Random Time Changes for Sock-Sorting and Other Stochastic Process Limit Theorems

Abstract

A common technique in the theory of stochastic process is to replace a discrete time coordinate by a continuous randomized time, defined by an independent Poisson or other process. Once the analysis is complete on this poissonized process, translating the results back to the original setting may be nontrivial. It is shown here that, under fairly general conditions, if the process $S_n$ and the time change $\phi_n$ both converge, when normalized by the same constant, to limit processes combined process $S_n(\phi_n(t))$ converges, when properly normalized, to a sum of the limit of the orginal process, and the limit of the time change multiplied by the derivative of $E S_n$. It is also shown that earlier results on the fine structure of the maxima are preserved by these time changes.