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

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 $(0,σ^2)$ 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 $κ_4$. Moreover, we show that upper bounds like $\bar{C}_δe^{-({1/4})t}ρ_δ(t)$ are valid for some $ρ_δ$ vanishing at infinity when $\int_{\mathbb{R}}|v|^{4+δ}f_0(v)\,dv<+\infty$ for some $δ$ in $[0,2[$ and $κ_4=0$. 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.