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 > 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.

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (3) ◽  
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 > 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.

scholarly journals Bounds for the Asymptotic Growth Rate of an Age-Dependent Branching Process

1969 ◽  
Vol 10 (1-2) ◽  
pp. 231-235 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Growth Rate ◽  
Population Size ◽  
Branching Process ◽  
Expected Number ◽  
Lattice Distribution ◽  
Asymptotic Growth ◽  
Age Dependent ◽  
The Mean ◽  
Image Position ◽  
Asymptotic Growth Rate

Let M(t) denote the mean population size at time t (conditional on a single ancestor of age zero at time zero) of a branching process in which the distribution of the lifetime T of an individual is given by Pr {T≦t} =G(t), and in which each individual gives rise (at death) to an expected number A of offspring (1λ A λ ∞). expected number A of offspring (1 < A ∞). Then it is well-known (Harris [1], p. 143) that, provided G(O+)-G(O-) 0 and G is not a lattice distribution, M(t) is given asymptotically by where c is the unique positive value of p satisfying the equation .

Growth in Baumslag–Solitar groups II: The Bass–Serre tree

2020 ◽  
Vol 30 (02) ◽  
pp. 339-378
Author(s):  
Jared Adams ◽  
Eric M. Freden
Keyword(s):  
Growth Rate ◽  
Minimal Length ◽  
Geometric Object ◽  
Asymptotic Growth ◽  
Growth Series ◽  
Context Sensitive ◽  
Counting Algorithms ◽  
Self Similar ◽  
Context Free ◽  
Asymptotic Growth Rate

Denote the Baumslag–Solitar family of groups as [Formula: see text]). When [Formula: see text] we study the Bass–Serre tree [Formula: see text] for [Formula: see text] as a geometric object. We suggest that the irregularity of [Formula: see text] is the principal obstruction for computing the growth series for the group. In the particular case [Formula: see text] we exhibit a set [Formula: see text] of normal form words having minimal length for [Formula: see text] and use it to derive various counting algorithms. The language [Formula: see text] is context-sensitive but not context-free. The tree [Formula: see text] has a self-similar structure and contains infinitely many cone types. All cones have the same asymptotic growth rate as [Formula: see text] itself. We derive bounds for this growth rate, the lower bound also being a bound on the growth rate of [Formula: see text].

scholarly journals Exponential Growth of Bifurcating Processes with Ancestral Dependence

2015 ◽  
Vol 47 (02) ◽  
pp. 545-564 ◽  
Author(s):  
Sana Louhichi ◽  
Bernard Ycart
Keyword(s):  
Growth Rate ◽  
Laplace Transform ◽  
Exponential Growth ◽  
Branching Process ◽  
Cell Kinetics ◽  
Growth Models ◽  
Branching Processes ◽  
Independent Random Variables ◽  
The Laplace Transform ◽  
Ergodicity Coefficients

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

Conditions for extinction in certain bisexual Galton–Watson branching processes

1984 ◽  
Vol 21 (02) ◽  
pp. 414-418
Author(s):  
David M. Hull
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Multitype Branching Process ◽  
Extinction Probabilities ◽  
Multitype Branching Processes ◽  
Necessary And Sufficient

A multitype branching process, the n-family community mating process, is introduced for the purpose of comparing extinction probabilities with those of bisexual Galton–Watson branching processes. Consideration of known properties of standard multitype branching processes leads to conditions which are both necessary and sufficient for extinction in a bisexual Galton–Watson branching process. An application is then made to the counterexample of the author's earlier paper.

scholarly journals Some Local Asymptotic Laws for the Cauchy Process on the Line

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
A. Chukwuemeka Okoroafor
Keyword(s):  
Growth Rate ◽  
Sojourn Time ◽  
Escape Rate ◽  
Time Process ◽  
Rate Process ◽  
Monotone Functions ◽  
Asymptotic Growth ◽  
Asymptotic Size ◽  
Asymptotic Growth Rate ◽  
Cauchy Process

This paper investigates the lim inf behavior of the sojourn time process and the escape rate process associated with the Cauchy process on the line. The monotone functions associated with the lower asymptotic growth rate of the sojourn time are characterized and the asymptotic size of the large values of the escape rate process is developed.

On the asymptotic growth rate of some spanning trees embedded in

2011 ◽  
Vol 39 (1) ◽  
pp. 44-48 ◽  
Author(s):  
Pedro M.M. de Castro ◽  
Olivier Devillers
Keyword(s):  
Growth Rate ◽  
Spanning Trees ◽  
Asymptotic Growth ◽  
Asymptotic Growth Rate

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.

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