THE ROLE OF THE CENTRAL LIMIT THEOREM IN DISCOVERING SHARP RATES OF CONVERGENCE TO EQUILIBRIUM FOR THE SOLUTION OF THE KAC EQUATION
成果类型：
Article
署名作者：
Dolera, Emanuele; Regazzini, Eugenio
署名单位：
University of Pavia
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/09-AAP623
发表日期：
2010
页码：
430-461
关键词：
摘要：
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, sigma(2)) has an upper bound which goes to zero with an exponential rate equal to 1/4 as t -> +infinity. 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 (C) over bar (delta)e(-(1/4)t) rho(delta)(t) are valid for some rho(delta) vanishing at infinity when fill integral R vertical bar v vertical bar(4+delta) f(0)(v) dv + infinity for some S 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.