On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space

Abstract

Assume that (X _n) are independent random variables in a Banach space, (b _n) is a sequence of real numbers, S_n= ∑ ₁ ⁿ b_iX_i, and B_n=∑ ₁ ⁿ b _i ² . Under certain moment restrictions imposed on the variablesX _n, the conditions for the growth of the sequence (b_n) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (S_n/B _n ln ln B_n)^1/2).