## The Annals of Probability

### Strong law of large numbers for multilinear forms

#### Abstract

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 $X^{(l)}$ .

#### Article information

Source
Ann. Probab., Volume 26, Number 2 (1998), 902-923.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855655

Digital Object Identifier
doi:10.1214/aop/1022855655

Mathematical Reviews number (MathSciNet)
MR1626539

Zentralblatt MATH identifier
0937.60016

Subjects
Primary: 60F15: Strong theorems

#### Citation

Gadidov, Anda. Strong law of large numbers for multilinear forms. Ann. Probab. 26 (1998), no. 2, 902--923. doi:10.1214/aop/1022855655. https://projecteuclid.org/euclid.aop/1022855655

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