关键词:  NA, 随机场, 对数律, 收敛性

Abstract:

Let d be a positive ingter and N ^d denote  the d-dimensional lattice of positive integers. Let {X_n , n ∈ N ^d}be a same distribution NA random fields, put S_n = ∑_{k≤ n} X_k, S_n^(k)=S_n-X_k, if r >2, EX₁ = 0 and σ²= Var(X₁}, then there exists a positive constant M:=100√(r-2)(1+σ²) such that the following is equivalent:

(I)   E |X₁|^r (log|X₁|)^d-1-r/2 < ∞;

(II)   ∑_{n∈ N^d} |n|^r/2-2 P(max_{1≤ k≤ n} |S _n^(k)| ≥ (2^d+1 )ε √|n| log | n |) < ∞, ∨ε  > M;
 
(III)   ∑_{n ∈ N ^d} |n|^r/2-2 P(max_{1≤ k≤ n} |S_k | ≥ ε √| n} log | n |) < ∞, ∨ε > M.

Key words: NA, Random fields, Law of , logarithm, Convergence rate