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Zbl 0553.44002
Sumita, Ushio
The matrix Laguerre transform. (English)
[J] Appl. Math. Comput. 15, 1-28 (1984). ISSN 0096-3003

The Laguerre transform, introduced by {\it J. Keilson} and {\it W. R. Nunn} [ibid. 5, 313-359 (1979; Zbl 0449.65086)], {\it J. Keilson}, {\it W. R. Nunn} and the author [ibid. 8, 137-174 (1981; Zbl 0457.42011)], and further studied by the author [Development of the Laguerre transform method for numerical exploration of applied probability models, Ph. D. Diss., Grad. School Management, Univ. Rochester (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 behind the "Laplacian curtain". 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. In a recent paper by the author and Kijima (1984) the bivariate Laguerre transform has also been developed for the study of bivariate processes and distributions.

MSC 2000:: *44A30 Multiple transforms
44A15 Special transforms
44A35 Convolution
60J25 Markov processes with continuous parameter
60K05 Renewal theory

Keywords: multiple convolutions; Laplacian curtain; matrix Laguerre transform; semi-Markov processes; Markov renewal processes

Citations: Zbl 0449.65086; Zbl 0457.42011

Cited in: Zbl 0783.60087 Zbl 0603.60085

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