Subcritical Branching Processes Research Articles

On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

Abstract The probability generating function (pgf) B ⁢ ( s ) {B(s)} of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) {(Z_{t}:t\geq 0)} (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 {B(0)=0} and B ′ ⁢ ( 0 ) > 0 {B^{\prime}(0)>0} . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 ( Z t = j ) {P_{1}(Z_{t}=j)} have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.

Subcritical multitype Markov branching processes with immigration generated by Poisson random measures

We investigate multitype subcritical Markov branching processes with immigration driven by Poisson random measures. Limiting distributions are established for various rates of the Poisson measures when they are asymptotically equivalent to exponential or regularly varying functions. Results analogous to a strong LLN are proved, and limiting normal distributions are obtained when the local intensity of the Poisson measure increases with time. When it decreases, conditional limiting distributions are established. When the intensity converges to a constant, a stationary limiting distribution is obtained. The asymptotic behavior of the first and second moments of the processes is also investigated.

Extended Sibuya Distribution in Subcritical Markov Branching Processes

The subcritical Markov branching process X(t) starting with one particle as the initial condition has the ultimate extinction probability q = 1. The branching mechanism in consideration is defined by the mixture of logarithmic distributions on the nonnegative integers. The purpose of the present paper is to prove that in this case the random number of particles X(t) alive at time t > 0 follows the shifted extended Sibuya distribution, with parameters depending on the time t > 0. The conditional limit probability is the logarithmic series distribution supported by the positive integers. 

Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations

Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). We study the flow of ancestral lineages as time goes to the past and show that, properly renormalized, it converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. The inverse subordinator is shown to be partitioning the current population into ancestral families with distinct common ancestors. When Grey’s condition is satisfied, the population comes from a discrete set of ancestors and the ancestral families have i.i.d. sizes distributed according to the quasi-stationary distribution of the CSBP conditioned on non-extinction. When Grey’s condition is not satisfied, the population comes from a continuum of ancestors which is described as the set of increase points S of the limiting inverse subordinator. The proof is based on a general result for stochastically monotone processes of independent interest, which relates θ-invariant measures and θ-invariant functions for a process and its Siegmund dual.

Long-Time Behavior for Subcritical Measure-Valued Branching Processes with Immigration

AbstractIn this work we study the long-time behavior for subcritical measure-valued branching processes with immigration on the space of tempered measures. Under some reasonable assumptions on the spatial motion, the branching and immigration mechanisms, we prove the existence and uniqueness of an invariant probability measure for the corresponding Markov transition semigroup. Moreover, we show that it converges with exponential rate to the unique invariant measure in the Wasserstein distance as well as in a distance defined in terms of Laplace transforms. Finally, we consider an application of our results to super-Lévy processes as well as branching particle systems on the lattice with noncompact spins.

Intermediately Subcritical Branching Process in a Random Environment: The Initial Stage of the Evolution

We consider branching process evolving in i.i.d. random environment. It is assumed that the process is intermediately subcritical. We investigate the initial stage of the evolution of the process given its survival for a long time.

Large Deviations of a Strongly Subcritical Branching Process in a Random Environment

Large Deviations of a Strongly Subcritical Branching Process in a Random Environment

Properties of multitype subcritical branching processes in random environment

Abstract We study properties of a p-type subcritical branching process in random environment initiated at moment zero by a vector z = (z 1, .., z p ) of particles of different types. For p = 1 the class of processes we consider corresponds to the so-called strongly subcritical case. It is shown that the survival probability of this process up to moment n behaves as C(z)λ n for large n, where the parameters λ ∈ (0, 1) and C(z) ∈ (0, ∞) are explicitly described in terms of the characteristics of the process. We also demonstrate that the distribution of the number of particles of different types at moment n → ∞ (given its survival up to this moment) does not asymptotically depend on the number and types of particles initiated the process.

Branching processes with immigration in atypical random environment

Motivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters An, n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of xi _{n}:=log ((1-A_{n})/A_{n}) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail mathbb {P}(Z_{n} ge m) of the n th population size Zn is asymptotically equivalent to noverline F(log m) as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 154, 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter α > 1. Further, for a subcritical branching process with subexponentially distributed ξn, we provide the asymptotics for the distribution tail mathbb {P}(Z_{n}>m) which are valid uniformly for all n, and also for the stationary tail distribution. Then we establish the “principle of a single atypical environment” which says that the main cause for the number of particles to be large is the presence of a single very small environmental parameter Ak.

Multitype weakly subcritical branching processes in random environment

Abstract A multi-type branching process evolving in a random environment generated by a sequence of independent identically distributed random variables is considered. The asymptotics of the survival probability of the process for a long time is found under the assumption that the matrices of the mean values of direct descendants have a common left eigenvector and the increment X of the associated random walk generated by the logarithms of the Perron roots of these matrices satisfies conditions E X < 0 and E XeX > 0.

