Critical branching as a pure death process coming down from infinity

Consider a self-replicating system evolving in the discrete time setting according to the next rules:

   -

the system is founded by a single individual, the founder born at time 0,

   -

the founder dies at a random age $L$ and gives a random number $N$ of births at random ages $\tau_{j}$ satisfying

$$1\leq\tau_{1}\leq\ldots\leq\tau_{N}\leq L,$$

   -

each new individual lives independently from others according to the same life law as the founder.

An individual which was born at time $t_{1}$ and dies at time $t_{2}$ is considered to be alive during the time interval $[t_{1},t_{2}-1]$. Letting $Z(t)$ stand for the number of individuals alive at time $t$, we study the random dynamics of the sequence

$$Z(0)=1,Z(1),Z(2),\ldots,$$

which is a natural extension of the well-known Galton-Watson process, or GW-process for short, see [13]. The process $Z(\cdot)$ is the discrete time version of what is usually called the Crump-Mode-Jagers process or the general branching process, see [5]. To emphasise the discrete time setting, we call it a GW-process with overlapping generations, or GWO-process for short.

Put $b:=\frac{1}{2}\mathrm{Var\hskip 0.56905pt}(N)$. This paper deals with the GWO-processes satisfying

$$\mathrm{E}(N)=1,\quad 0<b<\infty.$$

(1.1)

Condition $\mathrm{E}(N)=1$ says that the reproduction regime is critical, implying $\mathrm{E}(Z(t))\equiv 1$ and making extinction inevitable, provided $b>0$. According to [1, Ch I.9], given (1.1), the survival probability

$$Q(t):=\mathrm{P}(Z(t)>0)$$

of a GW-process satisfies the asymptotic formula $tQ(t)\to b^{-1}$ as $t\to\infty$ (this was first proven in [6] under a third moment assumption). A direct extension of this classical result for the GWO-processes,

$$tQ(ta)\to b^{-1},\quad t\to\infty,\quad a:=\mathrm{E}(\tau_{1}+\ldots+\tau_{N}),$$

was obtained in [3, 4] under conditions (1.1), $a<\infty$,

$$t^{2}\mathrm{P}(L>t)\to 0,\quad t\to\infty,$$

(1.2)

plus an additional extra condition. (Notice that by our definition, $a\geq 1$, and $a=1$ if and only if $L\equiv 1$, that is when the GWO-process in question is a GW-process). Treating $a$ as the mean generation length, see [5, 8], we may conclude that the asymptotic behaviour of the critical GWO-process with short-living individuals, see condition (1.2), is similar to that of the critical GW-process, provided time is counted generation-wise.

New asymptotical patterns for the critical GWO processes are found under the assumption

$$t^{2}\mathrm{P}(L>t)\to d,\quad 0\leq d<\infty,\quad t\to\infty,$$

(1.3)

which compared to (1.2), allows the existence of long-living individuals given $d>0$. Condition (1.3) was first introduced in the pioneering paper [12] dealing with the Bellman-Harris processes. In the current discrete time setting, the Bellman-Harris process is a GWO-process subject to two restrictions:

   -

$\mathrm{P}(\tau_{1}=\ldots=\tau_{N}=L)=1$, so that all births occur at the moment of individual’s death,

   -

the random variables $L$ and $N$ are independent.

For the Bellman-Harris process, conditions (1.1) and (1.3) imply $a=\mathrm{E}(L)$, $a<\infty$, and according to [12, Theorem 3], we get

$$tQ(t)\to h,\quad t\to\infty,\qquad h:=\frac{a+\sqrt{a^{2}+4bd}}{2b}.$$

(1.4)

As was shown in [11, Corollary B], see also [7, Lemma 3.2] for an adaptation to the discrete time setting, relation (1.4) holds even for the GWO-processes satisfying conditions (1.1), (1.3), and $a<\infty$.

The main result of this paper, Theorem 1 of Section 2, considers a critical GWO-process under the above mentioned neat set of assumptions (1.1), (1.3), $a<\infty$, and establishes the convergence of the finite-dimensional distributions conditioned on survival at a remote time of observation. A remarkable feature of this result is that its limit process is fully described by a single parameter $c:=4bda^{-2}$, regardless of complicated mutual dependencies between the random variables $\tau_{j},N,L$.

