2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221949Let $f(\cdot,t)$ be the probability density function which represents the solution of Kac's equation at time $t$, with initial data $f_0$, and let $g_σ$ be the Gaussian density with zero mean and variance $σ^2$, $σ^2$ being the value of the second moment of $f_0$. This is the first study which proves that the total variation distance between $f(\cdot,t)$ and $g_σ$ goes to zero, as $t\to +\infty$, with an exponential rate equal to -1/4. In the present paper, this fact is proved on the sole assumption that $f_0$ has finite fourth moment and its Fourier transform $φ_0$ satisfies $|φ_0(ξ)|=o(|ξ|^{-p})$ as $|ξ|\to+\infty$, for some $p>0$. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.Published in at http://dx.doi.org/10.1214/08-AAP538 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)Probability60F05, 82C40 (Primary)text