Reaching the best possible rate of convergence to equilibrium for solutions of Kac.s equation via central limit theorem

Let f(., t) be the probability density function which represents the solution of Kac.s equation at time t, with initial data f0, and let g. be the Gaussian density with zero mean and variance .2, .2 being the value of the second moment of f0. This is the first study which proves that the total variation distance between f(., t) and g. goes to zero, as t.+., with an exponential rate equal to .1/4. In the present paper, this fact is proved on the sole assumption that f0 has finite fourth moment and its Fourier transform .0 satisfies |.0(.)|=o(|.|.p) as |.|.+., 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.