The three assertions to be shown equivalent are: (i) $X$ is infinitely divisible; (ii) $X$ is $\lambda$-positive; (iii) there exists a sequence $\left(X_{i}\right)_{i \geqslant 1}$ of independent Poisson variables, as in II.C.3, such that $X \sim \sum_{i=1}^{\infty} i X_{i}$.
a) Show the result announced at the beginning of subsection III.C, namely that assertions (i), (ii), and (iii) are equivalent for an infinitely divisible random variable $X$ taking values in $\mathbb{N}$ with $\mathbb{P}(X=0)>0$.
b) How to adapt this result to random variables taking values in $\mathbb{N}^{*}$?
c) Let $X$ be a random variable following the geometric distribution $\mathcal{G}(p)$, where $p \in ]0,1[$: $$\forall k \in \mathbb{N}^{*} \quad \mathbb{P}(X = k) = (1-p)^{k-1}p$$ Is the random variable $X$ infinitely divisible?

Let $n$ be a non-zero natural integer. For $i \in \llbracket 1 , n \rrbracket$, $X _ { i }$ is a random variable on $(\Omega , \mathcal { A } , \mathbb { P })$ following a Bernoulli distribution with parameter $\lambda / n$. We assume that $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ are mutually independent and we set $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$.
For $t \in \mathbb { R }$, calculate $\lim _ { n \rightarrow + \infty } M _ { S _ { n } } ( t )$.

Let $n$ be a non-zero natural number and $t$ be a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_X(t) = \mathbb{E}(\mathrm{e}^{\mathrm{i}tX})$.
Show $$\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right).$$

Let $n$ be a non-zero natural number and $t$ a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_X(t) = \mathbb{E}(\mathrm{e}^{\mathrm{i}tX})$.
Show $$\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right).$$

Let $n$ be a non-zero natural number and $t$ be a real number. Using the result of Q1, deduce $$\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}.$$

Let $n$ be a non-zero natural number and $t$ a real number. We set $$\forall n \in \mathbb{N}^{\star}, \quad X_n = \sum_{k=1}^{n} \frac{\varepsilon_k}{2^k}$$ where $(\varepsilon_n)_{n \geqslant 1}$ is a sequence of independent random variables taking values in $\{-1,1\}$ with $\mathbb{P}(\varepsilon_n = 1) = \mathbb{P}(\varepsilon_n = -1) = 1/2$ for all $n \geqslant 1$, and $\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right)$.
Deduce $$\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}.$$

Let $n$ be a non-zero natural number and $t$ be a real number. We have $\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right)$ and $\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}$.
Determine the pointwise limit of the sequence of functions $\left(\Phi_{X_n}\right)_{n \geqslant 1}$.

We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:

• we randomly draw a ball from the urn;
• we replace the drawn ball in the urn;
• we add to the urn a ball of the same color as the drawn ball.

We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$, where $g_{n}(t) = \sum_{k=0}^{+\infty} P(X_{n} = k) t^{k}$.
Determine the distributions of $X_{1}, X_{2}$ and $X_{3}$ and then the functions $g_{1}, g_{2}$ and $g_{3}$.

We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:

• we randomly draw a ball from the urn;
• we replace the drawn ball in the urn;
• we add to the urn a ball of the same color as the drawn ball.

We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$.
Using the recurrence relation from question 8, deduce that, for all integers $n$ greater than or equal to 1 and all real $t$, $$g_{n}(t) = \frac{t^{2} - t}{n+1} g_{n-1}^{\prime}(t) + g_{n-1}(t)$$

We have an infinite supply of black and white balls. An urn initially contains one black ball and one white ball. We perform a sequence of draws according to the following protocol:

• we randomly draw a ball from the urn;
• we replace the drawn ball in the urn;
• we add to the urn a ball of the same color as the drawn ball.

We define the sequence $(X_{n})_{n \in \mathbb{N}}$ of random variables by $X_{0} = 1$ and, for all integers $n \geqslant 1$, $X_{n}$ gives the number of white balls in the urn after $n$ draws. We denote by $g_{n}$ the generating function of the random variable $X_{n}$.
Prove that, for all integers $n \in \mathbb{N}^{*}$ and all real $t$, $$g_{n}(t) = \frac{1}{n+1} \sum_{k=1}^{n+1} t^{k}$$

Explicitly calculate the generating function $G_{X_1}$ of the random variable $X_1$, where $X_1$ follows a Poisson distribution with parameter $1/2$.

