A characterization of a new type of strong law of large numbers

Deli Li, Yongcheng Qi, Andrew Rosalsky

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

Let 0 < p < 2 and 1 ≤ q < ∞. Let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set Sn = X1 + · · ·+Xn, n ≥ 1. We say X satisfies the (p, q)-type strong law of large numbers (and write X ∈ SLLN(p, q)) if (formula given) < ∞ almost surely. This paper is devoted to a characterization of X ∈ SLLN(p, q). By applying results obtained from the new versions of the classical Lévy, Ottaviani, and Hoffmann- Jørgensen (1974) inequalities proved by Li and Rosalsky (2013) and by using techniques developed by Hechner (2009) and Hechner and Heinkel (2010), we obtain sets of necessary and sufficient conditions for X ∈ SLLN(p, q) for the six cases: 1 ≤ q < p < 2, 1 < p = q < 2, 1 < p < 2 and q > p, q = p = 1, p = 1 < q, and 0 < p < 1 ≤ q. The necessary and sufficient conditions for X ∈ SLLN(p, 1) have been discovered by Li, Qi, and Rosalsky (2011). Versions of the above results in a Banach space setting are also given. Illustrative examples are presented.

Original languageEnglish (US)
Pages (from-to)539-561
Number of pages23
JournalTransactions of the American Mathematical Society
Volume368
Issue number1
DOIs
StatePublished - Jan 2016

Keywords

  • (p-q)-type strong law of large numbers
  • Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers
  • Rademacher type p Banach space
  • Real separable Banach space
  • Stable type p Banach space
  • Sums of i.i.d. random variables

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