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In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186-201] it isproved that the total variation distance between the solution $f(\cdot,t)$ ofKac's equation and the Gaussian density $(0,\sigma^2)$ has an upper bound whichgoes to zero with an exponential rate equal to -1/4 as $t\to+\infty$. In thepresent paper, we determine a lower bound which decreases exponentially to zerowith this same rate, provided that a suitable symmetrized form of $f_0$ hasnonzero 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 yieldthe insuperable barrier of -1 for the rate of convergence.

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