Period-Life of a Branching Process with Migration and Continuous Time

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

COVID-19 (SARS-CoV-2) Outbreaks in Rural United Kingdom: Patterns of Spread and Empirical Evidence for the 'Rule of Six'

Background: Multiple countries across the world are facing the second wave of severe acute respiratory syndrome coronavirus. Understanding the early transmission dynamics is crucial for outbreak control measures and evaluating public health policy. Methods: We used detailed contact tracing data for the rural county of Lincolnshire to quantify risk factors of infection and model disease spread. We used subcritical branching processes ( Findings: 158,390 individuals were contact traced between the periods of May 11, 2020 to September 14, 2020 during the early phase of the second wave with 1143 confirmed positives in Lincolnshire. We found empirical evidence in support of the “rule of 6” for secondary contacts and largest gathering component size of 60 people. There is highly-over dispersed offspring distribution k=0·35 95% CI [0·41-0·29] indicative of super spreading events. We found through multivariable analysis, evidence of young adult population driving the outbreak and protection increasing with age for those in isolation. There is an increased risk of 66% (IRR 1·66, 95% CI [1·02 - 2·20]), 50% (IRR 1·50, 95% CI [1·16 - 2·37]) associated with eating out and shopping followed by social events respectively when compared to house hold acquired covid-19 infections. Interpretation: Generating functions and random graph theory help model the local transmission heterogeneities from the very onset of the outbreak. Focusing on the secondary contacts and case cluster size distribution is a better method for detecting the stochastic beginnings of a pandemic, rather than decision-making being based on exceeding particular threshold. Contact tracing and viral testing is essential to provide the reliable data and help with the public health policy for effective interventions. Funding: NoneDeclaration of Interests: None.Ethics Approval Statement: Ethics permission has been granted by University of Lincoln Ethics Committee (Protocol ID: 4188).

Subcritical Branching Processes in Random Environment with Immigration: Survival of a Single Family

We consider a subcritical branching process in an independent and identically distributed (i.i.d.) random environment, where one immigrant arrives at each generation. We consider the event $\mathca...

Subcritical Branching Processes in Random Environment with Immigration Stopped at Zero

We consider the subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are change of measure and some conditional limit theorems for random walks.

On central limit theorems for branching processes with dependent immigration

In this paper we consider subcritical and supercritical discrete time branching processes with generation dependent immigration. We prove central limit theorems for fluctuation of branching processes with immigration when the mean of immigrating individuals tends to infinity with the generation number and immigration process is m−dependent. The first result states on weak convergence of the fluctuation subcritical branching processes with m−dependent immigration to standard normal distribution. In this case, we do not assume that the mean and variance of immigration process are regularly varying at infinity. In contrast, in Theorem 3.2, we suppose that the mean and variance are to be regularly varying at infinity. The proofs are based on direct analytic method of probability theory.

The Initial Evolution Stage of a Weakly Subcritical Branching Process in a Random Environment

A conditional limit theorem is proved describing the distribution of the population size in the initial evolution stage of a weakly subcritical branching process in a random environment given its s...

The Survival Probability for a Class of Multitype Subcritical Branching Processes in Random Environment

The asymptotic behavior of the survival probability for multi-type branching processes in a random environment is studied. In the case where all particles are of one type, the class of processes under consideration corresponds to intermediately subcritical processes. Under fairly general assumptions on the form of the generating functions of the laws of reproduction of particles, it is proved that the survival probability at a remote instant n of time for a process that started at the zero instant of time from one particle of any type is of the order of λnn−1/2, where λ ∈ (0, 1) is a constant defined in terms of the Lyapunov exponent for products of the mean-value matrices of the laws of reproduction of particles.

Multitype weakly subcritical branching processes in random environment

Рассматривается многотипный ветвящийся процесс Гальтона-Ватсона в случайной среде, задаваемой последовательностью независимых одинаково распределенных случайных величин. В предположении, что приращение $X$ сопровождающего случайного блуждания, порожденного логарифмами перроновых корней матриц средних этого процесса, удовлетворяет условиям $\mathbf{E}X<0$ и $\mathbf{E}Xe^{X}>0,$ а матрицы средних процесса имеют общий неслучайный левый собственный вектор, найдена асимптотика вероятности невырождения процесса в далекий момент времени.

Multi-type subcritical branching processes in a random environment

Abstract We study the asymptotic behavior of the survival probability of a multi-type branching process in a random environment. In the one-dimensional situation, the class of processes considered corresponds to the strongly subcritical case. We also prove a conditional limit theorem describing the distribution of the number of particles in the process given its survival for a long time.

A subcritical decomposable branching process in a mixed environment

AbstractA two-type decomposable branching process is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. The reproduction law of the first type particles is specified by a random environment. The reproduction law of the second type particles is one and the same for all generations.A limit theorem is proved describing the conditional distribution of the number of particles in the process at timent,t∈ (0,1], given the survival of the process up to momentn→ ∞.