 An extension of Feller’s strong law of large numbersDeli Li, Han-Ying Liang and Andrew Rosalsky Statistics & Probability Letters, 2018, vol. 132, issue C, 83-90 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,Xn;n≥1} be a sequence of independent and identically distributed Banach space valued random variables and set Sn=∑i=1nXi,n≥1. Let {an;n≥1} and {bn;n≥1} be increasing sequences of positive real numbers such that limn→∞an=∞ and bn∕an;n≥1 is a nondecreasing sequence. We show that Sn−nEXI{‖X‖≤bn}bn→0almost surelyfor every Banach space valued random variable X with ∑n=1∞P(‖X‖>bn)<∞ if Sn∕an→0 almost surely for every symmetric Banach space valued random variable X with ∑n=1∞P(‖X‖>an)<∞. 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. 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 (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:132:y:2018:i:c:p:83-90 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spl.2017.09.011 Access Statistics for this article Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul More articles in Statistics & Probability Letters from Elsevier |
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