(1):=-R^{n}(0)-2d,\quad l^{n}(1):=-l^{n}(0),$ (6.12)

In other words, the random walk changes its label and and the radius gets
reflected symmetrically with respect to the point $-d$. Notice that the new
radius lies in ${\mathbb{N}}$.

Recall $\widetilde{\tau}_{0}$ defined in (LABEL:eq:defn_of_exits_entra). If
$R^{n}(0)=x\in\\{-2d,\dots,0\\}$ so that $x+d\in\\{-d,\dots,d\\}$, and
$y\in{\mathbb{N}}$, then we set

$\displaystyle{\mathop{\rm P}}((R^{n}(1),l^{n}(1)):=(y,+)\ |\
(R^{n}(0),l^{n}(0))=(x,+)):={\mathop{\rm P}}(X(\widetilde{\tau}_{0})=y+d\
|X(0)=x+d),$ (6.13) $\displaystyle{\mathop{\rm P}}((R^{n}(1),l^{n}(1)):=(y,-)\
|\ (R^{n}(0),l^{n}(0))=(x,+)):={\mathop{\rm P}}(X(\widetilde{\tau}_{0})=-y-d\
|X(0)=x+d),$ $\displaystyle{\mathop{\rm P}}((R^{n}(1),l^{n}(1)):=(y,-)\ |\
(R^{n}(0),l^{n}(0))=(x,-)):={\mathop{\rm P}}(X(\widetilde{\tau}_{0})=-y-d\
|X(0)=-x-d),$ $\displaystyle{\mathop{\rm P}}((R^{n}(1),l^{n}(1)):=(y,+)\ |\
(R^{n}(0),l^{n}(0))=(x,-)):={\mathop{\rm P}}(X(\widetilde{\tau}_{0})=y+d\
|X(0)=-x-d).$

We define the initial condition $\mathcal{X}^{n}(0)=(R^{n}(0),l^{n}(0))$ as
follows:

$\displaystyle(R^{n}(0),l^{n}(0))=\begin{cases}(X^{n}(0)-d,+),&\text{ if
}X^{n}(0)\geq-d,\\\ (-X^{n}(0)-d,-),&\text{ if }X^{n}(0)<-d,\end{cases}$
(6.14)

where $(X^{n}(k))_{k\geq 0}$ is the original perturbed random walk.

Then

$\displaystyle\Big{(}\frac{{R^{n}(0)}}{v_{-}\sqrt{n}}1\\!\\!\,{\rm
I}_{l^{n}(0)=-},\frac{{R^{n}(0)}}{v_{+}\sqrt{n}}1\\!\\!\,{\rm
I}_{l^{n}(0)=+}\Big{)}\Rightarrow(x_{-},x_{+}),\ n\to\infty,$ (6.15)

where $x_{+}=x\vee 0$, $x_{-}=(-x)\vee 0$, and $x$ is from (3.36).

Theorem 3.3 and Corollary 2.3 imply that

$\displaystyle\varphi\Big{(}\frac{l^{n}(n\cdot)R^{n}([n\cdot])}{\sqrt{n}}\Big{)}\Rightarrow
W^{\text{skew}}_{\gamma}(\cdot,x),\quad n\to\infty,$ (6.16)

where $\varphi$ is defined in (1.8). The permeability parameter $\gamma$
equals

$\displaystyle\gamma$ $\displaystyle=\frac{{{\mathop{\rm
E}}}_{\pi_{\text{entrance}}}(R(0)-R(\sigma))(v_{+}^{-1}1\\!\\!\,{\rm
I}_{l(0)=+}-v_{-}^{-1}1\\!\\!\,{\rm I}_{l(0)=-})}{{{\mathop{\rm
E}}}_{\pi_{\text{entrance}}}(R(0)-R(\sigma))v_{l(0)}^{-1}}$ (6.17)

see Theorem 3.3 and Corollary 2.3. The stopping time $\sigma$ was defined in
(3.8) and the stationary measure $\pi_{\text{entrance}}$ was defined in Lemma
3.1.

