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July 1996 Strong law of large numbers for sums of products
Cun-Hui Zhang
Ann. Probab. 24(3): 1589-1615 (July 1996). DOI: 10.1214/aop/1065725194


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.


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Cun-Hui Zhang. "Strong law of large numbers for sums of products." Ann. Probab. 24 (3) 1589 - 1615, July 1996.


Published: July 1996
First available in Project Euclid: 9 October 2003

zbMATH: 0868.60024
MathSciNet: MR1411507
Digital Object Identifier: 10.1214/aop/1065725194

Primary: 60F15
Secondary: 60G50

Keywords: Decoupling , Marcinkiewicz–Zygmund law , maximum of products , Quadratic forms , Strong law of large numbers , U-statistics

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 3 • July 1996
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