Subcritical Branching Research Articles

Markovian paths to extinction

Subcritical Markov branching processes {Zt} die out sooner or later, say at time T < ∞. We give results for the path to extinction {ZuT, 0 ≤ u ≤ 1} that include its finite dimensional distributions and the asymptotic behaviour of xu−1ZuT, as Z0=x → ∞. The limit reflects an interplay of branching and extreme value theory. Then we consider the population on the verge of extinction, as modelled by ZT-u, u > 0, and show that as Z0= x → ∞ this process converges to a Markov process {Yu}, which we describe completely. Emphasis is on continuous time processes, those in discrete time displaying a more complex behaviour, related to Martin boundary theory.

On the path to extinction

Populations can die out in many ways. We investigate one basic form of extinction, stable or intrinsic extinction, caused by individuals on the average not being able to replace themselves through reproduction. The archetypical such population is a subcritical branching process, i.e., a population of independent, asexually reproducing individuals, for which the expected number of progeny per individual is less than one. The main purpose is to uncover a fundamental pattern of nature. Mathematically, this emerges in large systems, in our case subcritical populations, starting from a large number, x, of individuals. First we describe the behavior of the time to extinction T: as x grows to infinity, it behaves like the logarithm of x, divided by r, where r is the absolute value of the Malthusian parameter. We give a more precise description in terms of extreme value distributions. Then we study population size partway (or u-way) to extinction, i.e., at times uT, for 0 < u < 1, e.g., u = 1/2 gives halfway to extinction. (Note that mathematically this is no stopping time.) If the population starts from x individuals, then for large x, the proper scaling for the population size at time uT is x into the power u - 1. Normed by this factor, the population u-way to extinction approaches a process, which involves constants that are determined by life span and reproduction distributions, and a random variable that follows the classical Gumbel distribution in the continuous time case. In the Markov case, where an explicit representation can be deduced, we also find a description of the behavior immediately before extinction.

A class of subcritical branching processes with immigration and infinite number of types of particles

We consider a subcritical branching process with immigration, infinite number of types T1, T2, ... of particles, and discrete time. The state of the process at the moment of time t is the set of vectors

LARGE AND MODERATE DEVIATIONS FOR OCCUPATION TIMES OF IMMIGRATION SUPERPROCESSES

Large and moderate deviation principles are proved for the occupation time process of a subcritical branching superprocess with immigration.

Functional limit theorems for strongly subcritical branching processes in random environment

For a strongly subcritical branching process ( Z n ) n ⩾ 0 in random environment the non-extinction probability at generation n decays at the same exponential rate as the expected generation size and given non-extinction at n the conditional distribution of Z n has a weak limit. Here we prove conditional functional limit theorems for the generation size process ( Z k ) 0 ⩽ k ⩽ n as well as for the random environment. We show that given the population survives up to generation n the environmental sequence still evolves in an i.i.d. fashion and that the conditioned generation size process converges in distribution to a positive recurrent Markov chain.

LIMIT THEOREMS FOR SUBCRITICAL AGE-DEPENDENT BRANCHING PROCESSES WITH TWO TYPES OF IMMIGRATION

ABSTRACT For the classical subcritical age-dependent branching process the effect of the following two-type immigration pattern is studied. At a sequence of renewal epochs a random number of immigrants enters the population. Each subpopulation stemming from one of these immigrants or one of the ancestors is revived by new immigrants and their offspring whenever it dies out, possibly after an additional delay period. All individuals have the same lifetime distribution and produce offspring according to the same reproduction law. This is the Bellman-Harris process with immigration at zero and immigration of renewal type (BHPIOR). We prove a strong law of large numbers and a central limit theorem for such processes. Similar conclusions are obtained for their discrete-time counterparts (lifetime per individual equals one), called Galton-Watson processes with immigration at zero and immigration of renewal type (GWPIOR). Our approach is based on the theory of regenerative processes, renewal theory and occupation measures and is quite different from those in earlier related work using analytic tools.

Limit Theorem for an Intermediate Subcritical Branching Process in a Random Environment

The asymptotic behavior of the survival probability of an intermediate subcritical branching process $Z_n$ in a random environment is found when a transformation of the reproduction law of the offspring number is attracted to a stable law $\alpha\in (1,2]$. It is shown that the distribution of the random variable $\{Z_n\}$ given $Z_n>0$ converges to a nondegenerate distribution as $n\to\infty$.

Limit theorems for subcritical branching processes in random environment

Let ( Z n ) n⩾0 be a branching process in random environment represented by a sequence of i.i.d. generating functions ( f n ) n⩾0 . In the subcritical case, Elog f′(1)<0, the non-extinction probability at generation n decays exponentially fast, the rate depending on whether E[ f′(1)log f′(1)] is less, equal or greater than 0. We determine the exact asymptotic of the non-extinction probability P( Z n >0) in all three cases under suitable integrability assumptions. Moreover, we show that Z n conditioned on Z n >0 has a non-degenerate limit law.

