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

TL;DR

This paper proves a new form of the strong law of large numbers with general normalizing sequences for negatively associated and i.i.d. random variables under a moment condition, extending previous results and addressing open questions.

Contribution

It introduces a comprehensive convergence result for weighted sums with general normalizing constants, applicable to negatively associated and i.i.d. variables, under minimal moment assumptions.

Findings

Complete convergence established for weighted sums.

New strong law results for negatively associated variables.

Examples include a law for variables similar to the St. Petersburg game.

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.