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

Tatpon Siripraparat and Kritsana Neammanee

Communications in Statistics - Theory and Methods, 2018, vol. 47, issue 1, 42-54

Abstract: Let (Xn) be a sequence of independent identically distributed random variables with P(X1=1)=P(X1=-1)=12$P(X_1=1)=P(X_1=-1)=\frac{1}{2}$. 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 Kn=|{k∈N|1≤k≤nandSk=0}|$K_{n}=|\lbrace k\in \mathbb {N}|1\le k\le n \ \text{and} \ S_k=0\rbrace |$. 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 Cn$\frac{C}{\sqrt{n}}$. After that, Sama-ae, Neammanee, and Chaidee (2016) gave a non uniform bound in terms of C(1+z)3n$\frac{C}{(1+z)^3\sqrt{n}}$. 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.

Date: 2018
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DOI: 10.1080/03610926.2017.1300286

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