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This article is cited in 8 scientific papers (total in 8 papers)
Exact maximal inequalities for exchangeable systems of random variables
S. Chobanyana, H. Salehib a Muskhelishvili Institute of Computational Mathematics
b Michigan State University, MI, USA
Abstract:
Given an exchangeable finite system of Banachspace valued random variables $(\xi_1,\ldots,\xi_n)$ with $\sum_1^n\xi_i=0$, we prove that $\mathbf{E}\Phi(\max_{k\le n}\|\xi_1+\cdots+\xi_k\|)$ is equivalent to $\mathbf{E}\Phi(\|\sum_1^n\xi_ir_i\|)$ for any increasing and convex $\Phi\colon\mathbf{R}^+\to\mathbf{R}^+$, $\Phi(0)=0$, where $(r_1,\ldots,r_n)$ is a system of Rademacher random variables independent of $(\xi_1,\ldots,\xi_n)$. We also establish the equivalence of the tails of the related distributions. The results seem to be new also for scalar random variables. As corollaries we find best estimations for the average of $\max_{k\le n}\|a_{\pi(1)}+\cdots+a_{\pi(k)}\|$ with respect to permutations $\pi$ of nonrandom vectors $a_1,\ldots,a_n$ from a normed space.
Keywords:
exchangeable random variables, Banach space, maximal inequality, permutations.
Received: 31.03.1998
Citation:
S. Chobanyan, H. Salehi, “Exact maximal inequalities for exchangeable systems of random variables”, Teor. Veroyatnost. i Primenen., 45:3 (2000), 555–567; Theory Probab. Appl., 45:3 (2001), 424–435
Linking options:
https://www.mathnet.ru/eng/tvp485https://doi.org/10.4213/tvp485 https://www.mathnet.ru/eng/tvp/v45/i3/p555
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