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WEAK CONVERGENCE OF CONDITIONED BIRTH-DEATH PROCESSES IN DISCRETE TIME
 

Summary: WEAK CONVERGENCE OF CONDITIONED BIRTH-DEATH
PROCESSES IN DISCRETE TIME
PAULINE SCHRIJNER AND
ERIK A. VAN DOORN , University of Twente
Abstract
We consider a discrete-time birth-death process on the nonnegative integers
with -1 as an absorbing state and study the limiting behaviour as n of the
process conditioned on nonabsorption until time n. By proving that a condition
recently proposed by Mart´inez and Vares is vacuously true, we establish that
the conditioned process is always weakly convergent when all self-transition
probabilities are zero. In the aperiodic case we obtain a necessary and sufficient
condition for weak convergence.
BIRTH-DEATH PROCESS, CONDITIONED PROCESS, RATIO LIMIT, WEAK CONVERGENCE
AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J80
1 Introduction
Roberts and Jacka [3] consider a continuous-time birth-death process on the positive
integers with absorption at zero and study the limiting behaviour as T of the
process conditioned to remain positive until time T. Under the assumptions of
nonexplosiveness, irreducibility and certain absorption of the original process, they
show that the conditioned process converges weakly to a time-homogeneous birth-

  

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

 

Collections: Engineering