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

Simple item page
dc.creatorDolera, Emanuele
dc.creatorGabetta, Ester
dc.creatorRegazzini, Eugenio
dc.date2009-03-02
dc.date.accessioned2026-07-07T12:48:08Z
dc.date.available2026-07-07T12:48:08Z
dc.descriptionLet $f(\cdot,t)$ 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.
dc.descriptionPublished 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)
dc.identifierhttps://arxiv.org/abs/0903.0255
dc.identifierhttp://arxiv.org/abs/0903.0255
dc.identifierAnnals of Applied Probability 2009, Vol. 19, No. 1, 186-209
dc.identifierdoi:10.1214/08-AAP538
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/221949
dc.subjectProbability
dc.subject60F05, 82C40 (Primary)
dc.titleReaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem
dc.typetext

Files

Collections

arXiv