THE LAW OF THE ITERATED LOGARITHM FOR B-VALUED I. I. D. R. V.’ S INDEXED BY T_θ^d

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.