Analytic birth--death processes: A Hilbert-space approach

Markus Kreer

Stochastic Processes and their Applications, 1994, vol. 49, issue 1, 65-74

Abstract: Methods of Hilbert space theory together with the theory of analytic semigroups lead to an alternative approach for discussing an analytic birth and death process with the backward equations gk = [lambda]k - 1gk - 1 - ([mu]k+[lambda]k)gk+[mu]k+1, k = 0, 1, 2, ..., where [lambda]- 1 = 0 = [mu]0. For rational growing forward and backward transition rates [lambda]k = O(k[gamma]), [mu]k = O(k[gamma]) (as k --> [infinity]), with 0 0) can be proved under fairly general conditions; so can the discreteness of the spectrum. Even in the critical case of asymptotically symmetric transition rates [lambda]k ~ [mu]k ~ k[gamma] one obtains for rational growing transition rates with 0

Keywords: infinite; tridiagonal; matrices; discreteness; of; spectrum; analytic; semigroups (search for similar items in EconPapers)
Date: 1994
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