Some Useful Functions for Functional Limit Theorems

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Abstract
Many useful descriptions of stochastic models can be obtained from functional limit theorems (invariance principles or weak convergence theorems for probability measures on function spaces). These descriptions typically come from standard functional limit theorems via the continuous mapping theorem. This paper facilitates applications of the continuous mapping theorem by determining when several important functions and sequences of functions preserve convergence. The functions considered are composition, addition, composition plus addition, multiplication, supremum, reflecting barrier, first passage time and time reversal. These functions provide means for proving new functional limit theorems from previous ones. These functions are useful, for example, to establish the stability or continuity of queues and other stochastic models.

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