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arXiv · 2103.00114

The Marcinkiewicz--Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences

Vu Thi Ngoc Anh·Nguyen Thi Thanh Hien·Lê Vǎn Thành·Vo Thi Hong Van
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Abstract

This paper establishes complete convergence for weighted sums and the Marcinkiewicz--Zygmund-type strong law of large numbers for sequences of negatively associated and identically distributed random variables $\{X,X_n,n\ge1\}$ with general normalizing constants under a moment condition that $ER(X)<\infty$, where $R(\cdot)$ is a regularly varying function. The result is new even when the random variables are independent and identically distributed (i.i.d.), and a special case of this result comes close to a solution to an open question raised by Chen and Sung (Statist Probab Lett 92:45--52, 2014). The proof exploits some properties of slowly varying functions and the de Bruijin conjugates. A counterpart of the main result obtained by Martikainen (J Math Sci 75(5):1944-1946, 1995) on the Marcinkiewicz--Zygmund-type strong law of large numbers for pairwise i.i.d. random variables is also presented. Two illustrated examples are provided, including a strong law of large numbers for pairwise negatively dependent random variables which have the same distribution as the random variable appearing in the St. Petersburg game.

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Vu Thi Ngoc Anh, Nguyen Thi Thanh Hien, Lê Vǎn Thành, Vo Thi Hong Van. 2021-02-27. The Marcinkiewicz--Zygmund-Type Strong Law of Large Numbers with General Normalizing Sequences. https://arxiv.org/abs/2103.00114

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