Invariance Principles for Independent and Weakly Dependent Random Variables

During the last three decades the concept “invariance principle” has undergone several subtle changes. In the early 1950’s an invariance principle was a result that nowadays often would be called a functional central limit theorem (FCLT). At present the term “invariance principle” generally stands as a synonym for an approximation theorem: A given process, such as a partial sum process, an empirical process, an extremal process, a U-statistic, etc. is approximated in distribution, in probability, in LP or almost surely by a canonical process, such as a Brownian motion, a Kiefer process, a special extremal process or in case of a U-statistic by a multiple stochastic integral.

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