On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space - Matsak, I. K.; Plichko, A. M. (Ukrains’kyi Matematychnyi Zhurnal, 1993)
Assume that (X n) are independent random variables in a Banach space, (b n) is a sequence of real numbers, Sn= ∑ 1 n biXi, and Bn=∑ 1 n b i 2 . Under certain moment restrictions imposed on the variablesX n, the conditions for the growth of the sequence (bn) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (Sn/B n ln ln Bn)1/2).