A note on simple branching processes with infinite mean

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.

Some limit theorems for a supercritical branching process allowing immigration

We consider the Bienaymé–Galton–Watson model of population growth in which immigration is allowed. When the mean number of offspring per individual, α, satisfies 1 < α < ∞, a well-known result proves that a normalised version of the size of the n th generation converges to a finite, positive random variable iff a certain condition is satisfied by the immigration distribution. In this paper we obtain some non-linear limit theorems when this condition is not satisfied. Results are also given for the explosive case, α = ∞.

Almost sure convergence in Markov branching processes with infinite mean

If {Zn} is a Galton–Watson branching process with infinite mean, it is shown that under certain conditions there exist constants {cn} and a function L, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, whose distribution function satisfies a certain functional equation. The method is then extended to a continuous-time Markov branching process {Zt} with infinite mean, where it is shown that there is always a function φ, slowly varying at 0, such that converges almost surely to a non-degenerate random variable, and a necessary and sufficient condition is given for this convergence to be equivalent to convergence of for some constant α > 0.

The simple branching process: a note on convergence when the mean is infinite

Let denote the simple branching process with Z 0 = 1 and let G denote the distribution function of Z 1 . Suppose G satisfies x −α−γ(x)≦1 − G(x) ≦ x −α+γ(x) for large x, where (i) 0 &lt; α &lt; 1, (ii) γ (x) is non-negative and non-increasing, (iii) xγ (x) is non-decreasing and (iv) Then lim n→∞ α n log (Zn + 1) converges almost surely to a non-degenerate finite random variable W satisfying P(W = 0) = q = probability of extinction of the process.

Asymptotic behaviour of continuous time, continuous state-space branching processes

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.

The simple branching process: a note on convergence when the mean is infinite

Letdenote the simple branching process withZ0= 1 and letGdenote the distribution function ofZ1.SupposeGsatisfiesx−α−γ(x)≦1 −G(x) ≦x−α+γ(x)for largex, where (i) 0 < α < 1, (ii)γ(x) is non-negative and non-increasing, (iii)xγ(x)is non-decreasing and (iv)Then limn→∞αnlog (Zn+ 1) converges almost surely to a non-degenerate finite random variableWsatisfyingP(W= 0) =q= probability of extinction of theprocess.

Some limit theorems for a supercritical branching process allowing immigration

We consider the Bienaymé–Galton–Watson model of population growth in which immigration is allowed. When the mean number of offspring per individual, α, satisfies 1 &lt; α &lt; ∞, a well-known result proves that a normalised version of the size of the n th generation converges to a finite, positive random variable iff a certain condition is satisfied by the immigration distribution. In this paper we obtain some non-linear limit theorems when this condition is not satisfied. Results are also given for the explosive case, α = ∞.

Asymptotic behaviour of continuous time, continuous state-space branching processes

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.

A representation for the limiting random variable of a branching process with infinite mean and some related problems

It is known that for a Bienaymé– Galton–Watson process {Zn } whose mean m satisfies 1 &lt; m &lt; ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem.

This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.

Regularly varying functions in the theory of simple branching processes

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

Asymptotic properties of supercritical branching processes I: The Galton-Watson process

We obtain results connecting the distributions of the random variables Z 1 and W in the supercritical Galton-Watson process. For example, if a &gt; 1, and converge or diverge together, and regular variation of the tail of one of Z 1, W with non-integer exponent α &gt; 1 is equivalent to regular variation of the tail of the other.

On the asymptotic behaviour of branching processes with infinite mean

The paper deals with the asymptotic behaviour of infinite mean Galton–Watson processes (denoted by {Zn }). We show that these processes can be classified as regular or irregular. The regular ones are characterized by the property that for any sequence of positive constants {Cn }, for which a.s. exists, The irregular ones, which will be shown by examples to exist, have the property that there exists a sequence of constants {Cn } such that In Part 1 we study the properties of {Zn /Cn } and give some characterizations for both regular and irregular processes. Part 2 starts with an a.s. convergence result for {yn (Zn )}, where {yn } is a suitable chosen sequence of functions related to {Zn }. Using this, we then derive necessary and sufficient conditions for the a.s. convergence of {U(Zn )/Cn }, where U is a slowly varying function. The distribution function of the limit is shown to satisfy a Poincaré functional equation. Finally we show that for every process {Zn } it is possible to construct explicitly functions U, such that U(Zn )/en converges a.s. to a non-degenerate proper random variable. If the process is regular, all these functions U are slowly varying. The distribution of the limit depends on U, and we show that by appropriate choice of U we may get a limit distribution which has a positive and continuous density or is continuous but not absolutely continuous or even has no probability mass on certain intervals. This situation contrasts strongly with the finite mean case.

Regularly varying functions in the theory of simple branching processes

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.

Reduced Branching Processes with Very Heavy Tails

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavy-tailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.