Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices

Let be a real separable Banach space and {X, X_{n, m}; (n, m) N²} B-valued i.i.d. random variables. Set . In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N² is studied. There is a gap between the moment conditions for CLIL(N¹) and those for CLIL(N²). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence to be almost surely conditionally compact in B, where, for 0, 1 r 2, N^r(, ) = {(n, m) N²; n m n exp{(log n)^r–1 (n)}} and (·) is any positive, continuous, nondecreasing function such that (t)/(log log t) is eventually decreasing as t , for some > 0.