Strong law of large numbers for sums of products

Let X, X(n), n greater than or equal to 1, be a sequence of independent identically distributed random variables. We give necessary and sufficient conditions for the strong law of large numbers for k = 2 without regularity conditions on X, [GRAPHICS] for k greater than or equal to 3 in three cases: (i) symmetric X, (ii) P{X greater than or equal to 0} = 1 and (iii) regularly varying P{\X\ > x} as x --> infinity, 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.