# Lund University Publications

## LUND UNIVERSITY LIBRARIES

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

(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)
author
and
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
2016-04-01 16:35:22
date last changed
2020-11-22 06:20:27
```@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},
url          = {http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1741&layout=abstract},
volume       = {12},
year         = {2007},
}

```