Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
Show that, if $X$ and $Y$ are two independent random variables taking values in $\mathbb{N}$, then $G_{X+Y}=G_X G_Y$.

Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
By admitting that, for all $k\in\mathbb{N}$, $S_k$ is independent of $X_{k+1}$, prove that, for all $k\in\mathbb{N}$, $G_{S_k}=(G_X)^k$.

Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
By admitting that, for all $n\in\mathbb{N}$, $T$ and $S_n$ are independent, show that $$\forall t\in\left[0,1\left[,\forall K\in\mathbb{N}\quad G_S(t)=\sum_{k=0}^K P(T=k)\left(G_X(t)\right)^k+\sum_{n=0}^\infty\left(\sum_{k=K+1}^\infty P(T=k)P\left(S_k=n\right)t^n\right)$$

Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
For $K\in\mathbb{N}$ and $t\in\left[0,1\left[$, we set $R_K=\sum_{n=0}^\infty\left(\sum_{k=K+1}^\infty P(T=k)P\left(S_k=n\right)t^n\right)$.
Show that $0\leqslant R_K\leqslant\frac{1}{1-t}\sum_{k=K+1}^\infty P(T=k)$.

Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$.
Conclude that $G_S=G_T\circ G_X$.

Let $(X_n)_{n\in\mathbb{N}^*}$ be a sequence of random variables, mutually independent, with the same distribution taking values in $\mathbb{N}$, and let $T$ be a random variable taking values in $\mathbb{N}$ independent of the previous ones. We denote by $G_X$ the generating function common to all the $X_n$. For $n\in\mathbb{N}$ and $\omega\in\Omega$, we set $S_n(\omega)=\sum_{k=1}^n X_k(\omega)$ and $S_0(\omega)=0$, then $S(\omega)=S_{T(\omega)}(\omega)$. We have shown $G_S=G_T\circ G_X$.
Deduce that, if $T$ and the $X_n$ have finite expectation, then so does $S$ and $E(S)=E(T)E(X_1)$.

Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with random variables $(X_{n,i})_{(n,i)\in\mathbb{N}\times\mathbb{N}^*}$ independent and all following distribution $\mu$, with $Y_0=1$ and $$\begin{cases} Y_{n+1}(\omega)=0 & \text{if }Y_n(\omega)=0\\ Y_{n+1}(\omega)=\sum_{i=1}^{Y_n(\omega)}X_{n,i}(\omega) & \text{if }Y_n(\omega)\neq 0\end{cases}$$ We denote by $f$ the generating function of $\mu$ and, for $n\in\mathbb{N}$, $\varphi_n$ the generating function of $Y_n$. We have $\varphi_0(t)=t$ for $t\in[0,1]$.
Show that, for all $n\in\mathbb{N}$, $\varphi_{n+1}=\varphi_n\circ f$.

Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We assume that every random variable following distribution $\mu$ has expectation equal to $m$ and a variance.
Express, for $n\in\mathbb{N}$, the expectation of $Y_n$ in terms of $m$ and $n$.

Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We denote by $f$ the generating function of $\mu$ and $\varphi_n$ the generating function of $Y_n$.
a) Verify that the probability of extinction is equal to the limit of the sequence $(\varphi_n(0))_{n\geqslant 0}$.
b) Verify that we can apply the results of Part I to the sequence $(\varphi_n(0))_{n\geqslant 0}$.

Let $\mu$ be a probability distribution characterized by the sequence $(p_k)_{k\in\mathbb{N}}$ with $\sum_{k=0}^{+\infty}p_k=1$ and $p_0+p_1<1$. We define the Galton-Watson process with $Y_0=1$ and the recurrence above. We denote by $f$ the generating function of $\mu$ and $m$ the expectation of $\mu$.
If $m\leqslant 1$, show that the probability of extinction is equal to 1.

We consider the Galton-Watson process with extinction time $T$ defined by: $$\omega\in\Omega\quad\begin{cases}T(\omega)=\min\{n\in\mathbb{N}\mid Y_n(\omega)=0\} & \text{if there exists }n\in\mathbb{N}\text{ such that }Y_n(\omega)=0\\ T(\omega)=-1 & \text{otherwise}\end{cases}$$
We assume $m<1$. Verify that $T$ has a finite expectation.