Our proof of Theorem 1, requiring an intricate asymptotic analysis of multi-dimensional probability generating functions, for the sake of readability, is split into two sections. Section 3 presents a new proof of (1.4) inspired by the proof of [12]. The crucial aspect of this approach, compared to the proof of [7, Lemma 3.2], is that certain essential steps do not rely on the monotonicity of the function $Q(t)$. In Section 4, the technique of Section 3 is further developed to finish the proof of Theorem 1.

We conclude this section by mentioning the illuminating family of GWO-processes called the Sevastyanov processes [9]. The Sevastyanov process is a generalised version of the Bellman-Harris process, with possibly dependent $L$ and $N$. In the critical case, the mean generation length of the Sevastyanov process, $a=\mathrm{E}(LN)$, can be represented as

$$a=\mathrm{Cov\hskip 0.56905pt}(L,N)+\mathrm{E}(L).$$

Thus, if $L$ and $N$ are positively correlated, the average generation length $a$ exceeds the average life length $\mathrm{E}(L)$.

Turning to a specific example of the Sevastyanov process, take

$$\mathrm{P}(L=t)=p_{1}t^{-3}(\ln\ln t)^{-1},\quad\mathrm{P}(N=0|L=t)=1-p_{2},\quad\mathrm{P}(N=n_{t}|L=t)=p_{2},\ t\geq 2,$$

where $n_{t}:=\lfloor t(\ln t)^{-1}\rfloor$ and $(p_{1},p_{2})$ are such that

$$\sum_{t=2}^{\infty}\mathrm{P}(L=t)=p_{1}\sum_{t=2}^{\infty}t^{-3}(\ln\ln t)^{-1}=1,\quad\mathrm{E}(N)=p_{1}p_{2}\sum_{t=2}^{\infty}n_{t}t^{-3}(\ln\ln t)^{-1}=1.$$

In this case, for some positive constant $c_{1}$,

$$\mathrm{E}(N^{2})=p_{1}p_{2}\sum_{t=1}^{\infty}n_{t}^{2}t^{-3}(\ln\ln t)^{-1}<c_{1}\int_{2}^{\infty}\frac{d(\ln t)}{(\ln t)^{2}\ln\ln t}<\infty,$$

implying that condition (1.1) is satisfied. Clearly, condition (1.3) holds with $d=0$. At the same time,

$$a=\mathrm{E}(NL)=p_{1}p_{2}\sum_{t=1}^{\infty}n_{t}t^{-2}(\ln\ln t)^{-1}>c_{2}\int_{2}^{\infty}\frac{d(\ln t)}{(\ln t)(\ln\ln t)}=\infty,$$

where $c_{2}$ is a positive constant. This example demonstrates that for the GWO-process, unlike the Bellman-Harris process, conditions (1.1) and (1.3) do not automatically imply the condition $a<\infty$.

The finite dimensional distributions of the limiting process $\eta(\cdot)$ are given below in terms of the $k$-dimensional probability generating functions $\mathrm{E}(z_{1}^{\eta(y_{1})}\cdots z_{k}^{\eta(y_{k})})$, $k\geq 1$, assuming

$$0=y_{0}<y_{1}<\ldots<y_{j}<1\leq y_{j+1}<\ldots<y_{k}<y_{k+1}=\infty,\quad 0\leq j\leq k,\quad 0\leq z_{1},\ldots,z_{k}<1.$$

(2.1)

Here the index $j$ highlights the pivotal value 1 corresponding to the time of observation $t$ of the underlying GWO-process.

