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

作者：Siripraparat Tatpon; Neammanee Kritsana^*

来源：Communications in Statistics - Theory and Methods, 2018, 47(1): 42-54.

DOI：10.1080/03610926.2017.1300286

• Concentration inequality approach
• half-normal distribution
• Stein's method
• symmetric simple random walk
• the number of returns to the origin
• uniform and non uniform bounds
• 60F05

Let (X n) be a sequence of independent identically distributed random variables with P(X 1 = 1) = P(X 1 = -1) = 12. A symmetric simple random walk is a discrete-time stochastic process (S n) n= 0 defined by S 0 = 0 and S n = ni = 1 X i for n = 1. K n is called the number of returns to the origin if K n = |{k. N | 1 = k = n and S k = 0}|. Dobler (2015) showed that the distribution of K n can be approximated by half-normal distribution and he also gave a uniform bound in terms of C v n. After that, Sama-ae, Neammanee, and Chaidee (2016) gave a non uniform bound in terms of C (1+ z) 3v 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.

• 出版日期2018