The Lagrange Distributions and Branching Processes

Abstract

The supply of explicit analytic forms of interesting multivariate distributions can be enriched by considering multivariate Lagrange distributions which are known to be the distributions of the sizes of trees in branching processes. The (univariate) Lagrange distributions (the Borel–Tanner distribution being an example) were so named by Consul and Shenton, but they overlooked how intimate the connection is with branching processes, a connection that, for example, instantly explains their convolution theorem. Their device for obtaining the moments is found to extend to the multivariate case by using appropriate multivariate notations, and alternative methods of calculating the moments are also given. The results are likely to be of value in the analysis of data in biological, queueing and other areas where there is an underlying branching mechanism at work.