Facultade de Fisioterapia

A probability inequality for sums of independent Banach space valued random variables

Li, Deli; Liang, Han-Ying; Rosalsky, Andrew
Abstract:
Let (B, parallel to center dot parallel to) be a real separable Banach space. Let phi(center dot) and psi(center dot) be two continuous and increasing functions defined on [0, infinity) such that phi(0) = psi(0) = 0, lim(t ->infinity) phi(t) = infinity, and psi(center dot)/phi(center dot) is a nondecreasing function on [0,infinity). Let {V-n; n >= 1} be a sequence of independent and symmetric B-valued random variables. In this note, we establish a probability inequality for sums of independent B-valued random variables by showing that for every n >= 1 and all t >= 0, P (parallel to Sigma(n)(i=1)Vi parallel to > tb(n)) <= 4P (parallel to Sigma(n)(i=1)phi(psi(-1)(parallel to V-i parallel to))/parallel to V-i parallel to parallel to > ta(n)) + Sigma P-n(i=1)(parallel to V-i parallel to > b(n)), where a(n) = phi(n) and b(n) = psi(n), n >= 1. As an application of this inequality, we establish what we call a comparison theorem for the weak law of large numbers for independent and identically distributed B-valued random variables.
Year:
2018
Type of Publication:
Article
Keywords:
Probability inequality; sums of independent Banach; space valued random variables; weak law of large numbers; LAWS
Journal:
STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES
Volume:
90
Number:
2
Pages:
214-223
DOI:
10.1080/17442508.2017.1318878
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