Therefore, we also have the convergence

$\displaystyle\varphi\Big{(}\frac{l^{n}([n\cdot])(R^{n}([n\cdot])+d)}{\sqrt{n}}\Big{)}\Rightarrow
W^{\text{skew}}_{\gamma}(\cdot,x),\quad n\to\infty.$ (6.18)

Step 2. Note that the mapping $(R,l)\mapsto l\cdot(R+d)$ is a bijection of the
sets $\mathbb{N}\times\\{+\\}$ and $\\{d+1,d+2,\dots\\}$ and also of the sets
$\mathbb{N}\times\\{-\\}$ and $\\{\dots,-d-2,-d-1\\}$.

Notice the following behavior of the sequence
$(l^{n}(\cdot)(R^{n}(\cdot)+d))$. Assume that for some $k$ the sequence
$R^{n}$ overjumps the set $\\{-2d,\dots,0\\}$ from above:

$\displaystyle R^{n}(k)>0\quad\text{and}\quad R^{n}(k+1)<-2d.$ (6.19)

Then by definition of transition probabilities (6.12) we have

$\displaystyle l^{n}(k)(R^{n}(k)+d)=l^{n}(k+1)(R^{n}(k+1)+d),$ (6.20)

i.e., in this case the sequence $(l^{n}(\cdot)(R^{n}(\cdot)+d))$ spends two
time steps at the same point. Let us construct a new sequence
$(\widehat{X}^{n}(k))_{k\geq 0}$ from the sequence
$(l^{n}(k)(R^{n}(k)+d))_{k\geq 0}$ that skips one of these steps.

We define a random time change as follows. Set

$\displaystyle\widehat{\lambda}_{n}(0):=0,$ (6.21)
$\displaystyle\widehat{\lambda}_{n}(k):=\sum_{j=1}^{k}\Big{(}1-1\\!\\!\,{\rm
I}(R^{n}(k+1)>0,\ R^{n}(k)<-2d)\Big{)},$
$\displaystyle\widehat{\lambda}_{n}^{-1}(k):=\inf\\{j\geq
0\colon\widehat{\lambda}_{n}(j)=k\\},$
$\displaystyle\widehat{X}^{n}(k):=l^{n}(\widehat{\lambda}_{n}^{-1}(k))(R^{n}(\widehat{\lambda}_{n}^{-1}(k))+d).$

Notice that $\widehat{X}^{n}$ is a Markov chain with transition probabilities

$\displaystyle{\mathop{\rm P}}(\widehat{X}^{n}(1)=j\ |\
\widehat{X}^{n}(0)=i)=\begin{cases}{\mathop{\rm P}}(X(1)=j\ |\
X(0)=i),&|i|>d,\\\ {\mathop{\rm P}}(X(\hat{\tau}_{0})=j\ |\ X(0)=i),&|i|\leq
d,\end{cases}$ (6.22)

where $(X(k))_{k\geq 0}$ is the original perturbed random walk and
$\widehat{\tau}_{0}:=\inf\\{k\geq 0\colon|X(k)|>d\\}$, see also
(LABEL:eq:defn_of_exits_entra).

Let us show that the number of skipped steps is negligible as $n\to\infty$ and
convergence

$\displaystyle\varphi\Big{(}\frac{\widehat{X}^{n}([n\cdot])}{\sqrt{n}}\Big{)}\Rightarrow
W^{\text{skew}}_{\gamma}(\cdot,x),\quad n\to\infty.$ (6.23)

holds true.

Indeed, we have the inequality

$\displaystyle{}[nt]-\widehat{\lambda}_{n}([nt])\leq\sqrt{n}L^{n}(t)\max(v_{-},v_{+}),\quad
t>0,$ (6.24)

where $L^{n}$ is defined in (3.29). It follows from the weak convergence of
$(L^{n}(\cdot))$ established Theorem 3.3, that for all $t>0$

$\displaystyle\Big{|}\frac{\widehat{\lambda}_{n}([nt])}{n}-t\Big{|}\leq\frac{L^{n}(t)\max(v_{-},v_{+})}{\sqrt{n}}+\frac{1}{n}\overset{{\mathop{\rm
P}}}{\to}0,\ n\to\infty.$ (6.25)

Hence for any $T>0$ we have uniform convergence

$\displaystyle\sup_{t\in[0,T]}\Big{|}\frac{\widehat{\lambda}_{n}([nt])}{n}-t\Big{|}\overset{{\mathop{\rm
P}}}{\to}0,\ n\to\infty,$ (6.26)

and by monotonicity of $t\mapsto\widehat{\lambda}_{n}([nt])$

$\displaystyle\sup_{t\in[0,T]}\Big{|}\frac{\widehat{\lambda}_{n}^{-1}([nt])}{n}-t\Big{|}\overset{{\mathop{\rm
P}}}{\to}0,\ n\to\infty.$ (6.27)

Therefore convergence (6.23) follows by Corollary 6.2.

