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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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