Exact Strong Laws for Multidimensionally Indexed Random Variables

Consider independent and identically distributed random variables {X, Xn, n∈Zd+} with either EX=0 or E |X|=∞. We establish strong laws so that ∑|n|⩽N anXn/bN→1 almost surely. Our procedure selects the constants {an, n∈Zd+} and {bN, N⩾1} so that these strong laws obtain in almost any possible setting.