A New Proof for a Strong Law of Large Numbers of Kolmogorov's Type via Weak Convergence

Abstract

In terms of the Dirac representation of sample mean and the weak convergence of empirical distributions that holds almost surely, we construct a new proof for a strong law of large numbers of Kolmogorov's type with i.i.d. random variables $X_{1}, X_{2}, \dots$ such that $\lim_{c \to \infty}\sup_{n \in \mathbb{N}}n^{-1}\sum_{i=1}^{n}|X_{i}|\cdot \mathbb{I}_{[c,+\infty[}\circ |X_{i}| = 0$ almost surely. That each random variable $X_{i}$ is $L^{1}$ is also a conclusion. Our proof is independent of both Kolmogorov's strong law and its known proof(s), and potentially furnishes a new way to obtain a short proof of Kolmogorov's strong law.