Step 3. Now we consider the original Markov chain $(X^{n}(k))_{k\geq 0}$ and
define a random time change that removes times that $X^{n}$ spends in the set
$\\{-d,\dots,d\\}$ after entering this set. Denote

$\displaystyle\lambda_{n}(0)=0,$ (6.28)
$\displaystyle\lambda_{n}(k):=\sum_{j=1}^{k}\Big{(}1\\!\\!\,{\rm
I}_{|X^{n}(j)|>d}+1\\!\\!\,{\rm I}_{|X^{n}(j)|\leq d,\
|X^{n}(j-1)|>d}\Big{)},\quad k\geq 1,$
$\displaystyle\lambda^{-1}_{n}(k):=\inf\\{j\geq 0\colon\lambda_{n}(j)\geq
k\\},\quad k\geq 0.$

Then it is easy to see that

$\displaystyle\Big{(}\widehat{X}^{n}(k)\Big{)}_{k\geq
0}\overset{\mathrm{d}}{=}\Big{(}X^{n}(\lambda_{n}^{-1}(k))\Big{)}_{k\geq 0}.$
(6.29)

From now on, without loss of generality we assume that

$\displaystyle\Big{(}\frac{\widehat{X}^{n}([nt])}{\sqrt{n}}\Big{)}_{t\geq
0}=\Big{(}\frac{X^{n}(\lambda_{n}^{-1}([nt]))}{\sqrt{n}}\Big{)}_{t\geq 0}.$
(6.30)

Step 4. In order prove the weak convergence we will apply Corollary 6.6 and
use the following representation:

$\displaystyle\Big{(}\frac{X^{n}(\lambda_{n}^{-1}([nt]))}{\sqrt{n}}\Big{)}_{t\geq
0}=\Big{(}\frac{X^{n}(n\cdot)}{\sqrt{n}}\circ{\eta_{n}(t)}\Big{)}_{t\geq 0},$
(6.31)

where $\eta_{n}(t)=\frac{\lambda_{n}^{-1}([nt])}{n}$. Convergence (6.9) in
Corollary 6.6 holds true because the supremum corresponding to (6.9) does not
exceed $\frac{2d}{\sqrt{n}}$. Hence it is left to check the condition (6.7).

Let

$\displaystyle\widehat{N}^{n}(k)=\sum_{j=0}^{k}1\\!\\!\,{\rm
I}_{|\widehat{X}^{n}(j)|\leq d}$ (6.32)

be the number of visits of $\widehat{X}^{n}$ to the membrane, i.e., to the set
$\\{-d,\dots,d\\}$. We define

$\displaystyle\widehat{\sigma}_{0}^{n}:=0,$ (6.33)
$\displaystyle\widehat{\tau}_{k}^{n}:=\inf\\{j\geq\widehat{\sigma}_{k}^{n}\colon\
|X^{n}(j)|>d\\},$
$\displaystyle\widehat{\sigma}_{k+1}^{n}=\inf\\{j>\widehat{\tau}_{k}^{n}\colon|X^{n}(j)|\leq
d\\},\quad k\geq 0.$

Denote the time spent in the membrane by

$\displaystyle\zeta^{n}_{k}:=\widehat{\tau}_{k}^{n}-\widehat{\sigma}_{k}^{n}.$
(6.34)

Due to Assumption B2, the random variables $\zeta^{n}_{k}$ have finite moments
of all orders. We note that

$\displaystyle
k-\lambda_{n}(k)\leq\sum_{j=0}^{\widehat{N}^{n}(k)}\zeta^{n}_{j},$ (6.35)

so that

$\displaystyle\Big{|}t-\frac{\lambda_{n}([nt])}{n}\Big{|}\leq\frac{1}{n}+\frac{1}{n}\sum_{j=0}^{\widehat{N}^{n}([nt])}\zeta^{n}_{j}.$
(6.36)

