On the strong law of large numbers for weighted sums of random variables

For a positive function ϕϕ defined on (0,∞)(0,∞), an integral representation is obtained for each of the three series ∑n=1∞P(|X|>ϕ(n)),∑n=1∞ϕ(n)−1E|X|I(|X|>ϕ(n)), and ∑n=1∞ϕ2(n)−1EX2I(|X|≤ϕ(n)). Equivalence conditions for convergence of each of the series are also obtained. Using these results, we can give sufficient conditions for the strong law of large numbers for weighted sums of random variables.