On the extinction time distribution of a branching process in varying environments

1980 ◽  
Vol 12 (2) ◽  
pp. 350-366 ◽  
Author(s):  
Tetsuo Fujimagari
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Tail Probability ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Extinction Time ◽  
Asymptotic Formulae ◽  
Varying Environments

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q < 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q < 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.

On the extinction time distribution of a branching process in varying environments

1980 ◽  
Vol 12 (02) ◽  
pp. 350-366 ◽  
Author(s):  
Tetsuo Fujimagari
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Tail Probability ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Extinction Time ◽  
Asymptotic Formulae ◽  
Varying Environments

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q &lt; 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q &lt; 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.

scholarly journals On the extinction times of varying and random environment branching processes

1975 ◽  
Vol 12 (01) ◽  
pp. 39-46 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Extinction Time ◽  
Probability Of Extinction ◽  
Stochastically Bounded ◽  
Necessary And Sufficient ◽  
Varying Environment

Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.

scholarly journals On the extinction times of varying and random environment branching processes

1975 ◽  
Vol 12 (1) ◽  
pp. 39-46 ◽  
Author(s):  
Alan Agresti
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Extinction Time ◽  
Probability Of Extinction ◽  
Stochastically Bounded ◽  
Necessary And Sufficient ◽  
Varying Environment

Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.

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

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.

Continuity for multi-type branching processes with varying environments

1999 ◽  
Vol 36 (01) ◽  
pp. 139-145 ◽  
Author(s):  
Owen Dafydd Jones
Keyword(s):  
Linear Combination ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Main Tool ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Varying Environments

Conditions are derived for the components of the normed limit of a multi-type branching process with varying environments, to be continuous on (0, ∞). The main tool is an inequality for the concentration function of sums of independent random variables, due originally to Petrov. Using this, we show that if there is a discontinuity present, then a particular linear combination of the population types must converge to a non-random constant (Equation (1)). Ensuring this can not happen provides the desired continuity conditions.

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (01) ◽  
pp. 135-141 ◽  
Author(s):  
Niels Keiding ◽  
John E. Nielsen
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Elementary Proof ◽  
Branching Processes ◽  
Random Environments ◽  
Subcritical Case ◽  
Supercritical Case ◽  
Extinction Probabilities ◽  
Varying Environments

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (03) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r &gt; 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

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.

Sign in / Sign up

Export Citation Format

Share Document