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Conditions for the existence of quasi-stationary distributions for birth-death processes with killing
 

Summary: Conditions for the existence of quasi-stationary
distributions for birth-death processes with killing
Erik A. van Doorn
Department of Applied Mathematics, University of Twente
P.O. Box 217, 7500 AE Enschede, The Netherlands
E-mail: e.a.vandoorn@utwente.nl
August 18, 2011
Abstract. We consider birth-death processes on the nonnegative integers,
where {1, 2, . . . } is an irreducible class and 0 an absorbing state, with the ad-
ditional feature that a transition to state 0 (killing) may occur from any state.
Assuming that absorption at 0 is certain we are interested in additional condi-
tions on the transition rates for the existence of a quasi-stationary distribution.
Inspired by results of M. Kolb and D. Steinsaltz (Quasilimiting behaviour for
one-dimensional diffusions with killing, Annals of Probability, to appear) we
show that a quasi-stationary distribution exists if the decay rate of the process
is positive and exceeds at most finitely many killing rates. If the decay rate
is positive and smaller than at most finitely many killing rates then a quasi-
stationary distribution exists if and only if the process one obtains by setting
all killing rates equal to zero is recurrent.
Keywords: birth-death process with killing, orthogonal polynomials, quasi-

  

Source: Al Hanbali, Ahmad - Department of Applied Mathematics, Universiteit Twente

 

Collections: Engineering