Journal ArticleDOI
On calculating extinction probabilities for branching processes in random environments
TLDR
In this article, the authors considered a discrete time Markov chain whose state space is the nonnegative integers and whose transition probability matrix possesses the representation of a sequence of non-negative real numbers satisfying, and a corresponding sequence of probability generating functions.Abstract:
Consider a discrete time Markov chain { Z n } whose state space is the non-negative integers and whose transition probability matrix ║ P ij ║ possesses the representation
where { P r }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k , a finite positive integer.read more
Citations
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Journal ArticleDOI
Branching processes with varying and random geometric offspring distributions
Niels Keiding,John E. Nielsen +1 more
TL;DR: In this article, it was shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case, and examples were given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.
Journal ArticleDOI
The fractional linear probability generating function in the random environment branching process
D. R. Grey,Lu Zhunwei +1 more
TL;DR: In this paper, the authors assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters.
Journal ArticleDOI
Calculating extinction probabilities for the birth and death chain in a random environment
TL;DR: In this paper, conditions for extinction and instability of a stochastic process evolving in a random environment controlled by an irreducible Markov chain (Yn) with state space ǫ = {0, 1, ···, N} were given.
Journal ArticleDOI
On determining absorption probabilities for markov chains in random environments
TL;DR: The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed in this article, where an efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov Chain.
References
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Book
An Introduction to the Theory of Numbers
TL;DR: The fifth edition of the introduction to the theory of numbers has been published by as discussed by the authors, and the main changes are in the notes at the end of each chapter, where the author seeks to provide up-to-date references for the reader who wishes to pursue a particular topic further and to present a reasonably accurate account of the present state of knowledge.
Book
The Theory of Branching Processes
TL;DR: A review of the Galton and Watson mathematical model that applies probability theory to the effects of chance on the development of populations is given in this article, followed by a systematic development of branching processes, and a brief description of some of the important applications.
Journal ArticleDOI
Applications of the theory of matrices
Journal ArticleDOI
On Branching Processes in Random Environments
TL;DR: In this article, the branching process of a Markov chain is defined as a branching process such that each point in the chain is associated with a variable associated with an operating system.
Related Papers (5)
On Branching Processes with Random Environments: I: Extinction Probabilities
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Branching Processes with Random Environments, II: Limit Theorems
Krishna B. Athreya,Samuel Karlin +1 more