Branching processes

Let $Z_{n}$ be the size of the $n$th generation in an ordinary branching process with $Z_{0} = 1, \mathbb {E}\left[Z_{1}\right] = \mu$, and $\operatorname {Var}\left[Z_{1}\right] > 0$. Show that $\mathbb {E}\left[Z_{n} Z_{m}\right] = \mu^{n-m} \mathbb {E}\left[Z_{m}^{2}\right]$ for $m \leq n$. Hence find the correlation coefficient $\rho \left(Z_{m}, Z_{n}\right)$ in terms of $\mu$.

Consider a branching process with generation sizes $Z_{n}$ satisfying $Z_{0} = 1$ and $\mathbb {P}\left(Z_{1} = 0\right) = 0$. Pick two individuals at random (with replacement) from the $n$th generation and let $L$ be the index of the generation which contains their most recent common ancestor. Show that $\mathbb {P}\left(L \geq r\right) \geq \mathbb {E}\left[Z_{r}^{-1}\right]$ for $0 \leq r < n$. Show when $r \neq 0$ that equality holds if and only if $Z_{1}$ is a.s. constant. What can be said if $\mathbb {P}\left(Z_{1} = 0\right) > 0$?

Consider a branching process whose family sizes have the geometric mass function $f(k) = q p^{k}$, $k \geq 0$, where $p+q = 1$, and let $Z_{n}$ be the size of the $n$th generation. Let $T = \min \left\{ n: Z_{n} = 0\right\}$ be the extinction time, and suppose that $Z_{0} = 1$. Find $\mathbb {P}\left(T = n\right)$. For what values of $p$ is it the case that $\mathbb {E}\left[T\right] < \infty$?

Let $Z_{n}$ be the size of the $n$th generation of a branching process, and assume $Z_{0} = 1$. Find an expression for the generating function $G_{n}$ of $Z_{n}$, in the cases when $Z_{1}$ has generating function:

Each generation of a branching process (with a single progenitor) is augmented by a random number of immigrants who are indistinguishable from the other members of the population. Suppose that the numbers of immigrants in different generations are independent of each other and of the past history of the branching process, each such number having probability generating function $H(s)$. Show that the probability generating function $G_{n}$ of the size of the $n$th generation satisfies $G_{n+1}(s) = G_{n}(G(s)) H(s)$, where $G$ is the probability generating function of a typical family of offspring.

Let $Z_{n}$ be the size of the $n$th generation in a branching process with $\mathbb {E}\left[s^{Z_{1}}\right] = (2-s)^{-1}$ and $Z_{0} = 1$. Let $V_{r}$ be the total number of generations of size $r$. Show that $\mathbb {E}\left[V_{1}\right] = \frac{1}{6} \pi^{2}$, and $\mathbb {E}\left[2 V_{2}-V_{3}\right] = \frac{1}{6} \pi^{2}-\frac{1}{90} \pi^{4}$.

Let $T$ be the total number of individuals in a branching process with family-size distribution $\operatorname {bin}(2, p)$, where $p \neq \frac{1}{2}$. Show that

What is the family-size distribution conditional on $T < \infty$?

Let $Z$ be a branching process with $Z_{0} = 1, \mathbb {E}\left[Z_{1}\right] = \mu > 1$, and $\operatorname {Var}\left[Z_{1}\right] = \sigma^{2}$. Use the Paley-Zygmund inequality to show that the extinction probability $\eta_{n} = \mathbb {P}\left(Z_{n} = 0\right)$ satisfies