Facultade de Fisioterapia

An extension of Feller's strong law of large numbers

Li, Deli; Liang, Han-Ying; Rosalsky, Andrew
Abstract:
This paper presents a general result that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and Feller's strong law of large numbers in a Banach space setting. Let {X, X-n; X >= 1} be a sequence of independent and identically distributed Banach space valued random variables and set. S-n = Sigma(n)(i=1) X-i, n >= 1. Let {a(n); n >= 1} and {b(n); n >= 1 be increasing sequences of positive real numbers such that lim(n ->infinity) a(n) = infinity and {b(n)/a(n); n >= 1} is a nondecreasing sequence. We show that S-n - nE(XI{||X|| <= b(n)})/b(n) -> 0 for every Banach space valued random variable X with Sigma(infinity)(n=1) P(||X|| > b(n)) < infinity if S-n/a(n) -> 0 almost surely for every symmetric Banach space valued random variable X with Sigma(infinity)(n=1) P(||X|| > a(n)) < infinity. To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables. (C) 2017 Elsevier B.V. All rights reserved.
Year:
2018
Type of Publication:
Article
Keywords:
Feller'; s strong law of large numbers; Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers; Rademacher type p Banach space; Sums of independent random variables
Journal:
Statistics & Probability Letters
Volume:
132
Pages:
83-90
Month:
January
DOI:
10.1016/j.spl.2017.09.011
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