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Journal of Probability and Statistics
Volume 2010, Article ID 730543, 23 pages
Research Article

Some Comments on Quasi-Birth-and-Death Processes and Matrix Measures

Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany

Received 12 August 2009; Revised 15 April 2010; Accepted 25 May 2010

Academic Editor: Nikolaos E. Limnios

Copyright © 2010 Holger Dette and Bettina Reuther. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We explore the relation between matrix measures and quasi-birth-and-death processes. We derive an integral representation of the transition function in terms of a matrix-valued spectral measure and corresponding orthogonal matrix polynomials. We characterize several stochastic properties of quasi-birth-and-death processes by means of this matrixmeasure and illustrate the theoretical results by several examples.