TY - JOUR

T1 - A characterization of a new type of strong law of large numbers

AU - Li, Deli

AU - Qi, Yongcheng

AU - Rosalsky, Andrew

N1 - Publisher Copyright:
© 2015 American Mathematical Society.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2016/1

Y1 - 2016/1

N2 - 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.

AB - 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.

KW - (p-q)-type strong law of large numbers

KW - Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers

KW - Rademacher type p Banach space

KW - Real separable Banach space

KW - Stable type p Banach space

KW - Sums of i.i.d. random variables

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U2 - 10.1090/tran/6390

DO - 10.1090/tran/6390

M3 - Article

AN - SCOPUS:84944790214

VL - 368

SP - 539

EP - 561

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -