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.

scholarly journals Branching processes with immigration in atypical random environment

Extremes ◽  
2021 ◽  
Author(s):  
Sergey Foss ◽  
Dmitry Korshunov ◽  
Zbigniew Palmowski
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Heavy Tails ◽  
Branching Processes ◽  
Environmental Parameters ◽  
Seminal Paper ◽  
Offspring Distribution ◽  
Tail Distribution ◽  
Distribution Tail ◽  
Varying Environment

AbstractMotivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters An, n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of $\xi _{n}:=\log ((1-A_{n})/A_{n})$ ξ n : = log ( ( 1 − A n ) / A n ) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail $\mathbb {P}(Z_{n} \ge m)$ ℙ ( Z n ≥ m ) of the n th population size Zn is asymptotically equivalent to $n\overline F(\log m)$ n F ¯ ( log m ) as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 154, 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter α > 1. Further, for a subcritical branching process with subexponentially distributed ξn, we provide the asymptotics for the distribution tail $\mathbb {P}(Z_{n}>m)$ ℙ ( Z n > m ) which are valid uniformly for all n, and also for the stationary tail distribution. Then we establish the “principle of a single atypical environment” which says that the main cause for the number of particles to be large is the presence of a single very small environmental parameter Ak.

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.

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.

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.

scholarly journals Transformations of Galton-Watson processes and linear fractional reproduction

2007 ◽  
Vol 39 (4) ◽  
pp. 1036-1053 ◽  
Author(s):  
F. C. Klebaner ◽  
U. Rösler ◽  
S. Sagitov
Keyword(s):  
Markov Chain ◽  
Time Reversal ◽  
Harmonic Functions ◽  
Invariant Measures ◽  
Graphical Representation ◽  
Simple Random Walk ◽  
Time Process ◽  
Offspring Distribution ◽  
Watson Process

By establishing general relationships between branching transformations (Harris-Sevastyanov, Lamperti-Ney, time reversals, and Asmussen-Sigman) and Markov chain transforms (Doob's h-transform, time reversal, and the cone dual), we discover a deeper connection between these transformations with harmonic functions and invariant measures for the process itself and its space-time process. We give a classification of the duals into Doob's h-transform, pathwise time reversal, and cone reversal. Explicit results are obtained for the linear fractional offspring distribution. Remarkably, for this case, all reversals turn out to be a Galton-Watson process with a dual reproduction law and eternal particle or some kind of immigration. In particular, we generalize a result of Klebaner and Sagitov (2002) in which only a geometric offspring distribution was considered. A new graphical representation in terms of an associated simple random walk on N2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

scholarly journals On a Continuous-State Population-Size-Dependent Branching Process and Its Extinction

2006 ◽  
Vol 43 (1) ◽  
pp. 195-207 ◽  
Author(s):  
Yuqiang Li
Keyword(s):  
Population Size ◽  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Size Dependent ◽  
State Population ◽  
Continuous State ◽  
Discrete States

A continuous-state population-size-dependent branching process {Xt} is a modification of the Jiřina process. We prove that such a process arises as the limit of a sequence of suitably scaled population-size-dependent branching processes with discrete states. The extinction problem for the population Xt is discussed, and the limit distribution of Xt / t obtained when Xt tends to infinity.

The age of a Galton-Watson population with a geometric offspring distribution

2002 ◽  
Vol 39 (4) ◽  
pp. 816-828 ◽  
Author(s):  
F. C. Klebaner ◽  
S. Sagitov
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Initial Population ◽  
Asymptotic Results ◽  
The Past ◽  
Offspring Distribution ◽  
Initial Population Size ◽  
Watson Process ◽  
Looking Back

Motivated by the question of the age in a branching population we try to recreate the past by looking back from the currently observed population size. We define a new backward Galton-Watson process and study the case of the geometric offspring distribution with parameter p in detail. The backward process is then the Galton-Watson process with immigration, again with a geometric offspring distribution but with parameter 1-p, and it is also the dual to the original Galton-Watson process. We give the asymptotic distribution of the age when the initial population size is large in supercritical and critical cases. To this end, we give new asymptotic results on the Galton-Watson immigration processes stopped at zero.

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.

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