Matrix measures and random walks with a block tridiagonal transition matrix

SIAM Journal on Matrix Analysis and Applications

Volumn 29, Issue 1, 2006, Pages 117-142

  Dette, Holger ^a   Reuther, Bettina ^a   Studden, W J ^b   Zygmunt, M ^c  

^a RUHR UNIVERSITY BOCHUM   (Germany)

^b PURDUE UNIVERSITY   (United States)

^c AGH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Poland)

Author keywords

Block tridiagonal transition matrix; Canonical moments; Chebyshev matrix polynomials; Markov chain; Matrix measure; Quasi birth and death process; Spectral measure

Indexed keywords

BLOCK TRIDIAGONAL TRANSITION MATRIX; CANONICAL MOMENTS; CHEBYSHEV MATRIX POLYNOMIALS; MATRIX MEASURE; SPECTRAL MEASURE;

MARKOV PROCESSES; NUMERICAL METHODS; PROBLEM SOLVING; QUEUEING THEORY; SPECTRUM ANALYSIS;

MATRIX ALGEBRA;

EID: 39449110777     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/050638230     Document Type: Article

References (35)

1. A. BASILEVSKY (1983), Applied Matrix Algebra in the Statistical Sciences, North-Holland, Amsterdam.
2. N. G. BEAN, P. K. POLLETT, AND P. G. TAYLOR (2000), Quasi-stationary distributions for leveldependent quasi-birth-and-death processes, Comm. Statist. Stochastic Models, 16, pp. 511-541.
3. JU. M. BEREZANSKII (1968), Expansions in Eigenfunctions of Selfadjoint Operators, Trans. Math. Monogr. 17, AMS, Providence, RI.
4. L. BRIGHT AND P. G. TAYLOR (1995), Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Comm. Statist. Stochastic Models, 11, pp. 497-525.
5. T. DAYAR AND F. QUESSETTE (2002), Quasi-birth-and- death processes with level-geometric distribution, SIAM J. Matrix Anal. Appl., 24, pp. 281-291.
6. H. DETTE (1996), On the generating functions of a random walk on the nonnegative integers, J. Appl. Probab., 33, pp. 1033-1052.
7. H. DETTE AND W. J. STUDDEN (1997), The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis, Wiley, New York.
8. H. DETTE AND W. J. STUDDEN (2001), Matrix measures, moment spaces, and Favard's theorem for the interval [0, 1] and [0, ∞), Linear Algebra Appl., 345, pp. 163-193.
9. A. J. DURAN (1995), On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math., 47, pp. 88-112.
10. A. J. DURAN (1996), Markov's theorem for orthogonal matrix polynomials, Canad. J. Math., 48, pp. 1180-1195.
11. A. J. DURAN (1999), Ratio asymptotics for orthogonal matrix polynomials, J. Approx. Theory, 100, pp. 304-344.
12. A. J. DURAN AND W. VAN ASSCHE (1995), Orthogonal matrix polynomials and higher-order recurrence relations, Linear Algebra Appl., 219, pp. 261-280.
13. W. FAIR (1971), Noncommutative continued fractions, SIAM J. Math. Anal., 2, pp. 226-232.
14. W. FELLER (1950), An Introduction to Probability Theory and Its Applications, Vol. I, John Wiley & Sons, New York.
15. D. P. GAVER, P. A. JACOBS, AND G. LATOUCHE (1984), Finite birth-and-death models in randomly changing environments, Adv. in Appl. Probab., 16, pp. 715-731.
16. B. HAJEK (1982), Birth-and-death processes on the integers with phases and general boundaries, J. Appl. Probab., 19, pp. 488-499.
17. 2-Spaces, Springer-Verlag, New York.
18. S. KARLIN AND J. MCGREGOR (1959), Random walks, Illionis J. Math., 3, pp. 66-81.
19. S. KARLIN AND H. M. TAYLOR (1975), A First Course in Stochastic Processes, Academic Press, New York.
20. G. LATOUCHE, C. E. M. PEARCE, AND P. G. TAYLOR (1998), Invariant measures for quasi-birth-and-death processes, Comm. Statist. Stochastic Models, 14, pp. 443-460.
21. G. LATOUCHE AND V. RAMASWAMI (1999), Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Ser. Stat. Appl. Probab. 5, SIAM, Philadelphia, Chapter 12.
22. Q. L. LI AND J. CAO (2004), Two types of RG-factorizations of quasi-birth-and-death processes and their applications to stochastic integral functionals, Stoch. Models, 20, pp. 299-340.
23. F. MARCELLÁN AND G. SANSIGRE (1993), On a class of matrix orthogonal polynomials on the real line, Linear Algebra Appl., 181, pp. 97-109.
24. I. MAREK (2003), Quasi-birth-and-death processes, level geometric distributions. An aggregation/disaggregation approach, J. Comput. Appl. Math., 152, pp. 277-288.
25. M. F. NEUTS (1981), Matrix Geometric Solutions in Stochastic Models. An Algorithmic Approach. The John Hopkins University Press, Baltimore.
26. M. F. NEUTS (1989), Structured Stochastic Matrices of M/G/1 Type and Their Applications, Marcel Dekker, New York.
27. A. OST (2001), Performance of Communication Systems: A Model-Based Approach with Matrix-Geometric Methods, Springer-Verlag, Berlin.
28. V. RAMASWAMI AND P. G. TAYLOR (1996), Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases, Comm. Statist. Stochastic Models, 12, pp. 143-164.
29. L. RODMAN (1990), Orthogonal matrix polynomials, in Orthogonal Polynomials, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 294, P. Nevai, ed., Kluwer, Dordrecht, The Netherlands, pp. 345-362.
30. A. SINAP AND W. VAN ASSCHE (1996), Orthogonal matrix polynomials and applications, J. Comput. Appl. Math., 66, pp. 27-52.
31. E. A. VAN DOORN AND P. SCHRUNER (1993), Random walk polynomials and random walk measures, J. Comput. Appl. Math., 49, pp. 289-296.
32. E. A. VAN DOORN AND P. SCHRIJNER (1995), Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes, J. Austral. Math. Soc. Ser. B, 37, pp. 121-144.
33. T. A. WHITEHURST (1982), An application of orthogonal polynomials to random walks, Pacific J. Math., 99, pp. 205-213.
34. W. WOESS (1985), Random walks and periodic continued fractions, Adv. in Appl. Probab., 17, pp. 67-84.
35. M. J. ZYGMUNT (2002), Matrix Chebyshev polynomials and continued fractions, Linear Algebra Appl., 340, pp. 155-168.