Convergence of Probability Measures

Convergence of Probability Measures is a graduate textbook in the field of mathematical probability theory. It was written by Patrick Billingsley and published by Wiley in 1968, with a second edition in 1999.

The subject weak convergence of measures involves rigorous study of how a continuous time (or space) stochastic process arises as a scaling limit of a discrete time (or space) process. A fundamental example, Donsker's theorem, is convergence of rescaled random walk to Brownian motion. The mathematical theory, combining probability and functional analysis, was first developed in the 1950s by Skorokhod and Prokhorov in the 1950s, but was regarded as a specialized advanced topic. This book's contribution was a self-contained treatment at a useful basic level of abstraction, that of Polish space. It covers key theory tools such as Prokhorov's theorem on relative compactness of measures and the Skorokhod space of càdlàg functions. Though criticized by Dudley for insufficient generality, by being widely accessible it was for many years the standard reference, as evidenced by over 22,000 citations on Google Scholar. In particular, the subject became a highly valuable tool within burgeoning fields of applied probability such as queuing theory and empirical process theory in Statistics.