Note that

$\displaystyle\frac{\widehat{N}^{n}([nt])}{\sqrt{n}}\leq
L^{n}(t)\max(v_{-},v_{+}),$ (6.37)

where $L^{n}$ is defined in (3.29) (compare with (6.24)). Similarly to (6.25),
for any $t\geq 0$ we have

$\displaystyle\frac{\widehat{N}^{n}([nt])}{n}\stackrel{{\scriptstyle{\mathop{\rm
P}}}}{{\to}}0,\quad n\to\infty.$ (6.38)

For $|x|\leq d$, denote

$\displaystyle f(x):={\mathop{\rm E}}[\zeta^{n}_{0}|X^{n}(0)=x]$ (6.39)

and note that $f(\cdot)$ does not depend on $n$ and is bounded. We write:

$\displaystyle\sum_{j=0}^{\widehat{N}^{n}([nt])}\zeta^{n}_{j}=\sum_{j=0}^{\widehat{N}^{n}([nt])}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}+\sum_{j=0}^{\widehat{N}^{n}([nt])}f(X^{n}(\widehat{\sigma}^{n}_{j})).$
(6.40)

Therefore we get

$\displaystyle\frac{1}{n}\sum_{j=0}^{\widehat{N}^{n}([nt])}f(X^{n}(\widehat{\sigma}^{n}_{j}))\stackrel{{\scriptstyle{\mathop{\rm
P}}}}{{\to}}0,\quad n\to\infty.$ (6.41)

The random sequence

$\displaystyle
k\mapsto\sum_{j=0}^{k}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}$
(6.42)

is a martingale, so that by Doob’s inequality for any $N\geq 1$

$\displaystyle{\mathop{\rm E}}\max_{0\leq k\leq
N}\Big{|}\sum_{j=0}^{k}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}\Big{|}^{2}$
$\displaystyle\leq 4{\mathop{\rm
E}}\sum_{j=0}^{N}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}^{2}$
(6.43) $\displaystyle\leq 4(N+1)\max_{|x|\leq d}{\mathop{\rm
E}}\Big{[}\Big{(}\zeta^{n}_{0}-f(X^{n}(\widehat{\sigma}^{n}_{0}))\Big{)}^{2}\Big{|}X^{n}(0)=x\Big{]}$
$\displaystyle\leq C(N+1),$

for some $C>0$. Hence for any $\delta>0$

$\displaystyle\frac{1}{n}\Big{|}\sum_{j=0}^{\widehat{N}^{n}([nt])}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}\Big{|}$
$\displaystyle\leq\frac{1}{n}1\\!\\!\,{\rm I}(\widehat{N}^{n}([nt])\leq
n\delta)\Big{|}\sum_{j=0}^{\widehat{N}^{n}([nt])}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}\Big{|}$
(6.44) $\displaystyle+\frac{1}{n}1\\!\\!\,{\rm
I}(\widehat{N}^{n}([nt])>n\delta)\Big{|}\sum_{j=0}^{\widehat{N}^{n}([nt])}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}\Big{|}$
$\displaystyle\leq\frac{1}{n}\max_{0\leq k\leq
n\delta}\Big{|}\sum_{j=0}^{k}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}\Big{|}$
$\displaystyle+\frac{1}{n}1\\!\\!\,{\rm
I}\Big{(}\frac{\widehat{N}^{n}([nt])}{n}>\delta\Big{)}\Big{|}\sum_{j=0}^{\widehat{N}^{n}([nt])}\Big{(}\zeta^{n}_{j}-f(X^{n}(\widehat{\sigma}^{n}_{j}))\Big{)}\Big{|}$

The first term in the latter formula converges to zero in mean square sense
due to (6.43) whereas the second term converges to zero in probability due to
(6.38). Combining all the above estimates yields

$\displaystyle\Big{|}\frac{\lambda_{n}([nt])}{n}-t\Big{|}\stackrel{{\scriptstyle{\mathop{\rm
P}}}}{{\to}}0,\quad n\to\infty.$ (6.45)

Due to monotonicity, for any $T>0$ we have convergence:

$\displaystyle\sup_{t\in[0,T]}\Big{|}\frac{\lambda_{n}([nt])}{n}-t\Big{|}\overset{{\mathop{\rm
P}}}{\to}0,\ n\to\infty,$ (6.46)
$\displaystyle\sup_{t\in[0,T]}\Big{|}\frac{\lambda_{n}^{-1}([nt])}{n}-t\Big{|}\overset{{\mathop{\rm
P}}}{\to}0,\ n\to\infty.$

