Geodesic
Two-boundary problem for a random walk in a random environment
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 3, pp. 417-430 Cet article a éte moissonné depuis la source Math-Net.Ru

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Given a sequence of independent identically distributed pairs of random variables $(p_i,q_i)$, $i\in\mathbf{Z}$, with $p_0+q_0=1$, and $p_0>0$ a.s., $q_0>0$ a.s., one considers a random walk in the random environment $(p_i,q_i)$, $i\in\mathbf{Z}$. This means that, for a fixed random environment, a walking particle transits from the state $i$ either to the state $(i+1)$ with probability $p_i$ or to the state $(i-1)$ with probability $q_i$. It is assumed that $\mathbf{E}\ln (p_0/q_0)=0$, that is, the walk is oscillating. We are concerned with the exit problem of the walk under consideration from the interval $(-\lfloor an\rfloor,\lfloor bn\rfloor)$, where $a$, $b$ are arbitrary positive constants. We find the asymptotics of the exit probability of the walk from the above interval from the right (the left). A limit theorem for the exit time of the walk from this interval is obtained.
Keywords: random walk in random environment, branching process in random environment with immigration, limit theorem. 
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     author = {V. I. Afanasyev},
     title = {Two-boundary problem for a random walk in a random environment},
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     pages = {417--430},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_3_a0/}
}
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V. I. Afanasyev. Two-boundary problem for a random walk in a random environment. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 3, pp. 417-430. http://geodesic.mathdoc.fr/item/TVP_2018_63_3_a0/

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