Statistical Equivalence and Stochastic Process 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). 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 talk 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. 1. Probability setting 1.1. Background Let F be the CDF for a probability on [0,1];. F abs. cont., with f(x) ∆ = ∂F ∂x on [0, 1]. Research supported in part by NSF-Division of Mathematical Sciences. Statistics Department, Wharton School, University of Pennsylvania, Philadelphia, PA 191046340, USA. E-mail: [email protected] 558 Lawrence D. Brown Let X1, . . . , Xn iid from F. F̂n denotes the sample CDF, F̂n(x) ∆ = 1 n n