As will be shown in Section 4.2, if $j=0$, then

$\displaystyle\mathrm{E}(z_{1}^{\eta(y_{1})}\cdots z_{k}^{\eta(y_{k})})=1-\frac{1+\sqrt{1+\sum\nolimits_{i=1}^{k}z_{1}\cdots z_{i-1}(1-z_{i})\Gamma_{i}}}{(1+\sqrt{1+c})y_{1}},\quad\Gamma_{i}:=c({y_{1}}/{y_{i}})^{2},$

and if $j\geq 1$,

$\displaystyle\mathrm{E}(z_{1}^{\eta(y_{1})}\cdots z_{k}^{\eta(y_{k})})=\frac{\sqrt{1+\sum_{i=1}^{j}z_{1}\cdots z_{i-1}(1-z_{i})\Gamma_{i}+cz_{1}\cdots z_{j}y_{1}^{2}}-\sqrt{1+\sum\nolimits_{i=1}^{k}z_{1}\cdots z_{i-1}(1-z_{i})\Gamma_{i}}}{(1+\sqrt{1+c})y_{1}}.$

In particular, for $k=1$, we have

	$\displaystyle\mathrm{E}(z^{\eta(y)})$	$\displaystyle=\frac{\sqrt{1+c(1-z)+czy^{2}}-\sqrt{1+c(1-z)}}{(1+\sqrt{1+c})y},\quad 0<y<1,$
	$\displaystyle\mathrm{E}(z^{\eta(y)})$	$\displaystyle=1-\frac{1+\sqrt{1+c(1-z)}}{(1+\sqrt{1+c})y},\quad y\geq 1.$

It follows that $\mathrm{P}(\eta(y)\geq 0)=1$ for $y>0$, and moreover, putting here first $z=1$ and then $z=0$, brings

	$\displaystyle\mathrm{P}(\eta(y)<\infty)$	$\displaystyle=\frac{\sqrt{1+cy^{2}}-1}{(1+\sqrt{1+c})y}\cdot 1_{\{0<y<1\}}+\Big{(}1-\frac{2}{(1+\sqrt{1+c})y}\Big{)}\cdot 1_{\{y\geq 1\}},$
	$\displaystyle\mathrm{P}(\eta(y)=0)$	$\displaystyle=\frac{y-1}{y}\cdot 1_{\{y\geq 1\}},$

implying that $\mathrm{P}(\eta(y)=\infty)>0$ for all $y>0$, and in fact, letting $y\to 0$, we may set $\mathrm{P}(\eta(0)=\infty)=1.$

To demonstrate that the process $\eta(\cdot)$ is indeed a pure death process, consider the function

$$\mathrm{E}(z_{1}^{\eta(y_{1})-\eta(y_{2})}\cdots z_{k-1}^{\eta(y_{k-1})-\eta(y_{k})}z_{k}^{\eta(y_{k})})$$

determined by

$\displaystyle\mathrm{E}(z_{1}^{\eta(y_{1})-\eta(y_{2})}\cdots z_{k-1}^{\eta(y_{k-1})-\eta(y_{k})}z_{k}^{\eta(y_{k})})$

$\displaystyle=\mathrm{E}(z_{1}^{\eta(y_{1})}(z_{2}/z_{1})^{\eta(y_{2})}\cdots(z_{k}/z_{k-1})^{\eta(y_{k})}).$

This function is given by two expressions

	$\displaystyle\frac{(1+\sqrt{1+c})y_{1}-1-\sqrt{1+\sum\nolimits_{i=1}^{k}(1-z_{i})\gamma_{i}}}{(1+\sqrt{1+c})y_{1}},\quad$	$\displaystyle\text{for }j=0,$
	$\displaystyle\frac{\sqrt{1+\sum\nolimits_{i=1}^{j-1}(1-z_{i})\gamma_{i}+(1-z_{j})\Gamma_{j}+cz_{j}y_{1}^{2}}-\sqrt{1+\sum\nolimits_{i=1}^{k}(1-z_{i})\gamma_{i}}}{(1+\sqrt{1+c})y_{1}},\quad$	$\displaystyle\text{for }j\geq 1,$

where $\gamma_{i}:=\Gamma_{i}-\Gamma_{i+1}$ and $\Gamma_{k+1}=0$. Setting $k=2$, $z_{1}=z$, and $z_{2}=1$, we deduce that the function

$$\mathrm{E}(z^{\eta(y_{1})-\eta(y_{2})};\eta(y_{1})<\infty),\quad 0<y_{1}<y_{2},\quad 0\leq z\leq 1,$$

(2.2)

is given by one of the following three expressions depending on whether $j=2$, $j=1$, or $j=0$,

