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On a Domination of Sums of Random Variables by Sums of Conditionally Independent Ones
Journal article   Open access   Peer reviewed

On a Domination of Sums of Random Variables by Sums of Conditionally Independent Ones

The Annals of probability, v 22(1), pp 453-468
01 Jan 1994
url
https://doi.org/10.1214/aop/1176988868View
Published, Version of Record (VoR) Open

Abstract

Banach space Increasing sequences Martingales Mathematical constants Mathematical functions Mathematical inequalities Random variables Stopping distances Tangents
It is known that if (Xn) and (Yn) are two (Fn)-adapted sequences of random variables such that for each k ≥ 1 the conditional distributions of Xkand Yk, given Fk-1, coincide a.s., then the following is true:$\|\sum X_k\|_p \leq B_p\| \sum Y_k\|_p, 1 \leq p < \infty$, for some constant Bpdepending only on p. The aim of this paper is to show that if a sequence (Yn) is conditionally independent, then the constant Bpmay actually be chosen to be independent of p. This significantly improves all hitherto known estimates on Bpand extends an earlier result of Klass on randomly stopped sums of independent random variables as well as our recent result dealing with martingale transforms of Rademacher sequences.

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Statistics & Probability
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