Supercritical Spatially Homogeneous Branching in Admits no Equilibria

At the beginning of investigations in spatially homogeneous branching processes in Euclidean space (Liemant [1]) it seemed to be obvious that the existence of equilibria implies criticality of branching. This prejudice was disproved by the example [2] of a subcritical homogeneous branching equilibrium in dimension one. We prove that supercritical homogeneous branching processes in Euclidean space and, more general, in a broad class of topological groups have no (non – void, homogeneous) equilibria.

Supercritical Spatially Homogeneous Branching in Admits no Equilibria

At the beginning of investigations in spatially homogeneous branching processes in Euclidean space (Liemant [1]) it seemed to be obvious that the existence of equilibria implies criticality of branching. This prejudice was disproved by the example [2] of a subcritical homogeneous branching equilibrium in dimension one. We prove that supercritical homogeneous branching processes in Euclidean space and, more general, in a broad class of topological groups have no (non – void, homogeneous) equilibria.

On the maximum of a subcritical branching process in a random environment

Let { ξ n } be a subcritical branching process in random environment with independent identically distributed generating functions f n ( s). It is shown that if there exists a positive number æ such that E(f 0′(1)) æ=1 then, for x→+∞, P sup n ξ n>x ∼Kx −æ, where K is a positive constant.

Limit theorems for intermediately subcritical and strongly subcritical branching processes in a random environment

Article Limit theorems for intermediately subcritical and strongly subcritical branching processes in a random environment was published on January 1, 2001 in the journal Discrete Mathematics and Applications (volume 11, issue 2).

Cryptanalysis of three mutually clock-controlled stop/go shift registers

A binary stream cipher consisting of three short linear-feedback shift registers (LFSRs) of total length 64 that are mutually clocked in the stop/go manner is cryptanalyzed in the known keystream sequence scenario. To reconstruct the internal state candidates at a known time from about 64 known keystream bits, two algorithms are developed. One is based on guessing a number of elements of the clock-control sequence and has a computational complexity of about 2/sup 40/ steps, where the average step complexity is comparable to the step complexity of the exhaustive search method. The other exploits a time-memory tradeoff based on the well-known birthday paradox and is successful if approximately T/spl middot/M/spl ges/2/sup 64/, where T is the required computational time in table lookups and M is the memory in 64-bit words. As the state-transition function is not one-to-one, to recover the initial state from the internal state candidates, two algorithms are introduced. One consists in guessing the number of clocks for each of the LFSRs. The other consists in the reversion of the internal states and is based on the theory of critical and subcritical branching processes.

Sequential point estimation for branching processes i, subcritical case

For a subcritical branching process with observable immigration, we consider the problem of finding a sample size for which the mean squared error of a natural estimator of the offspring mean is less than a prescribed value u. If the parameters of the process are known, then a minimum sample size achieving the above objective (approximately) exists. When the parameters are unknown, we propose a stopping rule and show that the resulting sequential procedure is asymptotically risk efficient as u approaches zero. Furthermore, we establish the asymptotic efficiency and normality of our stopping rule.

On the Constant in the Definition of Subexponential Distributions

Previous article Next article On the Constant in the Definition of Subexponential DistributionsB. A. RogozinB. A. Rogozinhttps://doi.org/10.1137/S0040585X97977665PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe prove that the constant in the definition of subexponential distributions is equal to 2.[1] V. P. Chistyakov, A theorem on sums of independent positive random variables and its applications to branching random processes, Theory Probab. Appl., 9 (1964), pp. 640–648. tba TPRBAU 0040-585X Theor. Probab. Appl. LinkGoogle Scholar[2] J. Chover, , P. Ney and , and S. Wainger, Degeneracy properties of subcritical branching processes, Ann. Probab., 1 (1973), pp. 663–673. anb APBYAE 0091-1798 Ann. Probab. CrossrefGoogle Scholar[3] J. Chover, , P. Ney and , and S. Wainger, Functions of probability measures, J. Anal. Math., 26 (1973), pp. 255–302. 278 JOAMAV 0099-698X J. Anal. Math. 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On maximum family size in branching processes

The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

On maximum family size in branching processes

The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

On maximum family size in branching processes

The number Y n of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y n and EY n provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

The distribution of time to extinction in subcritical branching processes: applications to outbreaks of infectious disease

We consider the distribution of the number of generations to extinction in subcritical branching processes, with particular emphasis on applications to the spread of infectious diseases. We derive the generation distributions for processes with Bernoulli, geometric and Poisson offspring, and discuss some of their distributional and inferential properties. We present applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.

The distribution of time to extinction in subcritical branching processes: applications to outbreaks of infectious disease

We consider the distribution of the number of generations to extinction in subcritical branching processes, with particular emphasis on applications to the spread of infectious diseases. We derive the generation distributions for processes with Bernoulli, geometric and Poisson offspring, and discuss some of their distributional and inferential properties. We present applications to the spread of infection in highly vaccinated populations, outbreaks of enteric fever, and person-to-person transmission of human monkeypox.