The matrix Laguerre transform

摘要

The Laguerre transform, introduced by Keilson and Nunn (1979) and Keilson, Nunn, Sumita (1981), provides an algorithmic basis for the computation of multiple convolutions in conjunction with other algebraic and summation operations. The methods enable one to evaluate numerically a variety of results in applied probability and statistics that have been available only formally. For certain more complicated models, the formulation must be extended. In this paper we establish the matrix Laguerre transform, appropriate for the study of semi-Markov processes and Markov renewal processes, as an extension of the scalar Laguerre transform. The new formalism enables one to calculate matrix convolutions and other algebraic operations in matrix form. As an application, a matrix renewal function is evaluated and its limit theorem is numerically exhibited.