Large time limit and local $L^2$-index theorems for families

Local Limit Theorems for Free

L2-index theorems, KK-theory, and connections

Let M be a compact manifold and D a Dirac type differential operator on M . Let A be a C∗-algebra. Given a bundle W (with connection) of A-modules over M , the operator D can be twisted with this bundle. One can then use a trace on A to define numerical indices of this twisted operator. We prove an explicit formula for these indices. Our result does complement the Mishchenko–Fomenko index theor...

Large deviations of combinatorial distributions II: Local limit theorems

This paper is a sequel to our paper [17] where we derived a general central limit theorem for probabilities of large deviations1 applicable to many classes of combinatorial structures and arithmetic functions; we consider corresponding local limit theorems in this paper. More precisely, given a sequence of integral random variables {Ωn}n≥1 each of maximal span 1 (see below for definition), we a...

Limit theorems for stationary Markov processes with L2-spectral gap

Let (Xt, Yt)t∈T be a discrete or continuous-time Markov process with state space X × R where X is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. (Xt, Yt)t∈T is assumed to be a Markov additive process. In particular, this implies that the first component (Xt)t∈T is also a Markov process. Markov random walks or additive f...

Local Limit Theorems for Free Groups

In this paper we obtain a local limit theorem for elements of a free group G under the abelianization map [·] : G → G/[G, G]. This is obtained via an analysis involving subshifts of finite type, where we obtain a result of independent interest. The case of fundamental groups of compact surfaces of genus ≥ 2 is also discussed. 0. Introduction Let G denote the free group on k ≥ 2 generators {a1, ...