Manfred Denker - Penn State

Title of talk: Von Mises statistics for a measure preserving transformation. Abstract: Let $T$ be a measure preserving transformation on a probability space. I will present three theorems on the almost sure and weak convergence of sums of the form $$ \sum_{0\le i_k <n, k=1,...,d} h(T^{i_1},...,h(T^{i_d}).$$ The difficulty here arises from the fact that the summands are not well defined as random variables on the probability space. Therefore I will explain how to describe reasonable subspaces of $L_2$ where these variables can be defined a.s. As a result I will state new ergodic theorems and new central limit theorems obtained from a suitable martingale approximation in the sense of Gordin's 1968 paper.