We just showed a conventional martingale analysis of the GWBP. But there is another line of attack to branching process theory using probability generating functions.
Returning to the notation of the previous subsection, the probability generating function (or pgf) of a probability distributionP={pk}k>0 supported on Z+, is
f(s) := ∞
X
k=0 pksk
The beauty of pgfs is that there are a couple different ways of looking at them. For one, pgfs are power series with coefficients in [0,1], so all the usual theorems apply. Secondly, if X ∼P then f(s) = E(sX). Immediate from these observations are results like
1. f(s) is continuous, and specifically continuous from the left ats = 1. 2. f0(s) =P
ksk−1p
3. µ=EX =f0(1)
But also, pgfs are especially relevant with respect to branching processes because we can compose individual offspring distribution pgfs to get pgfs of total population sizes:
Theorem 2.1.4. Let f denote the pgf of the offspring distribution in a GWBP. Also let fn:=E(sZn) denote the pgf of the distribution of Zn. Then
fn+1(s) = f(fn(s))
This gives us an easy way to analyze distributions of future generations of the process. And this can be just as useful as the martingale analysis when examining the behavior of the process in the limit. For example, the extinction theorem 2.1.1 is usually proved using pgfs and, as a side benefit, often make explicit calculations easy:
Theorem 2.1.5. For a supercritical GWBP, the probability of extinction is a solution to the fixed point equation f(s) = s.
Proof. Write Q:={∃n :Zn= 0} for the extinction event so that P(Q) is the probability of extinction. Also let Qn :={Zn= 0}.
Clearly, Qn ⊂ Qn+1 and so P(Qn) ↑ P(Q) as n → ∞. But also, P(Qn) = fn(0). Therefore by Theorem 2.1.4, fn+1(0) = f(fn(0)) which means that P(Qn+1) = f(P(Qn)).
Then since f is continuous,
f(P(Q)) = f( lim
n→∞fn(0)) = limn→∞f(fn(0)) = limn→∞fn+1(0) =P(Q)
So how does the picture change if we let the offspring distributions depend on the generation number? In the literature, there are two ways to let this happen.
1. Varying environment: Let the offspring distributions still be deterministic, but vary by generation.
That is, letφndenote the pgf of the offspring distribution for individuals in the (n−1)th generation. If {φn}n>0 is deterministic, then we are in a varying environment.
2. Random environment: Let the offspring distributions be random across generations, but iid (stationary, ergodic).
That is, let {ζn}n>0 be a sequence of iid (stationary, ergodic) random “environmental
variables” in some space Θ, where we associate with each point ζ ∈Θ a pgfφζ. If φζn is the pgf of the offspring distribution for individuals in the (n−1)th generation, then we are in a random environment.
The case of varying environment has been considered since the dawn of branching processes, and a concise summaries of main results can be found in [64] or [51]. In general, because there is no additional randomness in these processes their behavior can be well-understood so long as one can deal with the analysis of the generation functions.
First, varying environment processes behave similarly to GWBP in many ways. For example, a Kesten-Stigum theorem holds for a class of supercritical varying environment processes. Letµj be the mean of the offspring distribution of the jth generation and call the process uniformly supercritical if
n+k−1
Y
j=k
µj >Bcn for some B >0, c >1, and all n, k >0
Also letXnrepresent the random number of offspring of an individual in thenth generation. Then:
Theorem 3 from [45]). If the branching process is uniformly supercritical and is dominated in the sense that there exists a random variable X with EX <∞ such that
P(X > x)>P(Xn/µn> x) for all x,
then there exists a sequence of constants cn such that Zn/cn converges to an a.s. finite random variable W with {W = 0}={Zn→0}.
But there are some surprising differences in contrast to GWBPs. We have seen in the discussion of the Kesten-Stigum theorem that, on the set of non-extinction, a supercritical GWBP has essentially only one rate of growth (up to multiplicative shifts ): µn. As shown in the theorem above, this happens to be true for a large class of branching processes in varying environment. But it is not always so. For example, the authors of [77] construct a branching process in varying environment which is supercritical and grows like 2n on one part of the sample space and mn with m > 4 on another part, both with positive measure. We shall discuss these points further later on in the thesis.
The case of random environment was first introduced in [100] where the sequence{ζn}n>0
was taken to be iid. Their results were later extended to any stationary, ergodic sequence in [8] and [7] where extinction criteria and limit theorems for the process Zn were developed.
The main takeaway from these papers is basically that under some reasonable conditions on{ζn}n>0, we can see the same usual behavior of the ordinary GWBP, with slight obvious
modifications. For the sake of brevity we leave the specifics to the reference.
Theorem 1 from [7]. Under some mild assumptions about the environmental process {ζn}n>0 including an XlogX+ condition, essentially the same results as in the Kesten-
Stigum theorem for Galton-Watson processes apply.
We note at this point that the entire line of work above all share a common assumption: the mean of the offspring distributions is finite. The other line of the work on branching
offspring distributions are not taken to be varying, but have infinite mean. While technically these are just supercritical GWBPs, the conditions of the Kesten-Stigum theorem are not at all satisfied, so the limiting behavior is markedly different.
In [94], the authors adapt techniques from the study of finite-mean supercritical branch- ing processes in [95] and [58] to characterize infinite-mean branching processes as either regular orirregular depending on their limiting growth behavior.
Recall from the Kesten-Stigum theorem that, for finite-mean processes, the probability that the martingale limit limn→∞Zn/µn = M is not zero is positive if and only if the Xlog+X condition is satisfied. Therefore in the infinite-mean realm, we say a process is regular if for any sequence of positive constants {cn}n>0 for which limZn/cn a.s. exists, P(limZn/cn= 0 or ∞) = 1.
However, just like how branching processes in varying environment surprisingly can display growth at two different rates, infinite-mean branching processes display interesting exceptions to the finite-mean behavior.
Theorem [94]. There exist infinite-mean Galton-Watson processes such that for some positive deterministic sequence {cn}n>0, the martingale limit M := limn→∞Zn/cn has P(M >0)>0.
Call these theirregular processes. In [94], it is also shown that for all regular processes there exists a slowly-varying functionU(·) such thatU(Zn)/enconverges to a non-degenerate limit. In Chapter 5 we take the first step towards investigating these behaviors for a new, related type of branching process.