By embedding partial sum processes into Brownian motion, it is well known that the deMoivre-Laplace central limit theorem is a consequence of the strong law of large numbers. It is the purpose here to show that the embedding technique can be used to establish both the degenerate convergence criterion and the normal convergence criterion for triangular arrays of uniformly asymptotically negligible random variables.
"A Probabilistic Proof of the Normal Convergence Criterion." Ann. Probab. 1 (5) 867 - 869, October, 1973. https://doi.org/10.1214/aop/1176996854