# A birth-death process suggested by a chain sequence

TL;DR: In this article, the authors consider a birth-death process whose birth and death rates are suggested by a chain sequence and use an elegant transformation to find the transition probabilities in a simple closed form.

Abstract: We consider a birth-death process whose birth and death rates are suggested by a chain sequence. We use an elegant transformation to find the transition probabilities in a simple closed form. We also find an explicit expression for time-dependent mean. We find parallel results in discrete time. Finally, we show that the processes under investigation are transient, and hence, the stationary distribution does not exist.

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TL;DR: In this paper, the existence and uniqueness of Q-functions are discussed. But they focus on transition functions and do not address the uniqueness problem in the context of transition functions.

Abstract: Contents: Transition Functions and Resolvents.- Existence and Uniqueness of Q-Functions.- Examples of Continuous Time Markov Chains.- More on the Uniqueness Problem.- Classification of States and Invariant Measures.- Strong and Exponential Ergodicity.- Reversibility, Monotonictity, and Other Properties.- Birth and Death Processes.- Population Processes.- Bibliography.- Symbol Index.- Author Index.- Subject Index.

348 citations

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TL;DR: A method able to construct a new birth-death process M(t) defined on the same state-space and the connection between the proposed method and the notion of ?

18 citations

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TL;DR: The basic mathematical theory for BDPs is outlined and new tools for statistical inference using data from BDP's are demonstrated, showing thatlihood-based inference for previously intractable B DPs is much easier than previously thought and regression approaches analogous to Poisson regression are straightforward to derive.

Abstract: Many important stochastic counting models can be written as general birth-death processes (BDPs). BDPs are continuous-time Markov chains on the non-negative integers in which only jumps to adjacent states are allowed. BDPs can be used to easily parameterize a rich variety of probability distributions on the non-negative integers, and straightforward conditions guarantee that these distributions are proper. BDPs also provide a mechanistic interpretation - birth and death of actual particles or organisms - that has proven useful in evolution, ecology, physics, and chemistry. Although the theoretical properties of general BDPs are well understood, traditionally statistical work on BDPs has been limited to the simple linear (Kendall) process. Aside from a few simple cases, it remains impossible to find analytic expressions for the likelihood of a discretely-observed BDP, and computational difficulties have hindered development of tools for statistical inference. But the gap between BDP theory and practical methods for estimation has narrowed in recent years. There are now robust methods for evaluating likelihoods for realizations of BDPs: finite-time transition, first passage, equilibrium probabilities, and distributions of summary statistics that arise commonly in applications. Recent work has also exploited the connection between continuously- and discretely-observed BDPs to derive EM algorithms for maximum likelihood estimation. Likelihood-based inference for previously intractable BDPs is much easier than previously thought and regression approaches analogous to Poisson regression are straightforward to derive. In this review, we outline the basic mathematical theory for BDPs and demonstrate new tools for statistical inference using data from BDPs.

16 citations

### Cites background from "A birth-death process suggested by ..."

...Lenin and Parthasarathy (2000) and Parthasarathy, Lenin, Schoutens, and Van Assche (1998) discuss further some well-known continued fractions whose connection to BDPs previously went unappreciated....

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TL;DR: The LPM-ARTICLE-2008-017 was published in the journal Physleta (10.1016/j.physleta.2008.01.082).

13 citations

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TL;DR: A time-inhomogeneous BD chain symmetric with respect to zero-state is constructed starting from the restricted process, and closed form expressions for the transition probabilities and for the conditional moments are obtained.

11 citations

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3,151 citations

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TL;DR: In this paper, the existence and uniqueness of Q-functions are discussed. But they focus on transition functions and do not address the uniqueness problem in the context of transition functions.

Abstract: Contents: Transition Functions and Resolvents.- Existence and Uniqueness of Q-Functions.- Examples of Continuous Time Markov Chains.- More on the Uniqueness Problem.- Classification of States and Invariant Measures.- Strong and Exponential Ergodicity.- Reversibility, Monotonictity, and Other Properties.- Birth and Death Processes.- Population Processes.- Bibliography.- Symbol Index.- Author Index.- Subject Index.

348 citations

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TL;DR: In this article, the results of [1] were used to establish equivalences between properties of the stochastic process and properties of sequences { Xn }, { n,u } and to evaluate, in terms of these sequences, some of the interesting probabilistic quantities associated with the process.

Abstract: In the applications one is given the matrix A and it is required to construct P(t) and to study the properties of the corresponding stochastic process. The existence, uniqueness, and the analytic properties of P(t) have been discussed in detail in [I]. The objective of this paper is to use the results of [1] to establish equivalences between properties of the stochastic process and properties of the sequences { Xn }, { n,u } and to evaluate, in terms of these sequences, some of the interesting probabilistic quantities associated with the process.

329 citations

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07 Aug 1991

TL;DR: In this paper, the existence and uniqueness of Q-functions are discussed. But they focus on transition functions and do not address the uniqueness problem in the context of transition functions.

Abstract: Contents: Transition Functions and Resolvents.- Existence and Uniqueness of Q-Functions.- Examples of Continuous Time Markov Chains.- More on the Uniqueness Problem.- Classification of States and Invariant Measures.- Strong and Exponential Ergodicity.- Reversibility, Monotonictity, and Other Properties.- Birth and Death Processes.- Population Processes.- Bibliography.- Symbol Index.- Author Index.- Subject Index.

327 citations

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TL;DR: An elegant time-dependent solution for the number in the M/M/c queueing system is derived in a direct way.

Abstract: An elegant time-dependent solution for the number in the M/M/c queueing system is derived in a direct way.

48 citations