Stable limit laws for random walk in a sparse random environment I: moderate sparsity

27 Apr 2018
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Buraczewski Dariusz
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Dyszewski Piotr
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Iksanov Alexander
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Marynych Alexander
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Roitershtein Alexander

A random walk in a sparse random environment is a model introduced by
Matzavinos et al. [Electron. J. Probab...21, paper no. 72: 2016] as a
generalization of both a simple symmetric random walk and a classical random
walk in a random environment. A random walk $(X_n)_{n\in \mathbb{N}\cup\{0\}}$
in a sparse random environment $(S_k,\lambda_k)_{k\in\mathbb{Z}}$ is a nearest
neighbor random walk on $\mathbb{Z}$ that jumps to the left or to the right
with probability $1/2$ from every point of $\mathbb{Z}\setminus
\{\ldots,S_{-1},S_0=0,S_1,\ldots\}$ and jumps to the right (left) with the
random probability $\lambda_{k+1}$ ($1-\lambda_{k+1}$) from the point $S_k$,
$k\in\mathbb{Z}$. Assuming that $(S_k-S_{k-1},\lambda_k)_{k\in\mathbb{Z}}$ are
independent copies of a random vector $(\xi,\lambda)\in \mathbb{N}\times (0,1)$
and the mean $\mathbb{E}\xi$ is finite (moderate sparsity) we obtain stable
limit laws for $X_n$, properly normalized and centered, as $n\to\infty$. While
the case $\xi\leq M$ a.s.\ for some deterministic $M>0$ (weak sparsity) was
analyzed by Matzavinos et al., the case $\mathbb{E} \xi=\infty$ (strong
sparsity) will be analyzed in a forthcoming paper.(read more)