Multitype linear fractional branching processes

We complete a paper written by Edward Pollak in 1974 on a multitype branching process the generating functions of whose birth law are fractional linear functions with the same denominator. The main tool is a parameterization of these functions adapted using the mean matrix M and an element w of the first quadrant. We use this opportunity to give a self-contained presentation of Pollak's theory.

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Extinction Probabilities of Branching Processes with Countably Infinitely Many Types

Advances in Applied Probability ◽

10.1017/s0001867800006777 ◽

2013 ◽

Vol 45 (04) ◽

pp. 1068-1082 ◽

Cited By ~ 1

Author(s):

S. Hautphenne ◽

G. Latouche ◽

G. Nguyen

Keyword(s):

Population Size ◽

Iterative Methods ◽

Generating Functions ◽

Branching Process ◽

Branching Processes ◽

Sufficient Condition ◽

Probabilistic Interpretation ◽

Extinction Probabilities ◽

The Mean ◽

Partial Extinction

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.

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Extinction Probabilities of Branching Processes with Countably Infinitely Many Types

Advances in Applied Probability ◽

10.1239/aap/1386857858 ◽

2013 ◽

Vol 45 (4) ◽

pp. 1068-1082 ◽

Cited By ~ 8

Author(s):

S. Hautphenne ◽

G. Latouche ◽

G. Nguyen

Keyword(s):

Population Size ◽

Iterative Methods ◽

Generating Functions ◽

Branching Process ◽

Branching Processes ◽

Sufficient Condition ◽

Probabilistic Interpretation ◽

Extinction Probabilities ◽

The Mean ◽

Partial Extinction

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.

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Branching processes and functional differential equations determining steady-size distributions in cell populations

Journal of Applied Probability ◽

10.1017/s0021900200102529 ◽

1995 ◽

Vol 32 (01) ◽

pp. 1-10

Author(s):

Ziad Taib

Keyword(s):

Differential Equation ◽

Branching Process ◽

Branching Processes ◽

Functional Differential Equations ◽

Cell Populations ◽

Size Distributions ◽

Multitype Branching Process ◽

Functional Differential ◽

Intuitive Interpretation ◽

Infinite Sum

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

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Continuity for multi-type branching processes with varying environments

Journal of Applied Probability ◽

10.1017/s0021900200016910 ◽

1999 ◽

Vol 36 (01) ◽

pp. 139-145 ◽

Cited By ~ 1

Author(s):

Owen Dafydd Jones

Keyword(s):

Linear Combination ◽

Branching Process ◽

Branching Processes ◽

Random Variables ◽

Main Tool ◽

Concentration Function ◽

Independent Random Variables ◽

Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

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Asymptotic behaviour of immigration-branching processes by general set of types. II: Supercritical branching part

Advances in Applied Probability ◽

10.1017/s000186780004074x ◽

1975 ◽

Vol 7 (03) ◽

pp. 468-494

Author(s):

H. Hering

Keyword(s):

Branching Process ◽

Branching Processes ◽

Almost Sure Convergence ◽

Transition Function ◽

Mean Square ◽

Second Moments ◽

Mean Square Convergence ◽

The Mean ◽

Parameter Dependent ◽

Dependent Probability

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

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Modeling Y-Linked Pedigrees through Branching Processes

Mathematics ◽

10.3390/math8020256 ◽

2020 ◽

Vol 8 (2) ◽

pp. 256

Author(s):

Miguel González ◽

Cristina Gutiérrez ◽

Rodrigo Martínez

Keyword(s):

Hearing Loss ◽

Asymptotic Behavior ◽

Gibbs Sampler ◽

Mutant Allele ◽

Branching Process ◽

Branching Processes ◽

Dominant Allele ◽

Linked Gene ◽

Different Types ◽

The Mean

A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-linked pedigree associated with hearing loss is analyzed, concluding that this mutation will persist in the population although without dominating the pedigree.

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The Properties of Generalized Collision Branching Processes

Mathematical Problems in Engineering ◽

10.1155/2020/1398476 ◽

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.

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

Journal of Applied Probability ◽

10.1239/jap/1032374477 ◽

1999 ◽

Vol 36 (2) ◽

pp. 611-619 ◽

Cited By ~ 6

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.

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Contagions in random networks with overlapping communities

Advances in Applied Probability ◽

10.1239/aap/1449859796 ◽

2015 ◽

Vol 47 (4) ◽

pp. 973-988 ◽

Cited By ~ 1

Author(s):

Emilie Coupechoux ◽

Marc Lelarge

Keyword(s):

Phase Transition ◽

Random Graph ◽

Epidemic Model ◽

Branching Process ◽

Branching Processes ◽

Graph Model ◽

Single Individual ◽

Overlapping Communities ◽

Random Graph Model ◽

The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

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