Hence, the application of Corollary 6.6 implies the desired convergence

$\displaystyle\frac{X^{n}([n\cdot])}{\sqrt{n}}\Rightarrow
W^{\text{skew}}_{\gamma}(\cdot,x),\quad n\to\infty.$ (6.47)

Inspecting the construction steps of the Markov chains $\mathcal{X}^{n}$,
$\widehat{X}^{n}$ and $X^{n}$ we conclude that the parameter $\gamma$ defined
in (6.17) coincides with $\gamma$ in (3.38). ∎

##  7 Proofs of Theorem 1.1, Corollary 1.3 and Theorem 1.4

###### Proof of Theorem 1.1.

It is sufficient to check that condition $\mathbf{B}_{3}$ of Theorem 3.5
follows from conditions $2^{\circ}$ and $3^{\circ}$ of Theorem 1.1. Indeed,
for $|x|\leq d$ we have

$\displaystyle{\mathop{\rm E}}\Big{[}|X(\tilde{\tau}_{0})\,|\,X(0)=x\Big{]}$
$\displaystyle=\sum_{|y|\leq d}\sum_{k=1}^{\infty}{\mathop{\rm
P}}\Big{(}\widetilde{\tau}_{0}=k,X(k-1)=y\,|\,X(0)=x\Big{)}{\mathop{\rm
E}}[|X(k)|\,|\,\widetilde{\tau}_{0}=k,X(k-1)=y]$ (7.1)
$\displaystyle=\sum_{|y|\leq d}\sum_{k=1}^{\infty}{\mathop{\rm
P}}\Big{(}\widetilde{\tau}_{0}=k,X(k-1)=y\,|\,X(0)=x\Big{)}{\mathop{\rm
E}}[|X(1)|\,|\,\widetilde{\tau}_{0}=1,X(0)=y]$ $\displaystyle=\sum_{|y|\leq
d}\sum_{k=1}^{\infty}{\mathop{\rm
P}}\Big{(}\widetilde{\tau}_{0}=k,X(k-1)=y\,|\,X(0)=x\Big{)}\frac{{\mathop{\rm
E}}[|X(1)|1\\!\\!\,{\rm I}_{|X(1)|>d}\,|\,X(0)=y]}{{\mathop{\rm
P}}(|X(1)|>d\,|\,X(0)=y)}$ $\displaystyle\leq\sum_{|y|\leq
d}\frac{{\mathop{\rm E}}[|X(1)|1\\!\\!\,{\rm
I}_{|X(1)|>d}\,|\,X(0)=y]}{{\mathop{\rm P}}(|X(1)|>d\,|\,X(0)=y)}<\infty.$

In the last line we use the convention that $\frac{0}{0}=0$. ∎

###### Proof of Corollary 1.3.

Denote for brevity the inverse function

$\psi(x):=\varphi^{-1}(x)=x(v_{+}1\\!\\!\,{\rm I}_{x\geq 0}+v_{-}1\\!\\!\,{\rm
I}_{x<0})$ (7.2)

Recall the SDE (2.12) and apply the Itô-Tanaka formula with symmetric
semimartingale local time to $\psi(W^{\text{skew}}_{\gamma})$, see [27,
Exercise 1.25, Chapter VI]:

$\displaystyle Y(t)=\psi(W^{\text{skew}}_{\gamma}(t))$
$\displaystyle:=x+\int_{0}^{t}\Big{(}v_{+}1\\!\\!\,{\rm
I}_{W^{\text{skew}}_{\gamma}(s)>0}+v_{-}1\\!\\!\,{\rm
I}_{W^{\text{skew}}_{\gamma}(s)<0}\Big{)}\,\mathrm{d}W(s)$ (7.3)
$\displaystyle+\frac{1}{2}\gamma(v_{+}+v_{-})L_{0}^{W^{\text{skew}}_{\gamma}}(t)+\frac{1}{2}(v_{+}-v_{-})L_{0}^{W^{\text{skew}}_{\gamma}}(t)$
$\displaystyle=$ $\displaystyle x+\int_{0}^{t}\Big{(}v_{+}1\\!\\!\,{\rm
I}_{Y(s)>0}+v_{-}1\\!\\!\,{\rm I}_{Y(s)<0})\Big{)}\,\mathrm{d}W(s)$
$\displaystyle+\frac{1}{2}\gamma(v_{+}+v_{-})L_{0}^{W^{\text{skew}}_{\gamma}}(t)+\frac{1}{2}(v_{+}-v_{-})L_{0}^{W^{\text{skew}}_{\gamma}}(t)$

