On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, 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.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (3) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn} whose state space is the non-negative integers and whose transition probability matrix ║Pij║ possesses the representation where {Pr}, 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 Z0 = k, a finite positive integer.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (01) ◽  
pp. 135-141 ◽  
Author(s):  
Niels Keiding ◽  
John E. Nielsen
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Elementary Proof ◽  
Branching Processes ◽  
Random Environments ◽  
Subcritical Case ◽  
Supercritical Case ◽  
Extinction Probabilities ◽  
Varying Environments

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (1) ◽  
pp. 135-141 ◽  
Author(s):  
Niels Keiding ◽  
John E. Nielsen
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Elementary Proof ◽  
Branching Processes ◽  
Random Environments ◽  
Subcritical Case ◽  
Supercritical Case ◽  
Extinction Probabilities ◽  
Varying Environments

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

scholarly journals The Properties of Generalized Collision Branching Processes

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Juan Wang ◽  
Chunhao Cai
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Branching Processes ◽  
Hitting Times ◽  
Conditional Mean ◽  
Basic Properties ◽  
Extinction Probabilities ◽  
The Mean

We consider basic properties regarding uniqueness, extinction, and explosivity for the Generalized Collision Branching Processes (GCBP). Firstly, we investigate some important properties of the generating functions for GCB q-matrix in detail. Then for any given GCB q-matrix, we prove that there always exists exactly one GCBP. Next, we devote to the study of extinction behavior and hitting times. Some elegant and important results regarding extinction probabilities, the mean extinction times, and the conditional mean extinction times are presented. Moreover, the explosivity is also investigated and an explicit expression for mean explosion time is established.

Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

1999 ◽  
Vol 36 (2) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Environments ◽  
Real Numbers ◽  
Size Dependent ◽  
Recurrent Markov Chain ◽  
The Mean ◽  
Positive Recurrent

A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.

scholarly journals Frekuensi Hari Hujan Menurut Bulan di Kota Balikpapan dengan Rantai Markov Waktu Diskrit

2019 ◽  
Vol 1 (2) ◽  
pp. 5-10
Author(s):  
Muhammad Azka
Keyword(s):  
Markov Chain ◽  
Discrete Time ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Transition Diagram ◽  
Discrete Time Markov Chain ◽  
Applied Method ◽  
Frequency Rate

The problem proposed in this research is about the amount rainy day per a month at Balikpapan city and discretetime markov chain. The purpose is finding the probability of rainy day with the frequency rate of rainy at the next month if given the frequency rate of rainy at the prior month. The applied method in this research is classifying the amount of rainy day be three frequency levels, those are, high, medium, and low. If a month, the amount of rainy day is less than 11 then the frequency rate for the month is classified low, if a month, the amount of rainy day between 10 and 20, then it is classified medium and if it is more than 20, then it is classified high. The result is discrete-time markov chain represented with the transition probability matrix, and the transition diagram.

On Null-Recurrent Markov Chains

1960 ◽  
Vol 12 ◽  
pp. 278-288 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Invariant Measure ◽  
Discrete Time ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Positive Vector ◽  
Scalar Multiple ◽  
Basic Facts ◽  
Image Position

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or1this vector is often called the "invariant measure" of the Markov chain.The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components2In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

1999 ◽  
Vol 36 (02) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Environments ◽  
Real Numbers ◽  
Size Dependent ◽  
Recurrent Markov Chain ◽  
The Mean ◽  
Positive Recurrent

A population-size-dependent branching process {Z n } is considered where the population's evolution is controlled by a Markovian environment process {ξ n }. For this model, let m k,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Z n (ω) = 0 for some n} and q = P(B). The asymptotic behaviour of lim n Z n and is studied in the case where supθ|m k,θ − m θ| → 0 for some real numbers {m θ} such that infθ m θ &gt; 1. When the environmental sequence {ξ n } is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q &lt; 1) are studied.

scholarly journals On Packings of Spheres in Hilbert Space

1955 ◽  
Vol 2 (3) ◽  
pp. 145-146 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Hilbert Space ◽  
Unit Sphere ◽  
Infinite Sequence ◽  
Real Hilbert Space ◽  
Real Numbers ◽  
Image Position

A point x in real Hilbert space is represented by an infinite sequence (x1, x2, x3, …) of real numbers such thatis convergent. The unit “sphere“ S consists of all points × for which ‖x‖ ≤ 1. The sphere of radius a and centre y is denoted by Sa(y) and consists of all points × for which ‖x−y‖ ≤ a.

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