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 $(0,{\sigma }^2)$ 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.

Published in at http://dx.doi.org/10.1214/09-AAP623 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem
• Central limit theorem for the solution of the Kac equation

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