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

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.