Small deviation for Random walk with random environment in time

2018/03/23 by You Lv, Lv, You
Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Point processes and geometric inequalities #Probability (math.PR) #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.1803.08772

openalex publication_date 2018/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give the random environment version of Mogul'ski\vı estimation in quenched sense.Assume that \μ\_n∈\bfN (called environment) is a sequence of i.i.d. random probability measures on \bfR.~ Let \X_n\_n∈\bfN be a sequence of independent random variables, where X_n has law μ_n. We set S_n=∑_i=1ⁿX_i. Under some integrability conditions, we show that on the log scale, for any power function f. the decay rate of \bfP_μ(∀_{0≤ i≤ n} S_f(n)+i∈[g(i/n)n^α,h(i/n)n^α]|S_f(n)=x) is e^-cn^1-2α almost surely as n→+∞, where c>0,α∈(0,(1)/(2)), g,h\inC[0,1] (the set of all continuous functions defined on [0,1]), g(s)

Citations

• Brownian motion between two random trajectories
• Branching random walk in random environment with random absorption wall

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