caa a22 4500 467893241 CHVBK 20180406152755.0 cr unu---uuuuu 170328e20061101xx s 000 0 eng 10.1007/s10955-006-9187-7 doi (NATIONALLICENCE)springer-10.1007/s10955-006-9187-7 Some New Results for McKean's Graphs with Applications to Kac's Equation [Elektronische Daten] [Ester Gabetta, Eugenio Regazzini] The main goal of the present paper is to sharpen some results about the error made when the Wild sums, used to represent the solution of the Kac analog of Boltzmann's equation, are truncated at the n-th stage. More precisely, in Carlen, Carvalho and Gabetta (J. Funct. Anal. 220: 362-387 (2005)), one finds a bound for the above-mentioned error which depends on (an Λ+ε). On the one hand, it is shown that Λ, the least negative eigenvalue of the linearized collision operator, is the best possible exponent. On the other hand, ε is an extra strictly positive number and a a positive coefficient which depends on ε too. Thus, it is interesting to check whether ε can be removed from the above bound. According to the aforesaid reference, this problem is studied here by means of the probability distribution of the depth of a leaf in a McKean random tree. In fact, an accurate study of the probability generating function of such a depth leads to conclude that the above bound can be replaced with (a′ n Λ). Springer Science+Business Media, Inc., 2006 depth of a leaf nationallicence depth of a tree nationallicence Kac's equation nationallicence McKean binary tree (or graph) nationallicence rate of convergence of Wild sums nationallicence Stirling numbers (of the first kind) nationallicence Wild convolution nationallicence Wild sum nationallicence Gabetta Ester Dipartimento di Matematica "F. Casorati”, Universită degli Studi di Pavia, Via Ferrata 1, I-27100, Pavia, Italy aut Regazzini Eugenio Dipartimento di Matematica "F. Casorati”, Universită degli Studi di Pavia, Via Ferrata 1, I-27100, Pavia, Italy aut Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 125/4(2006-11-01), 943-970 0022-4715 125:4<943 2006 125 10955 https://doi.org/10.1007/s10955-006-9187-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-006-9187-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gabetta Ester Dipartimento di Matematica "F. Casorati”, Universită degli Studi di Pavia, Via Ferrata 1, I-27100, Pavia, Italy aut NATIONALLICENCE 700 1- Regazzini Eugenio Dipartimento di Matematica "F. Casorati”, Universită degli Studi di Pavia, Via Ferrata 1, I-27100, Pavia, Italy aut NATIONALLICENCE 773 0- Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 125/4(2006-11-01), 943-970 0022-4715 125:4<943 2006 125 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer