The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation

TL;DR

This paper investigates the precise exponential rate of convergence to equilibrium for solutions of Kac's equation, establishing both upper and lower bounds that depend on initial conditions and cumulants, with a focus on the role of the central limit theorem.

Contribution

It provides the first matching lower bound for the convergence rate, confirming the optimality of the -1/4 exponential rate under certain conditions, and explores the influence of initial data and cumulants.

Findings

Lower bound matches the upper bound, confirming the -1/4 rate.

The convergence rate depends on the fourth cumulant of the initial distribution.

Initial conditions with zero fourth cumulant can improve the convergence rate.

Abstract

In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186-201] it is proved that the total variation distance between the solution $f(\cdot,t)$ of Kac's equation and the Gaussian density $(0,\sigma^2)$ has an upper bound which goes to zero with an exponential rate equal to -1/4 as $t\to+\infty$. In the present paper, we determine a lower bound which decreases exponentially to zero with this same rate, provided that a suitable symmetrized form of $f_0$ has nonzero fourth cumulant $\kappa_4$. Moreover, we show that upper bounds like $\bar{C}_{\delta}e^{-({1/4})t}\rho_{\delta}(t)$ are valid for some $\rho_{\delta}$ vanishing at infinity when $\int_{\mathbb{R}}|v|^{4+\delta}f_0(v)\,dv<+\infty$ for some $\delta$ in $[0,2[$ and $\kappa_4=0$. Generalizations of this statement are presented, together with some remarks about non-Gaussian initial conditions which yield the insuperable barrier of -1 for the rate of convergence.