## The Annals of Probability

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

### Strong law of large numbers for sums of products

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 9 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1065725194

**Digital Object Identifier**

doi:10.1214/aop/1065725194

**Mathematical Reviews number (MathSciNet)**

MR1411507

**Zentralblatt MATH identifier**

0868.60024

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G50: Sums of independent random variables; random walks

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aop/1065725194