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.
Categories: math.PR