​Assume that a renormalized Birkhoff sum $S_n f/B_n$ converges in distribution to a nontrivial limit. What can one say about the sequence $B_n$? Most natural statements in the literature involve sequences $B_n$ of the form $B_n = n^\alpha L(n)$, where $L$ is slowly varying. We will discuss the possible growth rate of $B_n$ both in the probability preserving case and the conservative case. In particular, we will describe examples where $B_n$ grows superpolynomially, or where $B_{n+1}/B_n$ does not tend to $1$.[-]

The Gaussian functor associates to every orthogonal representation of a group G on a Hilbert space, a probability measure preserving action of G called a Gaussian action. This construction is a fundamental tool in ergodic theory and is the source of a large and interesting class of probability measure preserving actions. In this talk, I will present a generalization of the Gaussian functor which associates to every affine isometric action of G on a Hilbert space, a nonsingular Gaussian action which is not measure preserving. This provides a new and large class of nonsingular actions whose properties are related in a very subtle way to the geometry of the original affine isometric action. In some cases, such as affine isometric actions comming from groups acting on trees, a fascinating phase transition phenomenon occurs.This talk is based on a joint work with Yuki Arano and Yusuke Isono, as well as a more recent joint work with Stefaan Vaes.[-]