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

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 Sn and the time change φn both converge, when normalized by the same constant αn, to limit processes S̃ and Φ̃, then the combined process Sn ◦ φn converges to S̃ + Φ̃ · d dt ES(t) when properly normalized. It is also shown that earlier results on the fine structure of the maxima are preserved by these time changes. The remainder of the paper then applies these simple results to processes which arise in a natural way from sorting procedures, and from random allocations. The first example is a generalization of “socksorting”: Given a pile of n mixed-up pairs of socks, we draw out one at a time, laying it on a table if its partner has not yet been drawn, and putting completed pairs away. The question is: What is the distribution of the maximum number of socks ever on the table, for large n? Similarly, when randomly throwing balls into n (a large number) boxes, we examine the distribution of the maximum over all times of the number of boxes that have (for example) exactly one ball.