EM algorithm for statistical estimation of two-type branching processes – A focus on the multinomial offspring distribution

2020 ◽  
Author(s):  
A. Staneva ◽  
V. Stoimenova
Keyword(s):  
Em Algorithm ◽  
Statistical Estimation ◽  
Branching Processes ◽  
Offspring Distribution

scholarly journals On the Extinction of a Class of Population-Size-Dependent Bisexual Branching Processes

2005 ◽  
Vol 42 (01) ◽  
pp. 175-184 ◽  
Author(s):  
Yongsheng Xing ◽  
Yongjin Wang
Keyword(s):  
Growth Rate ◽  
Population Size ◽  
Branching Processes ◽  
Suitable Condition ◽  
Size Dependent ◽  
Offspring Distribution ◽  
Bisexual Branching Processes ◽  
The Law ◽  
Mating Unit

In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.

On population-size-dependent branching processes

1984 ◽  
Vol 16 (1) ◽  
pp. 30-55 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Stochastic Model ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Gamma Distribution ◽  
Branching Processes ◽  
Independent Random Variables ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

Geometric rate of growth in population-size-dependent branching processes

1984 ◽  
Vol 21 (01) ◽  
pp. 40-49 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Population Size ◽  
Process Model ◽  
Branching Process ◽  
Necessary Conditions ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a branching-process model {Zn }, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m &gt; 1 as n →∞. For a certain class of processes {Zn } necessary conditions for convergence in L 1 and L 2 and sufficient conditions for almost sure convergence and convergence in L 2 of Wn = Zn/mn are given.

Bounds for Certain Branching Processes

1969 ◽  
Vol 6 (01) ◽  
pp. 201-204 ◽  
Author(s):  
Edward Pollak
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Offspring Distribution

Summary We consider a branching process for which the offspring distribution has the generating function f(t) and mean f '(1) = m &lt; 1. The probability that a line descended from an individual still survives in generation n is asymptotically of the form cmn. A method is derived whereby good bounds for c may be obtained. This method makes use of the first three moments of the distribution of offspring.

On population-size-dependent branching processes

1984 ◽  
Vol 16 (01) ◽  
pp. 30-55 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Stochastic Model ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Gamma Distribution ◽  
Branching Processes ◽  
Independent Random Variables ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a stochastic model for the development in time of a population {Z n } where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m &lt; l, m = 1, m&gt; l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m &lt; 1 or m = 1 and mn approaches 1 not slower than n –2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n –1, then Zn /n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an , such that Zn /an converges in probability to a non-degenerate limit. If mn approaches m &gt; 1 not slower than n– α, α &gt; 0, and do not grow to ∞ faster than nß , β &lt;1 then Zn /mn converges almost surely and in L 2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

Self-normalized large deviation for supercritical branching processes

2018 ◽  
Vol 55 (2) ◽  
pp. 450-458
Author(s):  
Weijuan Chu
Keyword(s):  
Branching Process ◽  
Large Deviation ◽  
Branching Processes ◽  
Random Variables ◽  
The Self ◽  
Offspring Distribution ◽  
Multitype Branching Processes ◽  
Supercritical Branching Processes ◽  
Independent And Identically Distributed

Abstract We consider a supercritical branching process (Zn, n ≥ 0) with offspring distribution (pk, k ≥ 0) satisfying p0 = 0 and p1 > 0. By applying the self-normalized large deviation of Shao (1997) for independent and identically distributed random variables, we obtain the self-normalized large deviation for supercritical branching processes, which is the self-normalized version of the result obtained by Athreya (1994). The self-normalized large deviation can also be generalized to supercritical multitype branching processes.

scholarly journals On the Extinction of a Class of Population-Size-Dependent Bisexual Branching Processes

2005 ◽  
Vol 42 (1) ◽  
pp. 175-184 ◽  
Author(s):  
Yongsheng Xing ◽  
Yongjin Wang
Keyword(s):  
Growth Rate ◽  
Population Size ◽  
Branching Processes ◽  
Suitable Condition ◽  
Size Dependent ◽  
Offspring Distribution ◽  
Bisexual Branching Processes ◽  
The Law ◽  
Mating Unit

In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.

Geometric rate of growth in population-size-dependent branching processes

1984 ◽  
Vol 21 (1) ◽  
pp. 40-49 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Population Size ◽  
Process Model ◽  
Branching Process ◽  
Necessary Conditions ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a branching-process model {Zn}, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m > 1 as n →∞. For a certain class of processes {Zn} necessary conditions for convergence in L1 and L2 and sufficient conditions for almost sure convergence and convergence in L2 of Wn = Zn/mn are given.

A unifying approach to branching processes in a varying environment

2020 ◽  
Vol 57 (1) ◽  
pp. 196-220
Author(s):  
Götz Kersting
Keyword(s):  
Time Dependence ◽  
Population Size ◽  
Limit Distribution ◽  
Branching Processes ◽  
Type Result ◽  
Offspring Distribution ◽  
Watson Process ◽  
Square Integrability ◽  
Varying Environment

AbstractBranching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned on the event $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

A note on simple branching processes with infinite mean

1977 ◽  
Vol 14 (04) ◽  
pp. 836-842 ◽  
Author(s):  
Irene L. Hudson ◽  
E. Seneta
Keyword(s):  
Branching Processes ◽  
Random Variable ◽  
Offspring Distribution ◽  
Watson Process

We consider the Bienaymé–Galton–Watson process without and with immigration, and with offspring distribution having infinite mean. For such a process, {Zn } say, conditions are given ensuring that there exists a sequence of positive constants, {ρn }, such that {ρnU(Zn + 1)} converges almost surely to a proper non-degenerate random variable, where U is a function slowly varying at infinity, defined on [1, ∞), continuous and strictly increasing, with U(1) = 0, U(∞) = ∞. These results subsume earlier ones with U(t) = log t.

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