Mathematics – Probability
Scientific paper
2010-09-17
Annals of Applied Probability 2010, Vol. 20, No. 2, 430-461
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/09-AAP623 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/09-AAP623
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.
Dolera Emanuele
Regazzini Eugenio
No associations
LandOfFree
The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-442174