	$\displaystyle\frac{\sqrt{1+cy_{1}^{2}+c(1-z)(1-(y_{1}/y_{2})^{2})}-\sqrt{1+c(1-z)(1-(y_{1}/y_{2})^{2})}}{(1+\sqrt{1+c})y_{1}},\quad$	$\displaystyle y_{2}<1,$
	$\displaystyle\frac{\sqrt{1+cy_{1}^{2}+c(1-z)(1-y_{1}^{2})}-\sqrt{1+c(1-z)(1-(y_{1}/y_{2})^{2})}}{(1+\sqrt{1+c})y_{1}},\quad$	$\displaystyle y_{1}<1\leq y_{2},$
	$\displaystyle 1-\frac{1+\sqrt{1+c(1-z)(1-(y_{1}/y_{2})^{2})}}{(1+\sqrt{1+c})y_{1}},\quad$	$\displaystyle 1\leq y_{1}.$

Since generating function (2.2) is finite at $z=0$, we conclude that

$$\mathrm{P}(\eta(y_{1})<\eta(y_{2});\eta(y_{1})<\infty)=0,\quad 0<y_{1}<y_{2}.$$

This implies

$$\mathrm{P}(\eta(y_{2})\leq\eta(y_{1}))=1,\quad 0<y_{1}<y_{2},$$

meaning that unless the process $\eta(\cdot)$ is sitting at the infinity state, it evolves by negative integer-valued jumps until it gets absorbed at zero.

Consider now the conditional probability generating function

$$\mathrm{E}(z^{\eta(y_{1})-\eta(y_{2})}|\eta(y_{1})<\infty),\quad 0<y_{1}<y_{2},\quad 0\leq z\leq 1.$$

(2.3)

In accordance with the above given three expressions for (2.2), generating function (2.3) is specified by the following three expressions

	$\displaystyle\frac{\sqrt{1+cy_{1}^{2}+c(1-z)(1-(y_{1}/y_{2})^{2})}-\sqrt{1+c(1-z)(1-(y_{1}/y_{2})^{2})}}{\sqrt{1+cy_{1}^{2}}-1},\quad$	$\displaystyle y_{2}<1,$
	$\displaystyle\frac{\sqrt{1+cy_{1}^{2}+c(1-z)(1-y_{1}^{2})}-\sqrt{1+c(1-z)(1-(y_{1}/y_{2})^{2})}}{\sqrt{1+cy_{1}^{2}}-1},\quad$	$\displaystyle y_{1}<1\leq y_{2},$
	$\displaystyle 1-\frac{\sqrt{1+c(1-z)(1-(y_{1}/y_{2})^{2})}-1}{(1+\sqrt{1+c})y_{1}-2},\quad$	$\displaystyle 1\leq y_{1}.$

In particular, setting here $z=0$, we obtain

$$\mathrm{P}(\eta(y_{1})-\eta(y_{2})=0|\eta(y_{1})<\infty)=\left\{\begin{array}[]{llr}\frac{\sqrt{1+c(1+y_{1}^{2}-(y_{1}/y_{2})^{2})}-\sqrt{1+c(1-(y_{1}/y_{2})^{2})}}{\sqrt{1+cy_{1}^{2}}-1}&\text{for}&0<y_{1}<y_{2}<1,\\ \frac{\sqrt{1+c}-\sqrt{1+c(1-(y_{1}/y_{2})^{2})}}{\sqrt{1+cy_{1}^{2}}-1}&\text{for}&0<y_{1}<1\leq y_{2},\\ 1-\frac{\sqrt{1+c(1-(y_{1}/y_{2})^{2})}-1}{(1+\sqrt{1+c})y_{1}-2}&\text{for}&1\leq y_{1}<y_{2}.\end{array}\right.$$

Notice that given $0<y_{1}\leq 1$,

$$\mathrm{P}(\eta(y_{1})-\eta(y_{2})=0|\eta(y_{1})<\infty)\to 0,\quad y_{2}\to\infty,$$

which is expected because of $\eta(y_{1})\geq\eta(1)\geq 1$ and $\eta(y_{2})\to 0$ as $y_{2}\to\infty$.

