The exponential non-uniform bound on the half-normal approximation for the number of returns to the origin-Reference-Cited by-同舟云学术

The exponential non-uniform bound on the half-normal approximation for the number of returns to the origin

Published:2024 Issue:7 Volume:9 Page:19031-19048
ISSN:2473-6988
Container-title:AIMS Mathematics
language:
Short-container-title:MATH

Author:

Siripraparat Tatpon¹,Jongpreechaharn Suporn²

Affiliation:

1. Department of Social and Applied Science, College of Industrial Technology, King Mongkut's University of Technology North Bangkok, Pracharat 1 Road, Bangkok 10800, Thailand

2. Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, 254 Phayathai Road, Bangkok 10330, Thailand

Abstract

This research explored the number of returns to the origin within the framework of a symmetric simple random walk. Our primary objective was to approximate the distribution of return events to the origin by utilizing the half-normal distribution, which is chosen for its appropriateness as a limit distribution for nonnegative values. Employing the Stein's method in conjunction with concentration inequalities, we derived an exponential non-uniform bound for the approximation error. This bound signifies a significant advancement in contrast to existing bounds, encompassing both the uniform bounds proposed by Döbler [<a href="#b1" ref-type="bibr">1</a>] and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee [<a href="#b2" ref-type="bibr">2</a>], and Siripraparat and Neammanee [<a href="#b3" ref-type="bibr">3</a>].

Publisher

American Institute of Mathematical Sciences (AIMS)

Reference19 articles.

1. C. Döbler, Stein's method for the half-normal distribution with applications to limit theorems related to simple random walk, ALEA, Lat. Am. J. Probab. Math. Stat., 12 (2015), 171–191.

2. A. Sama-ae, N. Chaidee, K. Neammanee, Half-normal approximation for statistics of symmetric simple random walk, Commun. Stat.-Theor. M., 47 (2018), 779–792. https://doi.org/10.1080/03610926.2016.1139129

3. T. Siripraparat, K. Neammanee, A non uniform bound for half-normal approximation of the number of returns to the origin of symmetric simple random walk, Commun. Stat.-Theor. M., 47 (2018), 42–54. https://doi.org/10.1080/03610926.2017.1300286

4. W. Feller, An introduction to probability theory and its applications, 3 Eds., New York: Wiley, 1968.

5. C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, In: Proceeding of the sixth Berkeley symposium on mathematical statistics and probability, Berkeley: University of California Press, 1972,583–602.