On the Depth of the Trees in the Mckean Representation of Wild's Sums

In the McKean representation of the Wild expression for the solution of Kac's analog of Boltzmann's equation, an essential role is played by the probabilistic properties of a class of binary trees. In particular, some results in Carlen et al. (2005) lay special stress on the importance of the probability distribution of the depth of a McKean tree. In studying the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation, in the above‐mentioned paper that distribution is used to determine the weights in a fundamental convex decomposition of the McKean average over n‐fold Wild convolutions of the initial data. In fact, a lower bound is given for the probability q_n,k that the depth is greater than or equal to k (for any positive integer k), in order to prove that such a probability goes to 1 as n diverges. In this framework, the present paper gives exact expressions for the above probability, deriving it from a difference‐integral equation for the generating function of (q_n,k)_n≥2^k . Moreover, in this very same scheme of things, a new lower bound for (q_n,k)_n≥2^k is given.

Keywords:

• Depth of a McKean tree (probability distribution of)
• Gamma distribution
• Kac equation
• McKean tree
• Multiple hypergeometric function
• Wild convolution
• Wild sum

Acknowledgments

The authors are grateful to Aldo Belleni Morante, Eric Carlen, and Maria Carvalho for their very helpful discussions and comments. This work was partially supported by the Ministero dell'Istruzione, dell'Universitá e della Ricerca (MURST National Project “Problemi matematici delle teorie cinetiche”, COFIN 2004), the European Community Human Potential Program under Contract HPRN‐CT‐2002‐00282 (HYKE), and the IMATI (CNR—Pavia, Italy).