and

$\displaystyle|Y(t)|=|\psi(W^{\text{skew}}_{\gamma}(t))|$
$\displaystyle:=|x|+\int_{0}^{t}\Big{(}v_{+}1\\!\\!\,{\rm
I}_{W^{\text{skew}}_{\gamma}(s)>0}+v_{-}1\\!\\!\,{\rm
I}_{W^{\text{skew}}_{\gamma}(s)<0}\Big{)}\,\mathrm{d}W(s)$ (7.4)
$\displaystyle+\frac{1}{2}\gamma(v_{+}-v_{-})L_{0}^{W^{\text{skew}}_{\gamma}}(t)+\frac{1}{2}(v_{+}+v_{-})L_{0}^{W^{\text{skew}}_{\gamma}}(t).$

On the other hand, the process $Y$ is a continuous semimartingale itself.
Hence, the Itô-Tanaka formula applied to the process $Y$ yields

$\displaystyle|Y(t)|$ $\displaystyle=|x|+\int_{0}^{t}\Big{(}1\\!\\!\,{\rm
I}_{Y(s)>0}-1\\!\\!\,{\rm I}_{Y(s)<0}\Big{)}\,\mathrm{d}Y(s)+L_{0}^{Y}(t)$
(7.5) $\displaystyle=\int_{0}^{t}\Big{(}v_{+}1\\!\\!\,{\rm
I}_{W^{\text{skew}}_{\gamma}(s)>0}+v_{-}1\\!\\!\,{\rm
I}_{W^{\text{skew}}_{\gamma}(s)<0}\Big{)}\,\mathrm{d}W(s)+L_{0}^{Y}(t),$

where $L_{0}^{Y}(t)$ is the symmetric semimartingale local time defined by

$\displaystyle L_{0}^{Y}(t)=\lim_{\varepsilon\to
0+}\frac{1}{2\varepsilon}\int_{0}^{t}1\\!\\!\,{\rm
I}_{|Y(s)|<\varepsilon}\mathrm{d}\langle Y,Y\rangle_{s}=\lim_{\varepsilon\to
0+}\frac{1}{2\varepsilon}\int_{0}^{t}\Big{(}v_{+}^{2}1\\!\\!\,{\rm
I}_{Y(s)>0}+v_{-}^{2}1\\!\\!\,{\rm I}_{Y(s)<0}\Big{)}\,\mathrm{d}s.$ (7.6)

Comparing the representations (7.4) and (7.5) we conclude that

$\displaystyle
L_{0}^{Y}(t)=\frac{1}{2}\Big{[}\gamma(v_{+}-v_{-})+(v_{+}+v_{-})\Big{]}L_{0}^{W^{\text{skew}}_{\gamma}}(t).$
(7.7)

Substituting this formula into (7.3) and recalling that $Y$ spends zero time
at zero we get (1.9). ∎

###### Proof of Theorem 1.4.

Let $\mathcal{X}$ be a Markov chain on $\mathcal{N}^{m}\cup\\{0\\}$ from
Theorem 1.4. Since the processes from Theorems 1.4 and 3.3 live on different
state spaces, first we construct a Markov chain $\widehat{\mathcal{X}}$ on
$\mathcal{Z}^{m}$ related to $\mathcal{X}$ that satisfy assumptions of Theorem
3.3.

In assumption $\mathbf{A}_{1}$, let the distribution of $\xi^{(i)}$ coincide
with the distribution of $\xi^{(i)}_{k}$, $i=1,\dots,m$. In particular,
$\widehat{\mathcal{X}}$ may visit the set
$\\{\dots,-1,0\\}\times\\{1,\dots,m\\}$.

