The Analogy Between Statistical Equivalence and Stochastic Strong Limit Theorems

A classical limit theorem of stochastic process theory concerns the sample cumulative distribution function (CDF) from independent random variables. If the variables are uniformly distributed then these centered CDFs converge in a suitable sense to the sample paths of a Brownian Bridge. The so-called Hungarian construction of Komlos, Major and Tusnady provides a strong form of this result. In this construction the CDFs and the Brownian Bridge sample paths are coupled through an appropriate representation of each on the same measurable space, and the convergence is uniform at a suitable rate. Within the last decade several asymptotic statistical-equivalence theorems for nonparametric problems have been proven, beginning with Brown and Low (1996) and Nussbaum (1996). These theorems consider two sequences of statistical problems. The parameter spaces for these problems are infinite-dimensional; hence the problems are called nonparametric. The approach here to statistical-equivalence is firmly rooted within the asymptotic statistical theory created by L. Le Cam but in some respects goes beyond earlier results. This paper contains a survey of some of these statistical-equivalence results. It also demonstrates the analogy between these results and those from the coupling method for proving stochastic process limit theorems. These two classes of theorems possess a strong inter-relationship, and technical methods from each domain can profitably be employed in the other. Results in a recent paper by Carter, Low, Zhang and myself will be described from this perspective.