A non uniform bound for half-normal approximation of the number of returns to the origin of symmetric simple random walk

Abstract

ABSTRACTLet (Xn) be a sequence of independent identically distributed random variables with . A symmetric simple random walk is a discrete-time stochastic process (Sn)n ⩾ 0 defined by S0 = 0 and Sn = ∑ni = 1Xi for n ⩾ 1. Kn is called the number of returns to the origin if . Döbler (2015) showed that the distribution of Kn can be approximated by half-normal distribution and he also gave a uniform bound in terms of . After that, Sama-ae, Neammanee, and Chaidee (2016) gave a non uniform bound in terms of . Observe that, the exponent of z is 3. In this paper, we improve the exponent of z to be any natural number k which make the better constant than before.