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 $X^{(l)}$ .