Strong Law of Large Numbers Under a General Moment Condition

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 $\sum f(k)/k < \infty$ 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 (Erdos (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.