Strong Law of Large Numbers Under a General Moment Condition
Sergei Chobanyan, Georgian Academy of Sciences, Georgia
Shlomo Levental, Michigan State University, USA
Habib Salehi, Michigan State University, USA
Abstract
We use our maximum inequality for p-th order random variables (p>1)
to prove a strong law of large numbers (SLLN) for sequences of p-th
order random variables. In particular, in the case p=2 our result shows
that ∑ f(k)/k <∞
is a sufficient condition for SLLN for f-quasi-stationary sequences. It
was known that the above condition, under the additional assumption of
monotonicity of f, implies SLLN (Erdös
(1949), Gal and Koksma (1950), Gaposhkin (1977), Moricz (1977)). Besides getting
rid of the monotonicity condition, the inequality enables us to extend thegeneral result to p-th order random variables, as well as to the case of
Banach-space-valued random variables.
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