We consider the Galton-Watson process with extinction time $T$. We assume $m<1$.
a) Show that, for all integer $n$, $P(Y_n\geqslant 1)\leqslant m^n$.
b) Show that $E(T)=\sum_{n=0}^{+\infty}P(T>n)$.
c) Deduce an upper bound for $E(T)$.

We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$. We admit that $Z$ is a random variable defined on $\bigcup_{k\in\mathbb{N}}\{Y_k=0\}$.
Show that $Z$ is defined on a set of probability 1.

We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$.
a) Show that, for all $k\in\mathbb{N}$, $(P(Z_n\leqslant k))_{n\in\mathbb{N}^*}$ is a convergent sequence. Determine its limit.
b) Deduce that, for all $k\in\mathbb{N}$, $(P(Z_n=k))_{n\in\mathbb{N}^*}$ converges to $P(Z=k)$.
c) Show that, for all $s\in\left[0,1\left[$, all $n\in\mathbb{N}^*$ and $K\in\mathbb{N}$, $$\left|G_{Z_n}(s)-G_Z(s)\right|\leqslant\sum_{k=0}^K\left|P(Z_n=k)-P(Z=k)\right|+\frac{s^K}{1-s}$$
d) Deduce that the sequence of functions $(G_{Z_n})$ converges pointwise to $G_Z$ on $[0,1]$.

We consider the Galton-Watson process. We assume $m\leqslant 1$. We denote, for $n\in\mathbb{N}^*$, $Z_n=1+\sum_{i=1}^n Y_i$ and $Z=1+\sum_{n=1}^{+\infty}Y_n$. We denote by $f$ the generating function of $\mu$ and $m$ the expectation of $\mu$.
a) Express $G_{Z_1}$ in terms of $f$.
b) We admit that, for all natural integer $n$ greater than or equal to 2 and for all $s\in[0,1]$, $G_{Z_n}(s)=sf(G_{Z_{n-1}}(s))$.
Deduce that, for all $s\in\left[0,1\left[$, $G_Z(s)=sf(G_Z(s))$.
c) Show that $Z$ has finite expectation if and only if $m<1$. Calculate the expectation when this is the case.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Express, for $t\in[0,1]$, $f(t)$ and calculate $m$.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Verify that, for all $t\in\left[0,1\left[$, $\varphi_n(t)\neq 1$.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. We have $\varphi_n(t)\neq 1$ for $t\in[0,1[$, so we can set $a_n(t)=\frac{1}{\varphi_n(t)-1}$.
Show that, for $t\in\left[0,1\left[$, the sequence $(a_n(t))_{n\in\mathbb{N}}$ is arithmetic.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Deduce that, for $t\in\left[0,1\left[$ and $n\in\mathbb{N}$, $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. We have $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$.
Express, for $(n,k)\in\mathbb{N}^2$, $P(Y_n=k)$ in terms of $n$ and $k$.

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$. We have $\varphi_n(t)=\frac{n+(1-n)t}{1+n-nt}$. The extinction time $T$ is defined as before.
Express, in terms of $n\in\mathbb{N}^*$, the probability of the event $T>n$.
Does the variable $T$ have a finite expectation?

We assume that, for all $k\in\mathbb{N}$, $p_k=\frac{1}{2^{k+1}}$.
Express, for $s\in\left[0,1\left[$, $G_Z(s)$ in terms of $s$.
Deduce the distribution of $Z$.

We assume $m>1$. We study a slightly different problem: $k$ being a fixed strictly positive integer, we assume that there are $k$ individuals in generation 0. We denote by $W_n$ the number of individuals in the $n$-th generation and define $u_n$ as the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the first time at rank $n$: $$u_n=P\left((W_n=k)\cap\left(\bigcap_{i=1}^{n-1}(W_i\neq k)\right)\right)$$ For $n$ and $r$ non-zero natural integers, $u_n^{(r)}$ is the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ takes the value $k$ for the $r$-th time at rank $n$.
Verify that the series $\sum_{n\geqslant 1}u_n s^n$ and $\sum_{n\geqslant 1}u_n^{(r)}s^n$ converge when $s\in[-1,1]$.

We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$.
Show that $P(W_1>k)>0$.

We assume $m>1$. We study the Galton-Watson process starting with $k$ individuals in generation 0, with $W_n$ the number of individuals in generation $n$.
Show that the probability that the sequence $(W_n)_{n\in\mathbb{N}^*}$ does not take the value $k$ is non-zero; we denote this probability by $u$.
One may study separately the cases $p_0=0$ and $p_0>0$.