Strong Laws of Large Numbers for Weighted Sums of Random Elements in Normed Linear Spaces

Consider a sequence of independent random elements {Vn, n > in a real separable normed linear space (assumed to be a Banach space in most of the results), and sequences of constants {a,, n > and {ha, n with 0 < b, "[" oo. Sets of conditions are provided for {an(V EVn) n > to obey a general strong law of large numbers of the form aj(Vj EVj)/bn --> 0 almost certainly. The hypotheses involve the distributions of the j=l {V,, n > }, the growth behaviors of {a n > and {bn, n > }, and for some of the results impose a geometric condition on X. Moreover, Feller’s classical result generalizing Marcinkiewiez-Zygmund strong law of large numbers is shown to hold for random elements in a real separable Rademacher type p (1 < p < 2) Banach space.