For a general class of statistics expressible as extrema of certain sample functions, an almost sure invariance principle, particularly useful in the context of the law of iterated logarithm and the probabilities of moderate deviations, is established, and its applications are stressed.
"An Almost Sure Invariance Principle for the Extrema of Certain Sample Functions." Ann. Probab. 4 (1) 81 - 88, February, 1976. https://doi.org/10.1214/aop/1176996182