Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Abstract

Let ξ₁, ξ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional η _n =∑ ⁿ _k=o ζ(S _k ) where S₁=0, S_k=Σ ^k_i=1 ξ_i (k≥1) andθ(x)=1 for x≥0,θ(x) = 0 for x<0. It is readily seen that η_n is the time spent by the random walk S_n, n≥0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (η_n = k) for n → ∞ and k→∞, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n⁻¹ of the probabilities P(τh_n = nx) and P(nx₁ ≤ η_n≤ nx₂) for 0<δ₁, ≤ x = k/n ≤ δ₂<1, 0<δ₁≤x₁<x₂≤δ₂<1.