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(⋅, 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 ̅C_δe^−(1/4)tρ_δ(t) are valid for some ρ_δ 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.