The Annals of Probability

Strong law of large numbers for sums of products

Cun-Hui Zhang

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Let $X, X_n, n \ge 1$, be a sequence of independent identically distributed random variables. We give necessary and sufficient conditions for the strong law of large numbers

n^{-k/p} \sum_{1\lei_1 \le i_2 <\dots < i_k \le n} X_{i_1}X_{i_2}\dots X_{i_k} \to 0\quad\text{a.s.}

for $k =2$ without regularity conditions on $X$, for $k \geq 3$ in three cases: (i) symmetric X, (ii) $P \{X \leq 0\} =1 and (iii) regularly varying $P\{|X|}> x\}$ as $x \to \infty$, without further conditions, and for general X and k under a condition on the growth of the truncated mean of X. Randomized, centered, squared and decoupled strong laws and general normalizing sequences are also considered.

Article information

Ann. Probab., Volume 24, Number 3 (1996), 1589-1615.

First available in Project Euclid: 9 October 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60G50: Sums of independent random variables; random walks

Strong law of large numbers Marcinkiewicz–Zygmund law U-statistics quadratic forms decoupling maximum of products


Zhang, Cun-Hui. Strong law of large numbers for sums of products. Ann. Probab. 24 (1996), no. 3, 1589--1615. doi:10.1214/aop/1065725194.

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