Strong law of large numbers for sums of products

Let X, X_n, n ≥ 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 ∑_{1≤i1<i2<⋯<ik≤n} X_i1 X_i2 . . . X_ik → 0 a.s. for k = 2 without regularity conditions on X, for k ≥ 3 in three cases: (i) symmetric X, (ii) P{X ≥ 0} = 1 and (iii) regularly varying P{|X| > x} as x → ∞, 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.