Uniformly accurate quantile bounds via the truncated moment generating function: The symmetric case
(2007) In Electronic Journal of Probability 12. p.12761298 Abstract
 Let X1, X2,... be independent and symmetric random variables such that Sn = X1+...+ Xn converges to a finite valued random variable S a. s. and let S* = sup(1 <= n <infinity) Sn (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 Sy 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 18/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 X1, X2,... be independent and symmetric random variables such that Sn = X1+...+ Xn converges to a finite valued random variable S a. s. and let S* = sup(1 <= n <infinity) Sn (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 Sy 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 18/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 nonnegative 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:
https://lup.lub.lu.se/record/655371
 author
 Klass, Michael J. and Nowicki, Krzysztof ^{LU}
 organization
 publishing date
 2007
 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
 10836489
 language
 English
 LU publication?
 yes
 id
 4f5204a99f584e42ba3a43d94da1df6a (old id 655371)
 alternative location
 http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1741&layout=abstract
 date added to LUP
 20160401 16:35:22
 date last changed
 20220128 20:44:55
@article{4f5204a99f584e42ba3a43d94da1df6a, abstract = {{Let X1, X2,... be independent and symmetric random variables such that Sn = X1+...+ Xn converges to a finite valued random variable S a. s. and let S* = sup(1 <= n <infinity) Sn (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 Sy 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 18/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 nonnegative 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 = {{10836489}}, language = {{eng}}, pages = {{12761298}}, 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}}, }