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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A maximal inequality for partial sums
of finite exchangeable sequences
of random variables


Author: Alexander R. Pruss
Journal: Proc. Amer. Math. Soc. 126 (1998), 1811-1819
MSC (1991): Primary 60E15
MathSciNet review: 1443850
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Abstract: Let $X_1,X_2,\dots,X_{2n}$ be a finite exchangeable sequence of Banach space valued random variables, i.e., a sequence such that all joint distributions are invariant under permutations of the variables. We prove that there is an absolute constant $c$ such that if $S_j=\sum _{i=1}^j X_i$, then

\begin{displaymath}P\bigl(\sup _{1\le j\le 2n} \| S_j \| > \lambda\bigr) \le c P(\| S_n \| > \lambda/c), \end{displaymath}

for all $\lambda\ge 0$. This generalizes an inequality of Montgomery-Smith and Lata{\l}a for independent and identically distributed random variables. Our maximal inequality is apparently new even if $X_1,X_2,\dotsc$ is an infinite exchangeable sequence of random variables. As a corollary of our result, we obtain a comparison inequality for tail probabilities of sums of arbitrary random variables over random subsets of the indices.


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Additional Information

Alexander R. Pruss
Email: pruss@pitt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04254-3
PII: S 0002-9939(98)04254-3
Keywords: Sums of exchangeable random variables, maximal inequalities
Received by editor(s): August 2, 1996
Received by editor(s) in revised form: December 2, 1996
Communicated by: Stanley Sawyer
Article copyright: © Copyright 1998 American Mathematical Society