Laws of the iterated logarithm for weighted sums of independent random variables

Let [Lambda] = lim supn-->[infinity](2n log log n)-1/2 [Sigma]k=1n[latin small letter f with hook](k/n)Xk, where [latin small letter f with hook] is a function defined on [0,1] and {X, Xn;n[greater-or-equal, slanted]1} is an iid sequence. If X is real-valued, it is shown that [Lambda] = [latin small letter f with hook]2, the L2-norm of [latin small letter f with hook], for all functions [latin small letter f with hook] in a certain class of absolutely continuous functions if E(X) = 0 and E(X2) = 1. Conversely, if [Lambda] = [latin small letter f with hook]2 for some such [latin small letter f with hook] with [integral operator]01[latin small letter f with hook](t)dt [not equal to] 0, then E(X) = 0, E(X2) = 1. Necessary and sufficient conditions for the compact law of the iterated logarithm are given in the case when X takes values in a separable Banach space, and a law of the iterated logarithm for sums of weighted partial sums is obtained in a Banach space setting.