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, σ2) 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 κ4. Moreover, we show that upper bounds like ̅Cδe−(1/4)tρδ(t) are valid for some ρδ vanishing at infinity when ∫ℝ|v|4+δf0(v) dv<+∞ for some δ in [0, 2[ and κ4=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.