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

In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186-201] it is proved that the total variation distance between the solution f(?, t) of Kac's equation and the Gaussian density (0, ?2) has an upper bound which goes to zero with an exponential rate equal to -1/4 as t->+?. 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 f0 has nonzero fourth cumulant ?4. Moreover, we show that upper bounds like ?C?e-(1/4)t??(t) are valid for some ?? vanishing at infinity when ?R|v|4+?f0(v) dv<+? for some ? in [0, 2[ and ?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.