Central limit theorem in uniform metrics for generalized Kac equations

Abstract

The aim of this paper is to give explicit rates for the speed of convergence to equilibrium of the solution of the generalized Kac equation in two strong metrics: the total variation distance (TV) and the uniform metric between characteristic functions (χ0). A fundamental role in our study is played by the probabilistic representation of the solution of the generalized Kac equation as marginal law of a stochastic process which is a weighted random sum of i.i.d. random variables, where the weights are positive and dependent. Exponential bounds for the total variation distance between the solution and the gaussian stationary state of the Kac equation have been proved by Dolera, Gabetta and Regazzini (2009). In our more general setting the equilibrium states are scale mixtures of stable distributions and hence not necessarily gaussian. Therefore we develop new tools based on ideal metrics that are used in the literature for quantitative central limit theorems for i.i.d. random variables in the domain of attraction of a stable distribution. We obtain first exponential bounds in the so-called ”r-smoothed total variation” and in the weighted χr-metric for a suitable r, then we deduce rates of convergence with respect to the “corresponding” uniform metrics TV and χ0.