NA随机场对数律的收敛速度

设d是一个正整数, N ^d是d -维正整数格点.设{X_n , n∈N ^d} 是一同分布的负相伴随机场, 记S_n =∑_{k≤ n} X_k, S_n^(k)=S_n-X_k, 如果r >2, EX₁ = 0 和σ²= Var(X₁}, 则存在一个正数M:=100√(r-2)(1+σ²)使得下列条件等价(I)E |X₁|^r (log|X₁|)^d-1-r/2 <∞;(II)∑_{n∈ N^d} |n|^r/2-2P(max_{1≤ k≤ n} |S_n^(k)|≥ (2^d+1 )ε√|n| log |n |) <∞,∨ε > M;(III)∑_{n∈N ^d} |n|^r/2-2P(max_{1≤ k≤n} |S_k |≥ε√| n} log| n |) <∞,∨ε > M.(III) $sumlimits_{{{bf n}}in {{cal N}}^{d}} |n|^{r/2-2}P(maxlimits_{{bf 1}leq{bf k}leq{bf n}}|S_{{bf k}}|geq varepsilon sqrt{|{bf n}|log |{bf n}|})M$.

Convergence Rate in the Law of |Logarithm for NA Random Fields

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.