Uniformly accurate quantile bounds via the truncated moment generating function: The symmetric case

Summary, in English

Let X-1, X-2,... be independent and symmetric random variables such that S-n = X-1+...+ X-n converges to a finite valued random variable S a. s. and let S* = sup(1 <= n <infinity) S-n (which is finite a.s.). We construct upper and lower bounds for s(y) and s(y)*, the upper 1/y((th) under bar) quantile of S-y and S*, respectively. Our approximations rely on an explicitly computable quantity ((q) under bar)y for which we prove that 1/2 (q) under bar (y/2) < s(y)(*) < 2 (q) under bar (2y) and 1/2 (q) under bar (y/4(1+root 1-8/y) < s(y) < 2 (q) under bar (2y). The RHS's hold for y >= 2 and the LHS's for y >= 94 and y >= 97, respectively. Although our results are derived primarily for symmetric random variables, they apply to non-negative variates and extend to an absolute value of a sum of independent but otherwise arbitrary random variables.

• Author: Michael J. Klass; Krzysztof Nowicki
• Department/s: Department of Statistics
• Publishing year: 2007
• Language: English
• Pages: 1276-1298
• Publication/Series: Electronic Journal of Probability
• Volume: 12
• Links: http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1741&layout=abstract
• Document type: Journal article
• Publisher: UNIV WASHINGTON, DEPT MATHEMATICS
• Topic: Probability Theory and Statistics
• Status: Published
• ISBN/ISSN/Other: ISSN: 1083-6489