  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.
