Revisiting recurrence criteria of birth and death processes. Short proofs

IF 0.4 Q4 STATISTICS & PROBABILITY
O. Zakusylo
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引用次数: 0

Abstract

The paper contains several new transparent proofs of criteria appearing in classification of birth and death processes (BDPs). They are almost purely probabilistic and differ from the classical techniques of three-term recurrence relations, continued fractions and orthogonal polynomials. Let T ∞ {T^\infty } be the passage time from zero to ∞ \infty . The regularity criterion says that T ∞ > ∞ {T^\infty } > \infty if and only if E T ∞ > ∞ \mathbb {E}{T^\infty } > \infty . It is heavily based on a result of Gong, Y., Mao, Y.-H. and Zhang, C. [J. Theoret. Probab. 25 (2012), no. 4, 950–980]. We obtain the latter expectation by using a two-term recurrence relation. We observe that the recurrence criterion is an immediate consequence of the well-known recurrence criterion for discrete-time BDPs and a result of Chung K. L. [Markov Chains with Stationary Transition Probabilities, Springer-Verlag, New York (1967)]. We obtain the classical criterion of positive recurrence using technique of the common probability space. While doing so, we construct a monotone sequence of BDPs with finite state spaces converging to BDPs with an infinite state space.
重新审视出生和死亡过程的复发标准。简短的证明
本文包含了出生和死亡过程分类标准的几个新的透明证明。它们几乎是纯概率的,不同于三项递推关系、连分式和正交多项式的经典技术。设T∞{T^\infty}为从零到∞\infty的通过时间。正则性准则表明,T∞>∞{T^\infty}>infty当且仅当E T∞>∞\mathbb{E}{T^ \infty}>infty。这在很大程度上是基于龚、毛和张的一个结果。[J.Theoret.Probab.252012,no.4950-980]。我们通过使用两项递推关系得到了后一种期望。我们观察到,递推准则是众所周知的离散时间BDP递推准则的直接结果,也是Chung K.L.[具有平稳转移概率的马尔可夫链,Springer Verlag,纽约(1967)]的结果。利用公共概率空间技术,得到了正递推的经典判据。在这样做的同时,我们构造了一个具有有限状态空间的BDP的单调序列,该序列收敛于具有无限状态空间的BD。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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CiteScore
1.30
自引率
0.00%
发文量
22
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