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An improvement of Hoffmann-Jorgensen's inequality

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Author

Summary, in English

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 → ∞.

Publishing year

2000

Language

English

Pages

851-862

Publication/Series

Annals of Probability

Volume

28

Issue

2

Document type

Journal article

Publisher

Institute of Mathematical Statistics

Topic

  • Probability Theory and Statistics

Keywords

  • expo- nential inequalities
  • Tail probability inequalities
  • Hoffmann-Jorgensen's inequality
  • Banach space valued random variables

Status

Published

ISBN/ISSN/Other

  • ISSN: 0091-1798