指标在Tθd上变动的B-值I.I.D.R.V.’S的叠对数律

指标在T_θ^d上变动的B-值I.I.D.R.V.’S的叠对数律

摘要: 记B是可分Banach空间,X是B-值随机变量,\mathbbN^d=\left\\barn=n_1, \cdots, n_d\right) ; n_i=1,2,…,i=1,…,d,T_\theta^d=\left\\barn \in \mathbbN^d ; \barn=\left(n_1, \cdots, n_d\right)\right.,θn_i≤n_y≤θ^-1n₄,i≠j,i,j=1,…,d,其中d≥2,0<θ<1,本文研究指标在T_θ^d上变动的B-值i. i. d. r. v. ’s的四种类型的叠对数律(即BLIL^(θ,α)₁,BLIL^(θ,d)₂,CLIL^(θ.D)₁和GLIL^(θ,d)₂,获得了X∈BLIL^(θ,α)₁、X∈BLIL^(θ,d)₂、X∈CLIL^(θ.D)₁和X∈GLIL^(θ,d)₂的充要条件。

Abstract: Let B be a real separable Banach space and X a B-valued random variable. Set \mathbbN^d=\left\\barn=n_1, \cdots, n_d\right) ; n_i=1,2,…,i=1,…,d andT_\theta^d=\left\\barn \in \mathbbN^d ; \barn=\left(n_1, \cdots, n_d\right)\right.,θn_i≤n_y≤θ^-1n₄,i≠j,i,j=1,…,d, where d≥2 and 0＜θ＜1. In this paper, we discuss four forms of the LIL for B-valued i.i.d.r.v.’ s indexed by T_θⁱ (i.e. BLIL^(θ,α)₁,BLIL^(θ,d)₂,CLIL^(θ.D)₁andGLIL^(θ,d)₂) and obtain necessary and sufficient conditions for X to satisfy, respectively, BLIL^(θ,α)₁,BLIL^(θ,d)₂,CLIL^(θ.D)₁andGLIL^(θ,d)₂.