Sama-Ae, Al-Ameen; Chaidee, Nattakarn; Neammanee, Kritsana Half-normal approximation for statistics of symmetric simple random walk. (English) Zbl 1384.60064 Commun. Stat., Theory Methods 47, No. 4, 779-792 (2018). Summary: In [“Stein’s method for the half-normal distribution with application to limit theorems related to simple random walk”, Preprint, arXiv:1303.4592], C. Döbler used Stein’s method to obtain the uniform bounds in half-normal approximation for three statistics of a symmetric simple random walk; the maximum value, the number of returns to the origin and the number of sign changes up to a given time \(n\). In this paper, we give the non-uniform bounds for these statistics by using Stein’s method and the concentration inequality approach. Cited in 1 ReviewCited in 1 Document MSC: 60F05 Central limit and other weak theorems 60G50 Sums of independent random variables; random walks Keywords:concentration inequality; half-normal distribution; Stein’s method; no-uniform and non-uniform bounds; symmetric simple random walk PDFBibTeX XMLCite \textit{A.-A. Sama-Ae} et al., Commun. Stat., Theory Methods 47, No. 4, 779--792 (2018; Zbl 1384.60064) Full Text: DOI References: [1] Chen, L.H.Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3(3):534-545. · Zbl 0335.60016 [2] Chen, L.H.Y., Shao, Q.M. (2001). A non-uniform Berry-Esseen bound via Stein’s Method. Probab. Theory Relat. Fields 120(2):236-254. · Zbl 0996.60029 [3] Chen, L.H.Y., Goldstein, L., Shao, Q.M. (2011). Normal Approximation by Stein’s Method. Heidelberg: Springer. · Zbl 1213.62027 [4] De Gennes, P.G. (1979). Scaling Concepts in Polymer Physics. Ithaca and London: Cornell University Press. [5] Döbler, C. (2013). Stein’s Method for the Half-Normal Distribution with Application to Limit Theorems Related to Simple Random Walk. arXiv:1303.4592v2. · Zbl 1328.60063 [6] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I (3rd ed.). New York: John Wiley & Sons Inc. · Zbl 0155.23101 [7] Goel, N.S., Richter-Dyn, N. (1974). Stochastic Models in Biology. New York: Academic Press. [8] Pearson, K. (1905). The problem of the random walk. Nature 72:294. · JFM 36.0303.02 [9] Redner, S. (2001). A Guide to First-Passage Process. Cambridge, UK: Cambridge University Press. · Zbl 0980.60006 [10] Stein, C. (1986). Approximate Computation of Expectations, Vol. 7, Institute of Mathematical Statistics, Stanford university. · Zbl 0721.60016 [11] Stein, C. (1972). A Bound for Error in the Normal Approximation to the Distribution of a Sum of Dependent Random Variables. Proceeding of the sixth Berkeley Symposium on Mathematical Statistics and Probability (Vol. II, pp. 586-602), Berkeley: University of California Press. [12] Van Kampen, N.G. (1992). Stochastic Processes in Physics and Chemistry, revised and enlarged edition. Amsterdam: North-Holland. [13] Weiss, G.H. (1994). Aspects and Applications of the Random Walk. Amsterdam: North-Holland. · Zbl 0925.60079 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.