Advanced

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

Klass, Michael J. and Nowicki, Krzysztof LU (2007) In Electronic Journal of Probability 12. p.1276-1298
Abstract
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Electronic Journal of Probability
volume
12
pages
1276 - 1298
publisher
UNIV WASHINGTON, DEPT MATHEMATICS
external identifiers
  • wos:000250196100001
  • scopus:35548937413
ISSN
1083-6489
language
English
LU publication?
yes
id
4f5204a9-9f58-4e42-ba3a-43d94da1df6a (old id 655371)
alternative location
http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1741&layout=abstract
date added to LUP
2007-12-12 14:35:46
date last changed
2017-01-01 07:09:21
@article{4f5204a9-9f58-4e42-ba3a-43d94da1df6a,
  abstract     = {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 &lt;= n &lt;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) &lt; s(y)(*) &lt; 2 (q) under bar (2y) and 1/2 (q) under bar (y/4(1+root 1-8/y) &lt; s(y) &lt; 2 (q) under bar (2y). The RHS's hold for y &gt;= 2 and the LHS's for y &gt;= 94 and y &gt;= 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       = {Klass, Michael J. and Nowicki, Krzysztof},
  issn         = {1083-6489},
  language     = {eng},
  pages        = {1276--1298},
  publisher    = {UNIV WASHINGTON, DEPT MATHEMATICS},
  series       = {Electronic Journal of Probability},
  title        = {Uniformly accurate quantile bounds via the truncated moment generating function: The symmetric case},
  volume       = {12},
  year         = {2007},
}