195 research outputs found

### Branching structure for the transient random walk on a strip in a random environment

An intrinsic branching structure within the transient random walk on a strip
in a random environment is revealed. As applications, which enables us to
express the hitting time explicitly, and specifies the density of the
absolutely continuous invariant measure for the "environments viewed from the
particle".Comment: 16 page

### A note on the passage time of finite state Markov chains

Consider a Markov chain with finite state $\{0, 1, ..., d\}$. We give the
generation functions (or Laplace transforms) of absorbing (passage) time in the
following two situations : (1) the absorbing time of state $d$ when the chain
starts from any state $i$ and absorbing at state $d$; (2) the passage time of
any state $i$ when the chain starts from the stationary distribution supposed
the chain is time reversible and ergodic. Example shows that it is more
convenient compared with the existing methods, especially we can calculate the
expectation of the absorbing time directly

### Scaling limit of the local time of the $(1,L)-$random walk

It is well known (Donsker's Invariance Principle) that the random walk
converges to Brownian motion by scaling. In this paper, we will prove that the
scaled local time of the $(1,L)-$random walk converges to that of the Brownian
motion. The results was proved by Rogers (1984) in the case $L=1$. Our proof is
based on the intrinsic multiple branching structure within the $(1,L)-$random
walk revealed by Hong and Wang (2013)

### Branching structure for an (L-1) random walk in random environment and its applications

By decomposing the random walk path, we construct a multitype branching
process with immigration in random environment for corresponding random walk
with bounded jumps in random environment. Then we give two applications of the
branching structure. Firstly, we specify the explicit invariant density by a
method different with the one used in Br\'emont [3] and reprove the law of
large numbers of the random walk by a method known as the environment viewed
from particles". Secondly, the branching structure enables us to prove a stable
limit law, generalizing the result of Kesten-Kozlov-Spitzer [11] for the
nearest random walk in random environment. As a byproduct, we also prove that
the total population of a multitype branching process in random environment
with immigration before the first regeneration belongs to the domain of
attraction of some \kappa -stable law.Comment: 31 page

### Branching structure for the transient (1;R)-random walk in random environment and its applications

An intrinsic multitype branching structure within the transient (1;R)-RWRE is
revealed. The branching structure enables us to specify the density of the
absolutely continuous invariant measure for the environments seen from the
particle and reprove the LLN with an drift explicitly in terms of the
environment, comparing with the results in Br\'emont (2002).Comment: 25 page

### Tail asymptotic of the stationary distribution for the state dependent (1,R)-reflecting random walk: near critical

In this paper, we consider the $(1,R)$ state-dependent reflecting random walk
(RW) on the half line, allowing the size of jumps to the right at maximal $R$
and to the left only 1. We provide an explicit criterion for positive
recurrence and the explicit expression of the stationary distribution based on
the intrinsic branching structure within the walk. As an application, we obtain
the tail asymptotic of the stationary distribution in the "near critical"
situation

### Limit theorems for the minimal position of a branching random walk in random environment

We consider a branching system of random walk in random environment (in
location) in $\mathbb{N}$. We will give the exact limit value of
$\frac{M_{n}}{n}$, where $M_{n}$ denotes the minimal position of branching
random walk at time $n$. A key step in the proof is to transfer our branching
random walks in random environment (in location) to branching random walks in
random environment (in time), by use of Bramson's "branching processes within a
branching process"

### Scaling limit of the local time of the Sinai's random walk

We prove that the local times of a sequence of Sinai's random walks
convergence to those of Brox's diffusion by proper scaling, which is accord
with the result of Seignourel (2000). Our proof is based on the convergence of
the branching processes in random environment by Kurtz (1979)

### Limit theorems for supercritical MBPRE with linear fractional offspring distributions

We investigate the limit behavior of supercritical multitype branching
processes in random environments with linear fractional offspring distributions
and show that there exists a phase transition in the behavior of local
probabilites of the process affected by strongly and intermediately
supercritical regimes. Some conditional limit theorems can also be obtained
from the representation of generating functions.Comment: 25 page

### Light-tailed behavior of stationary distribution for state-dependent random walks on a strip

In this paper, we consider the state-dependent reflecting random walk on a
half-strip. We provide explicit criteria for (positive) recurrence, and an
explicit expression for the stationary distribution. As a consequence, the
light-tailed behavior of the stationary distribution is proved under
appropriate conditions. The key idea of the method employed here is the
decomposition of the trajectory of the random walk and the main tool is the
intrinsic branching structure buried in the random walk on a strip, which is
different from the matrix-analytic method

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