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

Tatpon Siripraparat; Suporn Jongpreechaharn; Tatpon Siripraparat; Suporn Jongpreechaharn

doi:10.3934/math.2024926

AIMS Mathematics

2024, Volume 9, Issue 7: 19031-19048. doi: 10.3934/math.2024926

Previous Article Next Article

Research article

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

Tatpon Siripraparat ¹,
Suporn Jongpreechaharn ^{2
,
,}

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

Received: 10 March 2024 Revised: 13 May 2024 Accepted: 29 May 2024 Published: 06 June 2024
MSC : 60F05

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 ^[1] and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee ^[2], and Siripraparat and Neammanee ^[3].
- half-normal approximation,
- the number of returns to the origin,
- Stein's method,
- symmetric simple random walk
Citation: Tatpon Siripraparat, Suporn Jongpreechaharn. The exponential non-uniform bound on the half-normal approximation for the number of returns to the origin[J]. AIMS Mathematics, 2024, 9(7): 19031-19048. doi: 10.3934/math.2024926

Related Papers:

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 ^[1] and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee ^[2], and Siripraparat and Neammanee ^[3].

References

[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 doi: 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 doi: 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.
[6]	L. H. Y. Chen, Poisson approximation for dependent trials, Ann. Probab., 3 (1975), 534–545. https://doi.org/10.1214/aop/1176996359 doi: 10.1214/aop/1176996359
[7]	A. N. Kumar, P. Vellaisamy, Binomial approximation to locally dependent collateralized debt obligations, Methodol. Comput. Appl. Probab., 25 (2023), 81. https://doi.org/10.1007/s11009-023-10057-8 doi: 10.1007/s11009-023-10057-8
[8]	A. N. Kumar, P. Kumar, A negative binomial approximation to the distribution of the sum of maxima of indicator random variables, Stat. Probabil. Lett., 208 (2024), 110040. https://doi.org/10.1016/j.spl.2024.110040 doi: 10.1016/j.spl.2024.110040
[9]	C. Döbler, Stein's method of exchangeable pairs for the beta distribution and generalizations, Electron. J. Probab., 20 (2014), 1–34. https://doi.org/10.1214/EJP.v20-3933 doi: 10.1214/EJP.v20-3933
[10]	R. Gaunt, Variance-gamma approximation via Stein's method, Electron. J. Probab., 19 (2014), 1–33. https://doi.org/10.1214/EJP.v19-3020 doi: 10.1214/EJP.v19-3020
[11]	J. Pike, H. Ren, Stein's method and the Laplace distribution, ALEA, Lat. Am. J. Probab. Math. Stat., 11 (2014), 571–587.
[12]	K. Barman, N. S. Upadhye, On Stein factors for Laplace approximation and their application to random sums, Stat. Probabil. Lett., 206 (2024), 109996. https://doi.org/10.1016/j.spl.2023.109996 doi: 10.1016/j.spl.2023.109996
[13]	J. Fulman, N. Ross, Exponential approximation and Stein's method of exchangeable pairs, ALEA, Lat. Am. J. Probab. Math. Stat., 10 (2013), 1–13.
[14]	S. Chatterjee, J. Fulman, A. Röllin, Exponential approximation by Stein's method and spectral graph theory, ALEA, Lat. Am. J. Probab. Math. Stat., 8 (2011), 197–223.
[15]	E. A. Peköz, A. Röllin, New rates for exponential approximation and the theorems of Rényi and Yaglom, Ann. Probab., 39 (2011), 587–608. https://doi.org/10.1214/10-AOP559 doi: 10.1214/10-AOP559
[16]	B. S. Zuo, C. C. Yin, Stein's lemma for truncated generalized skew-elliptical random vectors, AIMS Mathematics, 5 (2020), 3423–3433. https://doi.org/10.3934/math.2020221 doi: 10.3934/math.2020221
[17]	C. Stein, Approximate computation of expectations, Stanford University: Institute of Mathematical Statistics, 1986. https://doi.org/10.1214/lnms/1215466568
[18]	Z. Ahmad, R. Browne, R. Chowdhury, R. Das, Y. S. Huang, Y. M. Zhu, Fast American option pricing using nonlinear stencils, In: Proceedings of the 29th ACM SIGPLAN annual symposium on principles and practice of parallel programming, New York: Association for Computing Machinery, 2024,316–332. https://doi.org/10.1145/3627535.3638506
[19]	Y. Ratibenyakool, K. Neammanee, Rate of convergence of binomial formula for option pricing, Commun. Stat.-Theor. M., 49 (2020), 3537–3556. https://doi.org/10.1080/03610926.2019.1590600 doi: 10.1080/03610926.2019.1590600

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)