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April 1998 Strong law of large numbers for multilinear forms
Anda Gadidov
Ann. Probab. 26(2): 902-923 (April 1998). DOI: 10.1214/aop/1022855655


Let $m \geq 2$ be a nonnegative integer and let ${X^{(l)}, X_i^{(l)}}_{i \epsilon \mathbb{N}}, l = 1, \dots, m$, be $m$ independent sequences of independent and identically distributed symmetric random variables. Define $S_n = \Sigma_{1 \leq i_1, \dots, i_m \leq n} X_{i_1}^{(l)} \dots X_{i_m}^{(m)}$, and let ${\gamma_n}_{n \epsilon \mathbb{N}}$ be a nondecreasing sequence of positive numbers, tending to infinity and satisfying some regularity conditions. For $m = 2$ necessary and sufficient conditions are obtained for the strong law of large numbers $\gamma_n^{-1} S_n \to 0$ a.s. to hold, and for $m > 2$ the strong law of large numbers is obtained under a condition on the growth of the truncated variance of the $X^{(l)}$ .


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Anda Gadidov. "Strong law of large numbers for multilinear forms." Ann. Probab. 26 (2) 902 - 923, April 1998.


Published: April 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0937.60016
MathSciNet: MR1626539
Digital Object Identifier: 10.1214/aop/1022855655

Primary: 60F15

Keywords: $U$-statistics , martingale , maximal inequality , multilinear forms , strong laws

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.26 • No. 2 • April 1998
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