Modul: MAT971 Stochastische Prozesse

## Analytic combinatorics in several variables and application to multivariate local limit theorems

Talk by Dr. Linxiao Chen

**Date:** 29.09.21 **Time:** 17.15 - 18.15 **Room:** ETH HG G 19.1

In this talk I will present a recipe for reading the asymptotics of a multi-dimensional infinite array of numbers from its generating function. In the bivariate case, this means reading the asymptotics of a_{m,n} as m,n → ∞ and m/n^θ → s (where θ>0 is fixed and s>0 is a variable) from the function A(x,y)= Σ_{m,n≥0} a_{m,n} x^m y^n. I will explain why such asymptotics are useful for studying probabilistic models, in particular for establishing uniform local limit theorems with exotic limit distributions, with some examples from random map models. Previously, similar recipes were available only in the case where θ=1 and the function A is rational. In contrast, our method works for any θ>0 and algebraic function A, but under some additional conditions on the singularity structure of A. If time permits, I will explain why the old and new methods treat disjoint cases, and how they might be combined together in the future.