The Annals of Applied Probability

The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation

Emanuele Dolera and Eugenio Regazzini

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Abstract

In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186–201] it is proved that the total variation distance between the solution f(⋅, t) of Kac’s equation and the Gaussian density (0, σ2) has an upper bound which goes to zero with an exponential rate equal to −1/4 as t→+∞. 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 f0 has nonzero fourth cumulant κ4. Moreover, we show that upper bounds like ̅Cδe−(1/4)tρδ(t) are valid for some ρδ vanishing at infinity when |v|4+δf0(v) dv<+∞ for some δ in [0, 2[ and κ4=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.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 2 (2010), 430-461.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1268143429

Digital Object Identifier
doi:10.1214/09-AAP623

Mathematical Reviews number (MathSciNet)
MR2650038

Zentralblatt MATH identifier
1195.60033

Subjects
Primary: 60F05: Central limit and other weak theorems 82C40: Kinetic theory of gases

Keywords
Berry–Esseen inequalities central limit theorem Kac’s equation cumulants kurtosis coefficient total variation distance Wild’s sum

Citation

Dolera, Emanuele; Regazzini, Eugenio. The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation. Ann. Appl. Probab. 20 (2010), no. 2, 430--461. doi:10.1214/09-AAP623. https://projecteuclid.org/euclid.aoap/1268143429


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