Exact Weak Laws and One Sided Strong Laws

Abstract

This paper explores Exact Weak Laws and their almost sure counterparts. First we show that the weighted sums of our infinite mean random variables can be balanced by a carefully selected sequence, i.e., that the ratio converges to one in probability. Then it is shown that the upper limit is almost surely infinity, while the lower limit is one.

Exact laws of large numbers for independent Pareto random variables

“…Note that Adler [4] recently extends Theorem 1.1 by considering P(X j > x) = {log(x + j)} α /(x + j) for α > −1. …”