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

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- by Deli Li, Yongcheng Qi and Andrew Rosalsky PDF
- Trans. Amer. Math. Soc.
**368**(2016), 539-561 Request permission

## 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 $\sum _{n = 1}^{\infty } \frac {1}{n}\left (\frac {\left |S_{n}\right |}{n^{1/p}} \right )^{q} < \infty$ 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 LĂ©vy, Ottaviani, and Hoffmann-JĂžrgensen (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
- Email: dli@lakeheadu.ca
**Yongcheng Qi**- Affiliation: Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, Minnesota 55812
- Email: yqi@d.umn.edu
**Andrew Rosalsky**- Affiliation: Department of Statistics, University of Florida, Gainesville, Florida 32611
- Email: rosalsky@stat.ufl.edu
- 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 - © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 539-561 - MSC (2010): Primary 60F15; Secondary 60B12, 60G50
- DOI: https://doi.org/10.1090/tran/6390
- MathSciNet review: 3413873