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. 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.
Let fX n ; n 1g and fY n ; n 1g be two sequences of uniform random variables. We obtain various strong and weak laws of large numbers for the ratio of these two sequences. Even though these are uniform and naturally bounded random variables the ratios are not bounded and have an unusual behaviour creating Exact Strong Laws.
We study the almost sure convergence of weighted sums of dependent random variables to a positive and finite constant, in the case when the random variables have either mean zero or no mean at all. These are not typical strong laws and they are called exact strong laws of large numbers. We do not assume any particular type of dependence and furthermore consider sequences which are not necessarily identically distributed. The obtained results may be applied to sequences of negatively associated random variables.
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