Measure change in multitype branching

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like theXlogXcondition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

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On maximum family size in branching processes

Journal of Applied Probability ◽

10.1239/jap/1032374622 ◽

1999 ◽

Vol 36 (3) ◽

pp. 632-643 ◽

Cited By ~ 10

Author(s):

Ibrahim Rahimov ◽

George P. Yanev

Keyword(s):

Asymptotic Behaviour ◽

Sample Size ◽

Random Sample ◽

Family Size ◽

Limit Theorems ◽

Branching Processes ◽

Random Sample Size ◽

Convergence Results ◽

Watson Process ◽

Supercritical 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.

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On maximum family size in branching processes

Journal of Applied Probability ◽

10.1017/s0021900200017459 ◽

1999 ◽

Vol 36 (03) ◽

pp. 632-643 ◽

Cited By ~ 3

Author(s):

Ibrahim Rahimov ◽

George P. Yanev

Keyword(s):

Asymptotic Behaviour ◽

Sample Size ◽

Random Sample ◽

Family Size ◽

Limit Theorems ◽

Branching Processes ◽

Random Sample Size ◽

Convergence Results ◽

Watson Process ◽

Supercritical 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.

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Weak convergence of random processes with immigration at random times

Journal of Applied Probability ◽

10.1017/jpr.2019.88 ◽

2020 ◽

Vol 57 (1) ◽

pp. 250-265

Author(s):

Congzao Dong ◽

Alexander Iksanov

Keyword(s):

Weak Convergence ◽

Branching Processes ◽

Sufficient Conditions ◽

Random Processes ◽

Branching Random Walk ◽

General Point ◽

Shot Noise Process ◽

Finite Dimensional ◽

Strong Resemblance ◽

Random Times

AbstractBy a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

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Multitype branching processes in random environments

Journal of Applied Probability ◽

10.2307/3211834 ◽

1971 ◽

Vol 8 (1) ◽

pp. 17-31 ◽

Cited By ~ 22

Author(s):

Edward W. Weissner

Keyword(s):

Random Environment ◽

Branching Process ◽

Branching Processes ◽

Sufficient Conditions ◽

Random Environments ◽

Restrictive Assumption ◽

Multitype Branching Processes ◽

Watson Process ◽

Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

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Multitype branching processes in random environments

Journal of Applied Probability ◽

10.1017/s0021900200110897 ◽

1971 ◽

Vol 8 (01) ◽

pp. 17-31 ◽

Cited By ~ 5

Author(s):

Edward W. Weissner

Keyword(s):

Random Environment ◽

Branching Process ◽

Branching Processes ◽

Sufficient Conditions ◽

Random Environments ◽

Restrictive Assumption ◽

Multitype Branching Processes ◽

Watson Process ◽

Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

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Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

Journal of Applied Probability ◽

10.1017/s0021900200003119 ◽

2007 ◽

Vol 44 (02) ◽

pp. 492-505

Author(s):

M. Molina ◽

M. Mota ◽

A. Ramos

Keyword(s):

Branching Process ◽

Branching Processes ◽

Sufficient Conditions ◽

Positive Probability ◽

Limit Laws ◽

Bisexual Branching Processes ◽

Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

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Stability analysis of uncertain delay differential equations

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-202507 ◽

2021 ◽

pp. 1-11

Author(s):

Jian Wang ◽

Yuanguo Zhu

Keyword(s):

Differential Equations ◽

Delay Differential Equations ◽

Functional Differential Equations ◽

Sufficient Conditions ◽

Liu Process ◽

Uncertain Dynamical System ◽

Analytic Expression ◽

Functional Differential ◽

The One ◽

Delay Differential

Uncertain delay differential equation is a class of functional differential equations driven by Liu process. It is an important model to describe the evolution process of uncertain dynamical system. In this paper, on the one hand, the analytic expression of a class of linear uncertain delay differential equations are investigated. On the other hand, the new sufficient conditions for uncertain delay differential equations being stable in measure and in mean are presented by using retarded-type Gronwall inequality. Several examples show that our stability conditions are superior to the existing results.

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Backbone Decomposition of Multitype Superprocesses

Journal of Theoretical Probability ◽

10.1007/s10959-021-01076-7 ◽

2021 ◽

Author(s):

Dorottya Fekete ◽

Sandra Palau ◽

Juan Carlos Pardo ◽

Jose Luis Pérez

Keyword(s):

Branching Processes ◽

Easy Consequence ◽

Continuous State ◽

The One

AbstractIn this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.

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Limit of 𝑝-Laplacian obstacle problems

Advances in Calculus of Variations ◽

10.1515/acv-2019-0058 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Raffaela Capitanelli ◽

Maria Agostina Vivaldi

Keyword(s):

Asymptotic Behavior ◽

Uniform Convergence ◽

Explicit Expression ◽

Positive Measure ◽

Sufficient Conditions ◽

Obstacle Problems ◽

Dimensional Case ◽

Asymptotic Behavior Of Solutions ◽

One Dimensional ◽

The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

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