Using probability generating functions, we proved that the subcritical branching process will certainly die out, i.e., its extinction probability is 1. It is natural to ask how fast \(P(X_n=0\))$ get closer to 1, or, in other words, how fast the survival probability \(P(X_n>0)\) is approaching 0 as \(n\to \infty\).Below we will address this question.
Let us start with the following inequality.
Markov Inequality If \(Z\) is a non-negative random variable, not identically 0, then for every \(z>0\) \[ P(Z\ge z)\le \frac{E(Z)}{z}. \] Proof. For \(z>0\), we have \[ P(Z\ge z)=E(I_{Z\ge z})=\frac{E(zI_{Z\ge z})}{z}\le \frac{E(ZI_{Z\ge z})}{z}\le \frac{E(Z)}{z}, \] where \(I_{A}\) is the indicator of event \(A\).
Let us go back to branching processes. If the offspring mean \(\mu<1\) then Markov inequality implies \[ Q_n:= P(X_n>0\ |\ X_0=1)=P(X_n\ge 1)\le E(X_n)=\mu^n. \] Therefore, the non-extinction probability decayes exponentially. How sharp is the upper bound above? The following theorem provides the answer.
Denote \(\ln^+x=\ln x\) when \(x>1\) and \(\ln^+ x=0\) when \(x\le 1\).
Theorem 12.1 If \(\mu<1\), then \[ Q_n\sim K\mu^n (1+o(1)), \qquad K>0, \qquad n\to \infty, \] if and only if \[ E(X_1\ln^+ X_1)=\sum_{k=1}^\infty p_kk\ln k<\infty. \] The proof is omited.
It follows from Theorem 12.1 that \[ \frac{\mu^n}{Q_n}=\frac{E(X_n)}{P(X_n>0)}=E(X_n\ |\ X_n>0)\approx \frac{1}{K}, \qquad n\to \infty. \]
In English, under the assumptions of Theorem 12.1, if the subcritical process did not die out after long enough time \(n\), then the expected population size stabilizes around a constant \(1/K\).
The fact that the subcritical branching process will certainly die out suggests that a study of the asymptotic properties, as \(n\) increases, of this class processes must be done assuming that \(X_n>0\), i.e., under the condition that the \(n\)th generation is not empty. Theorem 12.1 establishes that the conditional expectation \(E(X_n\ |\ X_n>0)\) approaches a constnt as \(n\to \infty\). This, in turn, raises the question if the generation size \(X_n\), given non-extinction until time \(n\), converges in distribution to a discrete random variable. Next theorem has the affirmative answer.
Theorem 12.2 If \(\mu<1\), then \[ \lim_{n\to \infty} P(X_n=k\ |\ X_n>0)=p^*_k, \qquad \sum_{k=1}^\infty p^*_k=1. \] The probabiility generating function of the limiting distribution \[ f^*(s)=\sum_{k=1}^\infty p^*_ks^k \] is the solution of the equation \[ 1-f^*(\Pi_{X_0}(s))=\mu (1-f^*(s)), \] where recall that \(\Pi_{X_0}(s)\) is the offspring PGF.
The proof of the theorem is omited.
Next theorem answers the question when the limiting random variable in Theorem 12.2 has finite expectation.
Theorem 12.3 \[ \frac{d}{ds} f^*(s)\big|_{s=1}<\infty \qquad \mbox{if and only if}\qquad E(X_1\ln^+ X_1)<\infty. \]
Example Consider the subcritcal branching process with geometric offspring distribution and offspring mean \(\mu=p/q\). For the offspring PGF we write \[ \Pi_{X_0}(s)=\frac{q}{1-ps}=\frac{1}{1+\mu (1-s)}. \] Also, we know that the PGF of \(X_n\) satisfies \[ 1-\Pi_{X_n}(s)=\frac{\mu^n (\mu-1)(1-s)}{\mu (\mu^n -1)(1-s)+\mu -1} \] and hence \[ P(X_n>0)=1-\Pi_{X_n}(0)=\frac{\mu^n (\mu-1)}{\mu^{n+1}-1}. \] The following formula for the conditional expectation holds \[ E\left(s^{X_n}\ |\ X_n>0\right)= \sum_{k=1}^\infty s^k P(X_n=k\ |\ X_n>0)= \frac{1}{P(X_n>0)}\sum_{k=1}^\infty s^kP(X_n=k) \] \[ =\frac{1}{P(X_n>0)}\left[E\left(s^{X_n}\right)-P(X_n=0)\right]=\frac{\Pi_{X_n}(s)-\Pi_{X_n}(0)}{1-\Pi_{X_n}(0)} \] \[ \frac{[1-\Pi_{X_n}(0)]-[1-\Pi_{X_n}(s)]}{1-\Pi_{X_n}(0)}=1-\frac{1-\Pi_{X_n}(s)}{1-\Pi_{X_n}(0)}. \] Therefore, \[ \lim_{n\to \infty} E\left(s^{X_n}\ |\ X_n>0\right)=\lim_{n\to \infty}\left[1-\frac{1-\Pi_{X_n}(s)}{1-\Pi_{X_n}(0)}\right] \] \[ = 1-\frac{1-s}{1-\mu s}=\frac{s(1-\mu)}{1-\mu s}=\frac{s(q-p)}{q-ps} \] \[ = \frac{s}{1+\mu^* (1-s)}=f^*(s), \] where \(\mu^*=p/(q-p)\). This shows that the limiting distribution is geometric as well, but has support \(\{1,2,\ldots\}\) rather than \(\{0,1,2,\ldots \}\). Also the expected value of the limiting random variable converges to infinity if \(p\uparrow 1/2\), i.e., when the parameters of the subcritical process get closer to those of the critical case \(p=q=1/2\).