The jumps from the state $(0,i)$ are defined as follows:

$\displaystyle{\mathop{\rm
P}}\big{(}\widehat{\mathcal{X}}(1)=(\widehat{y},j)\,|\,\widehat{\mathcal{X}}(0)=(0,i)\big{)}:={\mathop{\rm
P}}\big{(}\mathcal{X}(1)=(\widehat{y},j)\,|\,\mathcal{X}(0)=0\big{)},\quad
i,j=1,\dots,m.$ (7.8)

For $\widehat{x}<0$ and $\widehat{y}\in{\mathbb{N}}$ we set

$\displaystyle{\mathop{\rm
P}}\big{(}\widehat{\mathcal{X}}(1)=(\widehat{y},j)\,|\,\widehat{\mathcal{X}}(0)=(\widehat{x},i)\big{)}:={\mathop{\rm
P}}(\mathcal{X}(1)=(\widehat{y},j)\,|\,R(0)+\xi^{(i)}_{1}=\widehat{x},l(0)=i\big{)},$
(7.9)

where $\mathcal{X}(0)=(R(0),l(0))$.

Let the initial distributions of $\widehat{\mathcal{X}}(0)$ and
$\mathcal{X}(0)$ coincide in the following sense:

$\displaystyle{\mathop{\rm
P}}\big{(}\widehat{\mathcal{X}}(0)=(\widehat{y},j)\big{)}:={\mathop{\rm
P}}(\mathcal{X}(0)=(\widehat{y},j)\big{)},\quad\widehat{y}\in{\mathbb{N}},\
j=1,\dots,m,$ (7.10) $\displaystyle{\mathop{\rm
P}}\big{(}\widehat{\mathcal{X}}(0)=(0,1)\big{)}:={\mathop{\rm
P}}(\mathcal{X}(0)=0\big{)},$

and ${\mathop{\rm P}}(\widehat{\mathcal{X}}(0)=(\widehat{y},j))=0$ in all
other cases.

According to Theorem 3.3, the scaling limit of $\widehat{\mathcal{X}}$ is a
WBM starting at 0.

Let

$\displaystyle\widehat{\lambda}(k)=\sum_{i=1}^{k}1\\!\\!\,{\rm
I}(\widehat{R}(i)\geq 0).$ (7.11)

The random time change $\widehat{\lambda}^{-1}(\cdot)$ removes time instants
when $\widehat{\mathcal{X}}$ visits the set
$\\{\dots,-1,0\\}\times\\{1,\dots,m\\}$. It follows from the construction of
$\widehat{\mathcal{X}}$ and $\widehat{\lambda}$ that

$\displaystyle\Big{(}\widehat{R}(\widehat{\lambda}^{-1}(k))1\\!\\!\,{\rm
I}_{l(\widehat{\lambda}^{-1}(k))=i},\ i=1,\dots,m\Big{)}_{k\geq
0}\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\Big{(}R(k)1\\!\\!\,{\rm
I}_{l(k)=i},\ i=1,\dots,m\Big{)}_{k\geq 0}.$ (7.12)

Following the argument of Step 2 in the proof of Theorem 3.5 we see that
scaling limit of the original Markov chain $\mathcal{X}$ is the same WBM as
for the scaling limit of $\widehat{\mathcal{X}}$. ∎