The random times

$$T=\sup\{u:\eta(u)=\infty\},\quad T_{0}=\operatorname*{\vphantom{p}inf}\{u:\eta(u)=0\},$$

are major characteristics of a trajectory of the limit pure death process. Since

$\displaystyle\mathrm{P}(T\leq y)=\mathrm{E}(z^{\eta(y)})\Big{|}_{z=1},\qquad\mathrm{P}(T_{0}\leq y)=\mathrm{E}(z^{\eta(y)})\Big{|}_{z=0},$

in accordance with the above mentioned formulas for $\mathrm{E}(z^{\eta(y)})$, we get the following marginal distributions

	$\displaystyle\mathrm{P}(T\leq y)$	$\displaystyle=\frac{\sqrt{1+cy^{2}}-1}{(1+\sqrt{1+c})y}\cdot 1_{\{0\leq y<1\}}+\Big{(}1-\frac{2}{(1+\sqrt{1+c})y}\Big{)}\cdot 1_{\{y\geq 1\}},$
	$\displaystyle\mathrm{P}(T_{0}\leq y)$	$\displaystyle=\frac{y-1}{y}\cdot 1_{\{y\geq 1\}}.$

The distribution of $T_{0}$ is free from the parameter $c$ and has the Pareto probability density function

$$f_{0}(y)=y^{-2}1_{\{y>1\}}.$$

In the special case (1.2), that is when (1.3) holds with $d=0$, we have $c=0$ and $\mathrm{P}(T=T_{0})=1$. If $d>0$, then $T\leq T_{0}$, and the distribution of $T$ has the following probability density function

$$f(y)=\left\{\begin{array}[]{llr}\frac{1}{(1+\sqrt{1+c})y^{2}}(1-\frac{1}{\sqrt{1+cy^{2}}})&\text{for}&0\leq y<1,\\ \frac{2}{(1+\sqrt{1+c})y^{2}}&\text{for}&y\geq 1,\end{array}\right.$$

having a positive jump at $y=1$ of size $f(1)-f(1-)=(1+c)^{-1/2}$. Observe that $\frac{f(1-)}{f(1)}\to\frac{1}{2}$ as $c\to\infty$.

Intuitively, the limiting pure death process counts the long-living individuals in the GWO-process, that is those individuals whose life length is of order $t$. These long-living individuals may have descendants, however none of them would live long enough to be detected by the finite dimensional distributions at the relevant time scale, see Lemma 2 below. Theorem 1 suggests a new perspective on Vatutin’s dichotomy, see [12], claiming that the long term survival of a critical age-dependent branching process is due to either a large number of short-living individuals or a small number of long-living individuals. In terms of the random times $T\leq T_{0}$, Vatutin’s dichotomy discriminates between two possibilities: if $T>1$, then $\eta(1)=\infty$, meaning that the GWO-process has survived due to a large number of individuals, while if $T\leq 1<T_{0}$, then $1\leq\eta(1)<\infty$ meaning that the GWO-process has survived due to a small number of individuals.

This section deals with the survival probability of the critical GWO-process

$$Q(t)=1-P(t),\quad P(t):=\mathrm{P}(Z(t)=0).$$

By its definition, the GWO-process can be represented as the sum

$$Z(t)=1_{\{L>t\}}+\sum\nolimits_{j=1}^{N}Z_{j}(t-\tau_{j}),\quad t=0,1,\ldots,$$

(3.1)

involving $N$ independent daughter processes $Z_{j}(\cdot)$ generated by the founder individual at the birth times $\tau_{j}$, $j=1,\ldots,N$ (here it is assumed that $Z_{j}(t)=0$ for all negative $t$). The branching property (3.1) implies the relation

$$1_{\{Z(t)=0\}}=1_{\{L\leq t\}}\prod\nolimits_{j=1}^{N}1_{\{Z_{j}(t-\tau_{j})=0\}},$$

saying that the GWO-process goes extinct by the time $t$ if, on one hand, the founder is dead at time $t$ and, on the other hand, all daughter processes are extinct by the time $t$. After taking expectations of both sides, we can write

$$P(t)=\mathrm{E}\Big{(}\prod\nolimits_{j=1}^{N}P(t-\tau_{j});L\leq t\Big{)}.$$

(3.2)

As shown next, this non-linear equation for $P(\cdot)$ entails the asymptotic formula (1.4) under conditions (1.1), (1.3), and $a<\infty$.