Limit theorems for bounded branching processes

2018 ◽  
Vol 28 (5) ◽  
pp. 285-292
Author(s):  
Gleb K. Kobanenko
Keyword(s):  
Discrete Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Random Moment ◽  
Upper Boundary

Abstract The conditions under which the nonextincting trajectories of a discrete time bounded branching process with probability 1: either only finitely many times hit the upper boundary, either infinitely often hit the upper boundary, or coincide with the upper boundary after some random moment

Decomposable branching processes with two types of particles

2018 ◽  
Vol 28 (2) ◽  
pp. 119-130 ◽  
Author(s):  
Vladimir A. Vatutin ◽  
Elena E. Dyakonova
Keyword(s):  
Asymptotic Behavior ◽  
Discrete Time ◽  
Conditional Distribution ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Infinite Variance ◽  
Tail Distribution

Abstract A two-type critical decomposable branching process with discrete time 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. Assuming that the offspring distributions of particles of both types may have infinite variance, the asymptotic behavior of the tail distribution of the random variable Ξ2, the total number of the second type particles ever born in the process is found. Limit theorems are proved describing (as N → ∞) the conditional distribution of the amount of the first type particles in different generations given that either Ξ2 = N or Ξ2 > N.

scholarly journals Alternating branching processes

2005 ◽  
Vol 42 (04) ◽  
pp. 1095-1108 ◽  
Author(s):  
Penka Mayster
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Extinction Probability ◽  
Thinning Operator

We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.

Functional limit theorems for high-level subcritical branching processes in random environment

2014 ◽  
Vol 24 (5) ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Moment Generating Function ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
The Moment ◽  
High Level ◽  
Passage Times

AbstractThe paper is concerned with subcritical branching process in random environment. It is assumed that the moment-generating function of steps of the associated random walk is equal to 1 for some positive value of the argument. Functional limit theorems for sizes of various generations and passage times to various levels are put forward.

Some limit theorems for Markov branching processes

1973 ◽  
Vol 10 (02) ◽  
pp. 299-306 ◽  
Author(s):  
J. R. Leslie
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Original Process ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Increasing Process ◽  
Watson Process ◽  
Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

A branching-process solution of the polydisperse coagulation equation

1984 ◽  
Vol 16 (01) ◽  
pp. 56-69 ◽  
Author(s):  
John L. Spouge
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Critical Time ◽  
Coagulation Equation ◽  
Multitype Branching Processes ◽  
Irreversible Aggregation ◽  
Image Position ◽  
Supercritical Branching Processes ◽  
Critical Branching Process

The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when The critical time tc when diverges corresponds to a critical branching process, while post-critical times t> tc correspond to supercritical branching processes.

A continuous time Markov branching model with random environments

1973 ◽  
Vol 5 (1) ◽  
pp. 37-54 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Population Model ◽  
Limit Theorems ◽  
Branching Process ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Markov Branching Process ◽  
Branching Model ◽  
Necessary And Sufficient

A population model is constructed which combines the ideas of a discrete time branching process with random environments and a continuous time non-homogeneous Markov branching process. The extinction problem is considered and necessary and sufficient conditions for extinction are determined. Also discussed are limit theorems for what corresponds to the supercritical case.

Conditional limit theorems for general branching processes

1977 ◽  
Vol 14 (3) ◽  
pp. 451-463 ◽  
Author(s):  
P. J. Green
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Conditional Limit Theorems ◽  
Special Case ◽  
Conditional Limit

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.

scholarly journals Limit theorems for supercritical age-dependent branching processes with neutral immigration

2011 ◽  
Vol 43 (1) ◽  
pp. 276-300 ◽  
Author(s):  
M. Richard
Keyword(s):  
Continuous Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Total Population ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Structured Populations ◽  
Total Population Size ◽  
Constant Rate ◽  
Age Dependent

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rateb. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1,P2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ /b.

scholarly journals Branching processes in random environment die slowly

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Vladimir Vatutin ◽  
Andreas Kyprianou
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Generating Functions ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Limit Distributions ◽  
Conditional Limit Theorems ◽  
International Audience ◽  
Conditional Limit

International audience Let $Z_n,n=0,1,\ldots,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $f_0(s),f_1(s),\ldots,$ and let $S_0=0$, $S_k=X_1+ \ldots +X_k,k \geq 1$, be the associated random walk with $X_i=\log f_{i-1}^{\prime}(1), \tau (m,n)$ be the left-most point of minimum of $\{S_k,k \geq 0 \}$ on the interval $[m,n]$, and $T=\min \{ k:Z_k=0\}$. Assuming that the associated random walk satisfies the Doney condition $P(S_n > 0) \to \rho \in (0,1), n \to \infty$, we prove (under the quenched approach) conditional limit theorems, as $n \to \infty$, for the distribution of $Z_{nt}, Z_{\tau (0,nt)}$, and $Z_{\tau (nt,n)}, t \in (0,1)$, given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt$.

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