## References

  * [1] M. T. Barlow, J. Pitman, and M. Yor.  On Walsh’s Brownian motions.  Séminaire de probabilités de Strasbourg, 23:275–293, 1989\.
  * [2] P. Billingsley.  Convergence of Probability Measures.  John Wiley & Sons, New York, second edition, 1999.
  * [3] V. Bogdanskii, I. Pavlyukevich, and A. Pilipenko.  Limit behaviour of random walks on $\mathbb{Z}^{m}$ with two-sided membrane.  ESAIM: Probability & Statistics, 26, 2022.
  * [4] C. Dong, O. Iksanov, and A. Pilipenko.  On a discrete approximation of a skew stable Lévy process.  arXiv preprint arXiv:2302.07298, 2023.
  * [5] N. Enriquez and Y. Kifer.  Markov chains on graphs and Brownian motion.  Journal of Theoretical Probability, 14(2):495–510, 2001.
  * [6] W. Feller.  An Introduction to Probability Theory and its Applications. Vol. II.  John Wiley & Sons, Inc., New York, 1966.
  * [7] M. Hairer and C. Manson.  Periodic homogenization with an interface: The one-dimensional case.  Stochastic Processes and Their Applications, 120(8):1589–1605, 2010\.
  * [8] J. M. Harrison and L. A. Shepp.  On skew Brownian motion.  The Annals of Probability, 9(2):309–313, 1981.
  * [9] I. S. Helland.  Convergence to diffusions with regular boundaries.  Stochastic Processes and their Applications, 12(1):27–58, 1982\.
  * [10] A. Iksanov and A. Pilipenko.  A functional limit theorem for locally perturbed random walks.  Probability and Mathematical Statistics, 36(2):353–368, 2016.
  * [11] A. Iksanov and A. Pilipenko.  On a skew stable Lévy process.  Stochastic Processes and their Applications, 156:44–68, 2023.
  * [12] A. Iksanov, A. Pilipenko, and B. Povar.  Functional limit theorems for random walks perturbed by positive alpha-stable jumps.  Bernoulli, 29(2):1638–1662, 2023.
  * [13] K. Itô and H. P. McKean, Jr.  Diffusion Processes and their Sample Paths.  Springer, 1965.
  * [14] J. Jacod and A. N. Shiryaev.  Limit Theorems for Stochastic Processes, volume 288 of Grundlehren der Mathematischen Wissenschaften.  Springer, Berlin, second edition, 2003.
  * [15] O. Kallenberg.  Foundations of Modern Probability.  Probability and its Applications. Springer, New York, second edition, 2002.
  * [16] J. Keilson and J. A. Wellner.  Oscillating Brownian motion.  Journal of Applied Probability, 15(2):300–310, 1978.
  * [17] A. Mijatović and V. Vysotsky.  Stationary entrance Markov chains, inducing, and level-crossings of random walks.  arXiv:1808.05010, 2018.
  * [18] R. A. Minlos and E. A. Zhizhina.  Limit diffusion process for a non-homogeneous random walk on a one-dimensional lattice.  Russian Mathematical Surveys, 52(2):327–340, 1997.
  * [19] H.-L. Ngo and M. Peigné.  Limit theorem for perturbed random walks.  Theory of Stochastic Processes, 24(40):61–78, 2019.
  * [20] A. Pilipenko.  An Introduction to Stochastic Differential Equations with Reflection, volume 11 of Lectures in Pure and Applied Mathematics.  Potsdam University Press, Potsdam, 2014.
  * [21] A. Pilipenko and Yu. Pryhod’ko.  Limit behavior of symmetric random walks with a membrane.  Theory of Probability and Mathematical Statistics, 85:93–105, 2012\.
  * [22] A. Pilipenko and A. Sarantsev.  Boundary approximation for sticky jump-reflected processes on the half-line.  arXiv:2303.02771, 2023.
  * [23] A. Y. Pilipenko and L. A. Sakhanenko.  On a limit behavior of a one-dimensional random walk with non-integrable impurity.  Theory of Stochastic Processes, 20(2):97–104, 2015.
  * [24] A. Yu. Pilipenko and Yu. E. Prikhod’ko.  On the limit behavior of a sequence of Markov processes perturbed in a neighborhood of the singular point.  Ukrainian Mathematical Journal, 67(4):564–583, 2015.
  * [25] A. Yu. Pilipenko and O. O. Prykhodko.  On a limit behaviour of a random walk penalised in the lower half-plane.  Theory of Stochastic Processes, 25(2):81–88, 2020.
  * [26] P. Ressel.  A topological version of Slutsky’s theorem.  Proceedings of the American Mathematical Society, 85(2):272–274, 1982.
  * [27] D. Revuz and M. Yor.  Continuous Martingales and Brownian Motion, volume 293 of Grundlehren der Mathematischen Wissenschaften.  Springer, Berlin, corrected third printing of the third edition, 2005\.
  * [28] A. V. Skorokhod.  Studies in the Theory of Random Processes.  Dover Publications, 1982.
  * [29] F. Spitzer.  Principles of Random Walk, volume 34 of Graduate Texts in Mathematics.  Springer, New York, second edition, 1976.
  * [30] V. Vysotsky.  Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem.  Stochastic Processes and their Applications, 125(5):1886–1910, 2015\.
  * [31] J. B. Walsh.  A diffusion with a discontinuous local time.  Astérisque, 52(53):37–45, 1978.
  * [32] W. Whitt.  Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues.  Springer, 2002.