Half-normal approximation for Number of returns to origin of random walks

Abstract

Let (Xn) be a sequence of independent identically distributed random variables with P(X₁=1) = p, P(X₁ = -1) for 0 < p < 1. A random walk is a discrete time stochastic process (Sn) defined by S₀=0 and [Equation] for n ≥1. Kn is called the number of returns to the origin if [Equation] and [Equation]. In case of symmetric random walk, i.e., [Equation] , D bler (2015) showed that the distribution of Kn can be approximated by half-normal distribution and he also gave a uniform bound of this approximation. After that Sama-ae et.al. (2016) gave non-uniform bounds. In this thesis, we improve a non-uniform bound of Sama-ae et.al. In case of asymmetric random walk, i.e., [Equation], we give a distribution of Kn and show that it is not convergent to half-normal distribution