A law of the single logarithm for weighted sums of i.i.d. random elements

摘要

Let {X, X-n, n >= 1} be a sequence of i.i.d. Banach space valued random elements with E (parallel to X parallel to(beta)/(log parallel to X parallel to)(beta/2)) < infinity, and {a(ni), 1 <= i <= n, n >= 1} an array of constants satisfying Sigma(n)(i=1) vertical bar a(ni)vertical bar(alpha) = O(n), where alpha > 0, beta > 0, and 1/2 = 1/alpha + 1/beta. In this paper, we obtain a law of the single logarithm for weighted sums Sigma(n)(i=1) a(ni)X(i). We also obtain a strong law of large numbers for weighted sums of i.i.d. Banach space valued random elements with a suitable moment condition. No assumptions are made concerning the geometry of the underlying Banach space.