A generalization of the Strassen converse

Hyung Tae Ha, Deli Li, Yongcheng Qi

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Abstract

This note is devoted to a generalization of the Strassen converse. Let gn:R ∞→[0,∞], n≥1 be a sequence of measurable functions such that, for every n≥1, gn(x+y2)≤C(gn(x)+gn(y)) and gn(x-y2)≤C(gn(x)+gn(y)) for all x,y R∞, where 0<C<∞ is a constant which is independent of n. Let {X,Xn;n≥1} be a sequence of i.i.d. random variables. Assume that there exist r≥1 and a function φ:[0,∞)→[0,∞) with limt→∞φ(t)=∞, depending only on the sequence {gn;n≥1} such that lim supn→∞gn(X1,X2,∞)=φ(E|X|r) a.s. whenever E|X|r<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X1,X2,∞)<∞ a.s. implies E|X|r<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,∞)=∞ for all c≠0). Some applications are provided to illustrate this result.

Original languageEnglish (US)
Pages (from-to)729-735
Number of pages7
JournalJournal of Mathematical Analysis and Applications
Volume374
Issue number2
DOIs
StatePublished - Feb 15 2011

Bibliographical note

Funding Information:
The authors are extremely grateful to the two referees for very carefully reading the manuscript and for offering several valuable comments and suggestions which enabled us to improve the presentation of this note. The research of Hyung-Tae Ha was supported by Kyungwon University Research Fund, the research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada, and the research of Yongcheng Qi was partially supported by NSF Grant DMS-1005345.

Keywords

  • Convergence of generalized moments
  • I.i.d. random variables
  • The Strassen converse
  • The law of the iterated logarithm

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