Abstract

Suppose a1 , a2 ,... is a sequence of real numbers with an → ∞. If lim sup(X1+ ... + Xn)/an = α a.s. for every sequence of independent nonnegative uniformly bounded random variables X1,X2,... satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables Y1,Y2, ... , lim sup(Y1+ ... + Yn)/an = α a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition, Σa-2n < ∞ , then the assumption of nonnegativity may be dropped.

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URL: http://digitalcommons.calpoly.edu/rgp_rsr/59