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

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- by Alexander R. Pruss PDF
- Proc. Amer. Math. Soc.
**126**(1998), 1811-1819 Request permission

## 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 \[ P\bigl (\sup _{1\le j\le 2n} \| S_j \| > \lambda \bigr ) \le c P(\| S_n \| > \lambda /c), \] for all $\lambda \ge 0$. This generalizes an inequality of Montgomery-Smith and Latała 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
- Received by editor(s): August 2, 1996
- Received by editor(s) in revised form: December 2, 1996
- Communicated by: Stanley Sawyer
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**126**(1998), 1811-1819 - MSC (1991): Primary 60E15
- DOI: https://doi.org/10.1090/S0002-9939-98-04254-3
- MathSciNet review: 1443850