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

Authors: Deli Li, Yongcheng Qi and Andrew Rosalsky
Journal: Trans. Amer. Math. Soc. 368 (2016), 539-561
MSC (2010): Primary 60F15; Secondary 60B12, 60G50
Published electronically: May 27, 2015
MathSciNet review: 3413873
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Abstract: Let $ 0 < p < 2$ and $ 1 \leq q < \infty $. Let $ \{X_{n};~n \geq 1 \}$ be a sequence of independent copies of a real-valued random variable $ X$ and set $ S_{n} = X_{1} + \cdots + X_{n}, ~n \geq 1$. We say $ X$ satisfies the $ (p, q)$-type strong law of large numbers (and write $ X \in SLLN(p, q)$) if $ \sum _{n = 1}^{\infty } \frac {1}{n}\left (\frac {\left \vert S_{n}\right \vert}{n^{1/p}} \right )^{q} < \infty $ almost surely. This paper is devoted to a characterization of $ X \in 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 \in SLLN(p, q)$ for the six cases: $ 1 \leq q < p < 2$, $ 1 < p = q < 2$, $ 1 < p < 2$ and $ q > p$, $ q = p = 1$, $ p = 1 < q$, and $ 0 < p < 1 \leq q$. The necessary and sufficient conditions for $ X \in 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.

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Additional Information

Deli Li
Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1

Yongcheng Qi
Affiliation: Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, Minnesota 55812

Andrew Rosalsky
Affiliation: Department of Statistics, University of Florida, Gainesville, Florida 32611

Keywords: Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers, $(p, q)$-type strong law of large numbers, sums of i.i.d. random variables, real separable Banach space, Rademacher type $p$ Banach space, stable type $p$ Banach space
Received by editor(s): June 10, 2013
Received by editor(s) in revised form: November 21, 2013
Published electronically: May 27, 2015
Additional Notes: The research of the first author was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada
The research of the second author was partially supported by NSF Grant DMS-1005345
Article copyright: © Copyright 2015 American Mathematical Society