Ӧ�ø���ͳ�� 2013, 29(6) 561-569 DOI:      ISSN: 1001-4268 CN: 31-1256

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Domain of Attraction of the Quasi-Stationary Distribution for the Linear Birth and Death Process with Killing
Zhang Hanjun, Zhu Yixia
School of Mathematics and Computational Science of Xiangtan University

The model of linear birth and death
processes with killing has been studied by Karlin and Tavar
(1982). This paper is concerned with three problems in connection
with quasi-stationary distributions (QSDs) for linear birth-death
process  with killing on a semi-infinite lattice of integers.
The first problem is to determine the decay parameter  of
. We have
are the birth, death and killing rates
in state , respectively. The second one is to prove the
uniqueness of the QSD which is a geometric distribution. It is
interesting to find that the unkilled process has a one-parameter
family of QSDs while the killed process has precisely one QSD. The
last one is to solve the domain of attraction problem, that is, we
obtain that any initial distribution is in the domain of attraction
of the unique QSD for . Our study is motivated by the
population genetics problem.

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1��������, �����.һ�൥�����̵ı�����[J]. Ӧ�ø���ͳ��, 2007,23(4): 377-383
2��������~~κ����.�����ʳ--����ʳģ��[J]. Ӧ�ø���ͳ��, 2009,25(4): 375-380
3��������,�ֹ���.������̵Ĺ���ṹ[J]. Ӧ�ø���ͳ��, 2013,29(1): 10-22

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