The Skorohod Metric on D[0, ∞)

Abstract

The uniform metric on D[0, 1] is the best choice for applications where the limit distribution concentrates on C[0, 1], or on some other separable subset of D[0,1]. It is well suited for convergence to brownian motion, brownian bridge, and the gaussian processes that appear as limits in the Empirical Central Limit Theorem. But it excludes, for example, poisson processes and other non-gaussian processes with independent increments, whose jumps are not constrained to lie in a fixed, countable subset of [0, 1]. To analyze such processes, Skorohod (1956) introduced four new metrics, all weaker than the uniform metric. Of these, the J₁ metric has since become the most popular. (Too popular in my opinion—too often it is dragged into problems for which the uniform metric would suffice.) But Skorohod’s J₁ metric on D[0, 1] will not be the main concern of this chapter. Instead we shall investigate a sort of J₁ convergence on compacta for D[0, ∞], the space where the interesting applications live.

… in which an alternative to the metric of uniform convergence on compacta is studied. With the new metric the limit processes need not confine their jumps to a countable set of time points. Amongst the convergence criteria developed is an elegant condition based on random increments, due to Aldous. The chapter might be regarded as an extended appendix to Chapter V.