NA随机场对数律的收敛速度

Abstract

Abstract:

Let d be a positive ingter and N ^ddenote 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/2-2P(max_{1≤ k≤ n}|S_k| ≥ ε √| n} log | n |) < ∞, ∨ε > M.

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

CLC Number:

60F15

Cite this article

JIN Jing-Sen. Convergence Rate in the Law of |Logarithm for NA Random Fields[J].Acta mathematica scientia,Series A, 2009, 29(4): 1138-1143.

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

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