An improvement of HoffmannJorgensen's inequality
(2000) In Annals of Probability 28(2). p.851862 Abstract
 Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set  x  = supΛ∈F Λ (x) (·  is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n Σk j=1 Xj  where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of { Xi }n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ( Sn ) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp... (More)
 Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set  x  = supΛ∈F Λ (x) (·  is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n Σk j=1 Xj  where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of { Xi }n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ( Sn ) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1766847
 author
 Klass, Michael J. and Nowicki, Krzysztof ^{LU}
 organization
 publishing date
 2000
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 expo nential inequalities, Tail probability inequalities, HoffmannJorgensen's inequality, Banach space valued random variables
 in
 Annals of Probability
 volume
 28
 issue
 2
 pages
 851  862
 publisher
 Institute of Mathematical Statistics
 external identifiers

 scopus:0034345596
 ISSN
 00911798
 language
 English
 LU publication?
 yes
 id
 743fac1c6501412aa36e3addac2f0c0b (old id 1766847)
 date added to LUP
 20160404 09:26:17
 date last changed
 20220129 17:49:22
@article{743fac1c6501412aa36e3addac2f0c0b, abstract = {{Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set  x  = supΛ∈F Λ (x) (·  is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n Σk j=1 Xj  where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of { Xi }n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ( Sn ) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞.}}, author = {{Klass, Michael J. and Nowicki, Krzysztof}}, issn = {{00911798}}, keywords = {{expo nential inequalities; Tail probability inequalities; HoffmannJorgensen's inequality; Banach space valued random variables}}, language = {{eng}}, number = {{2}}, pages = {{851862}}, publisher = {{Institute of Mathematical Statistics}}, series = {{Annals of Probability}}, title = {{An improvement of HoffmannJorgensen's inequality}}, url = {{https://lup.lub.lu.se/search/files/5324072/1770750}}, volume = {{28}}, year = {{2000}}, }