On constant tail behaviour for the limiting random variable in a supercritical branching process

1995 ◽  
Vol 32 (01) ◽  
pp. 267-273
Author(s):  
B. M. Hambly
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Explicit Computation ◽  
Left Tail ◽  
Tail Behaviour ◽  
Supercritical Branching Processes

We examine a family of supercritical branching processes and compute the density of the limiting random variable, W, for their normalized population size. In this example the left tail of W decays exponentially and there is no oscillation in this tail as typically observed. The branching process is embedded in the n-adic rational random walk approximation to Brownian motion on [0, 1]. This connection allows the explicit computation of the density of W.

On constant tail behaviour for the limiting random variable in a supercritical branching process

1995 ◽  
Vol 32 (1) ◽  
pp. 267-273 ◽  
Author(s):  
B. M. Hambly
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Explicit Computation ◽  
Left Tail ◽  
Tail Behaviour ◽  
Supercritical Branching Processes

We examine a family of supercritical branching processes and compute the density of the limiting random variable, W, for their normalized population size. In this example the left tail of W decays exponentially and there is no oscillation in this tail as typically observed. The branching process is embedded in the n-adic rational random walk approximation to Brownian motion on [0, 1]. This connection allows the explicit computation of the density of W.

On a functional equation for general branching processes

1973 ◽  
Vol 10 (1) ◽  
pp. 198-205 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Population Size ◽  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Slow Variation ◽  
Age Dependent ◽  
Class Of Functions

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.

On a functional equation for general branching processes

1973 ◽  
Vol 10 (01) ◽  
pp. 198-205 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Population Size ◽  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Slow Variation ◽  
Age Dependent ◽  
Class Of Functions

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW ) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y ) &lt;∞, for γ &gt; 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) &lt; ∞ for 0 β &lt; 1, where L is one of a class of functions of slow variation.

On the limit of a supercritical branching process

1988 ◽  
Vol 25 (A) ◽  
pp. 215-228 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Recent Progress ◽  
Branching Processes ◽  
Random Variable ◽  
Tail Behaviour ◽  
Tauberian Theory ◽  
The Right ◽  
Schröder Case ◽  
Böttcher Case

LetWbe the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

On the limit of a supercritical branching process

1988 ◽  
Vol 25 (A) ◽  
pp. 215-228 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Recent Progress ◽  
Branching Processes ◽  
Random Variable ◽  
Tail Behaviour ◽  
Tauberian Theory ◽  
The Right ◽  
Schröder Case ◽  
Böttcher Case

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

On the convergence of the supercritical branching processes with immigration

1977 ◽  
Vol 14 (2) ◽  
pp. 387-390 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Limit Distribution Function ◽  
Branching Process With Immigration ◽  
Necessary And Sufficient ◽  
Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

Large deviations in the supercritical branching process

1993 ◽  
Vol 25 (04) ◽  
pp. 757-772 ◽  
Author(s):  
J. D. Biggins ◽  
N. H. Bingham
Keyword(s):  
Distribution Function ◽  
Large Deviations ◽  
Population Size ◽  
Branching Process ◽  
Large Deviation ◽  
Moment Generating Function ◽  
Large Deviation Theory ◽  
Tail Behaviour ◽  
The Moment ◽  
Deviation Theory

The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution function of W decays exponentially quickly. In essence, knowledge of the behaviour of transforms can be combined with some ‘large-deviation' theory to get detailed information on the oscillation of the distribution function of W near zero or at infinity. In particular we show how an old result of Harris (1948) on the asymptotics of the moment-generating function of W translates to tail behaviour.

Stochastic modelling of age-structured population with time and size dependence of immigration rate

2018 ◽  
Vol 33 (5) ◽  
pp. 289-299 ◽  
Author(s):  
Boris J. Pichugin ◽  
Nikolai V. Pertsev ◽  
Valentin A. Topchii ◽  
Konstantin K. Loginov
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Stochastic Modelling ◽  
Random Variable ◽  
Structured Population Model ◽  
Immigration Rate ◽  
Structured Population ◽  
Mean And Variance ◽  
Age Structured ◽  
Age Structured Population

Abstract A stochastic age-structured population model with immigration of individuals is considered. We assume that the lifespan of each individual is a random variable with a distribution function which may differ fromthe exponential one. The immigration rate of individuals depends on the time and total population size. Upper estimates for the mean and variance of the population size are established based on the theory of branching processes with constant immigration rate. A Monte Carlo simulation algorithm of population dynamics is developed. The results of numerical experiments with the model are presented.

A connection between the limit and the maximum random variable of a branching process in varying environments

1982 ◽  
Vol 19 (03) ◽  
pp. 681-684 ◽  
Author(s):  
F. C. Klebaner ◽  
H.-J. Schuh
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Varying Environments

We show for a certain class of Galton–Watson branching processes in varying environments (Zn ) n that moments of the maximum random variable sup n Zn/Cn exist if and only if the same moments of lim nZn/Cn exist, where Cn is a sequence of suitable constants.

The xlogx condition for general branching processes

1998 ◽  
Vol 35 (03) ◽  
pp. 537-544
Author(s):  
Peter Olofsson
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Rate Of Growth

The xlogx condition is a fundamental criterion for the rate of growth of a general branching process, being equivalent to non-degeneracy of the limiting random variable. In this paper we adopt the ideas from Lyons, Pemantle and Peres (1995) to present a new proof of this well-known theorem. The idea is to compare the ordinary branching measure on the space of population trees with another measure, the size-biased measure.

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