Excursions of birth and death processes, orthogonal polynomials, and continued fractions

1999 ◽  
Vol 36 (3) ◽  
pp. 752-770 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Orthogonal Polynomial ◽  
Continued Fractions ◽  
Spectral Measure ◽  
Continued Fraction ◽  
Transition Probability ◽  
Polynomial System ◽  
Death Process ◽  
Probabilistic Interpretation ◽  
Birth And Death Process ◽  
Birth And Death Processes

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

1999 ◽  
Vol 36 (03) ◽  
pp. 752-770 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Orthogonal Polynomial ◽  
Continued Fractions ◽  
Spectral Measure ◽  
Continued Fraction ◽  
Transition Probability ◽  
Polynomial System ◽  
Death Process ◽  
Probabilistic Interpretation ◽  
Birth And Death Process ◽  
Birth And Death Processes

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

ZIPF AND LERCH LIMIT OF BIRTH AND DEATH PROCESSES

2009 ◽  
Vol 24 (1) ◽  
pp. 129-144 ◽  
Author(s):  
B. Klar ◽  
P. R. Parthasarathy ◽  
N. Henze
Keyword(s):  
Chemical Kinetics ◽  
Continued Fractions ◽  
Zipf’S Law ◽  
Death Process ◽  
Speed Of Convergence ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Zipf's Law ◽  
Wide Range ◽  
Epidemiology Data

Birth and death processes are useful in a wide range of disciplines from computer networks and telecommunications to chemical kinetics and epidemiology. Data from many different areas such as linguistics, music, or warfare fit Zipf's law surprisingly well. The Lerch distribution generalizes Zipf's law and is applicable in survival and dispersal processes. In this article we construct a birth and death process that converges to the Lerch distribution in the limit as time becomes large, and we investigate the speed of convergence. This is achieved by employing continued fractions. Numerical illustrations are presented through tables and graphs.

scholarly journals The limiting behavior of transient birth and death processes conditioned on survival

1968 ◽  
Vol 8 (4) ◽  
pp. 716-722 ◽  
Author(s):  
Phillip Good
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Death Process ◽  
Initial Condition ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Limiting Behavior ◽  
Image Position ◽  
Number Of Individuals ◽  
Over Time

The development of a population over time can often be simulated by the behavior of a birth and death process, whose transition probability matrix P(t) = (Pij(t), where X(t) denotes the number of individuals at time t, satisfies the differential equations and the initial condition

On a property of finite-state birth and death processes

1986 ◽  
Vol 23 (04) ◽  
pp. 859-866
Author(s):  
A. J. Branford
Keyword(s):  
Simple Proof ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Converse Result ◽  
Finite State ◽  
Event Times

A simple proof is given of the result that the ‘overflow' from a finite-state birth and death process is a renewal stream characterized by hyperexponential inter-event times. Our structure is utilized to give a converse result that any hyperexponential renewal stream can be so produced as the overflow from a finite-state birth and death process.

On the parity of individuals in birth and death processes

1982 ◽  
Vol 14 (03) ◽  
pp. 484-501
Author(s):  
S. K. Srinivasan ◽  
C. R. Ranganathan
Keyword(s):  
General Model ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth Rates ◽  
Birth And Death Processes ◽  
Age Dependent ◽  
The Mean ◽  
Number Of Individuals ◽  
Number Of Births

This paper deals with the parity of individuals in an age-dependent birth and death process. A more general model with parity and age-dependent birth rates is also considered. The mean number of individuals with parity 0, 1, 2, ·· ·is obtained for the two models. The first moments of the total number of births in the population up to time t and the sum of the parities of the individuals existing at time t are obtained. A brief discussion on the parity of individuals in a population including ‘twins' is also given.

Birth and death processes with random environments in continuous time

1981 ◽  
Vol 18 (01) ◽  
pp. 19-30 ◽  
Author(s):  
Robert Cogburn ◽  
William C. Torrez
Keyword(s):  
Discrete Time ◽  
Random Environment ◽  
Continuous Time ◽  
Death Process ◽  
Random Environments ◽  
Birth And Death Process ◽  
Time Model ◽  
Birth And Death Processes ◽  
Discrete Time Model ◽  
Extinction Probabilities

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.

Application of Stieltjes theory for S-fractions to birth and death processes

1983 ◽  
Vol 15 (03) ◽  
pp. 507-530 ◽  
Author(s):  
G. Bordes ◽  
B. Roehner
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Special Class ◽  
Jacobi Matrix ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Death Process ◽  
Nearest Neighbour ◽  
Birth And Death Process ◽  
Birth And Death Processes

We are interested in obtaining bounds for the spectrum of the infinite Jacobi matrix of a birth and death process or of any process (with nearest-neighbour interactions) defined by a similar Jacobi matrix. To this aim we use some results of Stieltjes theory for S-fractions, after reviewing them. We prove a general theorem giving a lower bound of the spectrum. The theorem also gives sufficient conditions for the spectrum to be discrete. The expression for the lower bound is then worked out explicitly for several, fairly general, classes of birth and death processes. A conjecture about the asymptotic behavior of a special class of birth and death processes is presented.

scholarly journals Birth and death processes in randomly fluctuating environments

2002 ◽  
Vol 166 ◽  
pp. 93-115
Author(s):  
Kanji Ichihara
Keyword(s):  
Random Environment ◽  
Time Dependent ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Fluctuating Environments

AbstractA birth and death process in a time-dependent random environment is introduced. We will discuss the recurrence and transience properties for the process.

Multivariate birth-and-death processes as approximations to epidemic processes

1973 ◽  
Vol 10 (1) ◽  
pp. 15-26 ◽  
Author(s):  
D. A. Griffiths
Keyword(s):  
Transient Process ◽  
Branching Process ◽  
Large Population ◽  
Disease Spread ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Epidemic Size

This paper presents the theory of a multivariate birth-and-death process and its representation as a branching process. The bivariate linear birth-and-death process may be used as a model for various epidemic situations involving two types of infective. Various properties of the transient process are discussed and the distribution of epidemic size is investigated. For the case of a disease spread solely by carriers when the two types of infective are carriers and clinical infectives the large population version of a model proposed by Downton (1968) is further developed and shown under appropriate circumstances to closely approximate Downton's model.

Weak convergence of conditioned birth and death processes

1994 ◽  
Vol 31 (1) ◽  
pp. 90-100 ◽  
Author(s):  
G. O. Roberts ◽  
S. D. Jacka
Keyword(s):  
Weak Convergence ◽  
Independent Interest ◽  
Death Process ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Monotonic Properties

We consider the problem of conditioning a non-explosive birth and death process to remain positive until time T, and consider weak convergence of this conditional process as T → ∞. By a suitable almost sure construction we prove weak convergence. The almost sure construction used is of independent interest but relies heavily on the strong monotonic properties of birth and death processes.

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