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Application of Stieltjes theory for S-fractions to birth and death processes

Published online by Cambridge University Press:  01 July 2016

G. Bordes  and
B. Roehner
G. Bordes*
Affiliation:
Laboratoire de Physique Corpusculaire, College de France
B. Roehner*
Affiliation:
Laboratoire de Physique Théorique et Hautes Energies, Université Paris VII
*
∗Postal address: Laboratoire de Physique Corpusculaire, College de France, Place Marcelin-Berthelot, 75005 Paris, France.
∗∗Postal address: Université Paris VII, Tour 33, Ier étage, 2, place Jussieu, 75251 Paris Cedex 05, France.
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Abstract

We are interested in obtaining bounds for the spectrum of the infinite Jacobi matrix of a birth and death process or of any process (with nearest-neighbour interactions) defined by a similar Jacobi matrix.

To this aim we use some results of Stieltjes theory for S-fractions, after reviewing them. We prove a general theorem giving a lower bound of the spectrum. The theorem also gives sufficient conditions for the spectrum to be discrete.

The expression for the lower bound is then worked out explicitly for several, fairly general, classes of birth and death processes. A conjecture about the asymptotic behavior of a special class of birth and death processes is presented.

Keywords

CONTINUED FRACTIONSSTIELTJES FRACTIONSSPECTRUMJACOBI MATRICES
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 3 , September 1983 , pp. 507 - 530
Copyright
Copyright © Applied Probability Trust 1983 

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References

Bailey, N. T. J. (1964) The Elements of Stochastic Processes. Wiley, New York.Google Scholar
Baker, G. A. Jr. (1975) Essentials of Padé Approximants. Academic Press, New York.Google Scholar
Dambrine, S. and Moreau, M. (1981) Note on the stochastic theory of a self-catalytic reaction II. Physica 106A, 574–588.Google Scholar
Gantmacher, F. R. and Krein, M. G. (1960) Oszillationsmatrizen, Oszillationskerne und Kleine Schwingungen Mechanischer Systeme. Akademie-Verlag, Berlin.Google Scholar
Gilewicz, J. (1977) Contribution à la théorie des approximants de Padé et à leur technique d'emploi. Thèse, Aix-en-Provence.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1965) Tables of Integrals and Products. Academic Press, New York.Google Scholar
John, P. W. M. (1957) Divergent time homogeneous birth and death processes. Ann. Math. Statist. 28, 514–517.Google Scholar
Jones, W. B. and Thron, W. J. (1980) Continued Fractions. Analytic Theory and Applications. Addison-Wesley, Reading, Ma.Google Scholar
Ledermann, W. and Reuter, G. E. H. (1954) Spectral theory for the differential equations of birth and death processes. Phil. Trans. R. Soc. London A 246, 321–369.Google Scholar
Magnus, W., Oberhettinger, F. and Soni, R. P. (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag, Berlin.Google Scholar
Maki, D. (1976) On birth-death processes with rational growth rates. SIAM J. Math. Anal. 7, 29–36.Google Scholar
Parzen, E. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
Perron, O. (1929) Die Lehre von den Kettenbrüchen. Chelsea, New York.Google Scholar
Pólya, G. and Szegö, G. (1972) Problems and Theorems in Analysis. Springer-Verlag, Berlin.Google Scholar
Reuter, G. E. H. (1957) Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97, 1–46.Google Scholar
Roehner, B. and Valent, G. (1982) Solving the birth and death processes with quadratic asymptotically symmetric transition rates. SIAM J. Appl. Math. 42, 1020–1046.Google Scholar
Stieltjes, T. J. (1894) Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, 1–122; 9A (1895) 1-47. (Reprinted in Mem. Acad. Sci. Paris 33, 1-196 and in Stieltjes (1918), 402-566.) Google Scholar
Stieltjes, T. J. (1918) Oeuvres complètes. Vol. 2. Noordhoff, Groningen.Google Scholar
Wall, H. S. (1967) Analytic Theory of Continued Fractions. Chelsea, New York.Google Scholar