On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices

摘要

Let｛X_1,X_n;n∈Ν~d｝he a field of independent identically distributed real random variables, a｛X_1,X_n;n∈Ν~d｝ triangular array of real numbers, where Ν~d is the is mensional lattice. Under Ihe minimal condition that sup_nk,k|a_nk| ＜∞ we show that |n|~(-1)/p∑_k≤nA_n,kX_k→0 as |n|→ ∞ if and only if E(|X|~p(L|X|)~(d-1) ＜∞ provided d≤2 In the above, if 1≤P ＜2. In the random variables aic needed to be centered at the mean. By establishing a certain law of the logarithm, we show that the Law of the Iterated Logarithm fails for the weighted sums ∑_≤na_(nk)X_kkunder the condilktns that EX=0, EX~2＜∞, and E(X~2(L|X|)d~(-1)/L_2|X|)＜∞ for almost all bounded families ｛X_1,X_n;n∈Ν~dK≤n｝ of numbers.