STRONG LAWS OF LARGE NUMBERS FOR WEIGHTED SUMS OF RANDOM ELEMENTS IN NORMED LINEAR SPACES

ABSTRACT. 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 con-stants {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. KEY WORDS AND PHRASES. Real separable Banach space, independent random elements, normed weighted sums, strong law of large numbers, almost certain convergence, stochastically dominated random elements, Rademacher type p, Beck-convex normed linear space, Schauder basis. (uniformly) tight sequence