Justify that $\forall t \in \mathbb{R}, G_{S_n}(t) = \left(G_{X_1}(t)\right)^n$.

We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ Show that the power series defining $F$ and $G$ have radius of convergence greater than or equal to 1. Justify then that the functions $F$ and $G$ are defined and of class $C^{\infty}$ on $]-1,1[$.
Show that $G$ is defined and continuous on $[-1,1]$ and that $$G ( 1 ) = P ( R \neq + \infty ) .$$

Let $n \in \mathbb { N } ^ { * }$ and $\left. p \in \right] 0,1 [$. We assume that $X : \Omega \rightarrow \mathbb { R }$ follows a binomial distribution $\mathcal { B } ( n , p )$ and we denote $q = 1 - p$. Show that, for all $t \in \mathbb { R } , \phi _ { X } ( t ) = \left( q + p \mathrm { e } ^ { \mathrm { i } t } \right) ^ { n }$.

We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ If $k$ and $n$ are positive integers such that $k \leq n$, show that $$P \left( \left( S _ { n } = 0 _ { d } \right) \cap ( R = k ) \right) = P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$ Deduce that $$\forall n \in \mathbb{N}^{*} , \quad P \left( S _ { n } = 0 _ { d } \right) = \sum _ { k = 1 } ^ { n } P ( R = k ) P \left( S _ { n - k } = 0 _ { d } \right) .$$

We consider the functions $F$ and $G$ defined by the formulas $$\begin{aligned} & \forall x \in ]-1,1[ , \quad F ( x ) = \sum _ { n = 0 } ^ { + \infty } P \left( S _ { n } = 0 _ { d } \right) x ^ { n } \\ & \forall x \in [ - 1,1 ] , \quad G ( x ) = \sum _ { n = 1 } ^ { + \infty } P ( R = n ) x ^ { n } \end{aligned}$$ Show that $$\forall x \in ]-1,1[ , \quad F ( x ) = 1 + F ( x ) G ( x ) .$$ Determine the limit of $F ( x )$ as $x$ tends to $1^{-}$, discussing according to the value of $P ( R \neq + \infty )$.

Let $\lambda > 0$. What is the characteristic function of a random variable following a Poisson distribution with parameter $\lambda$ ?

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is countable and we use the notations of question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Using the results of Q17--Q19, establish that $V _ { m } ( T ) \xrightarrow [ T \rightarrow + \infty ] { } \mathbb { P } ( X = m )$.

Let $X : \Omega \rightarrow \mathbb { R }$ and $Y : \Omega \rightarrow \mathbb { R }$ be two discrete random variables such that $\phi _ { X } = \phi _ { Y }$. Show that, for all $m \in \mathbb { R } , \mathbb { P } ( X = m ) = \mathbb { P } ( Y = m )$, in other words that $X$ and $Y$ have the same distribution.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. Let $k \in \mathbb { N } ^ { * }$. We assume that $X$ admits a moment of order $k$. Deduce an expression of $\mathbb { E } \left( X ^ { k } \right)$ in terms of $\phi _ { X } ^ { ( k ) } ( 0 )$.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ The generating series of $T$ is given by $G _ { T } ( t ) = \sum _ { n = 0 } ^ { \infty } \mathbb { P } ( T = n ) t ^ { n }$, defined if $t \in [ - 1,1 ]$. Using the previous questions, express $G _ { T }$ using $g$ and $\mathbb { P } ( T = 0 )$.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that there exists a sequence $\left( P _ { k } \right) _ { k \in \mathbf { N } }$ of polynomials with coefficients in $\mathbf { C }$, independent of $p$, such that
$$\forall \theta \in \mathbf { R } , \forall k \in \mathbf { N } , \Phi _ { X } ^ { ( k ) } ( \theta ) = p i ^ { k } e ^ { i \theta } \frac { P _ { k } \left( q e ^ { i \theta } \right) } { \left( 1 - q e ^ { i \theta } \right) ^ { k + 1 } } \quad \text { and } \quad P _ { k } ( 0 ) = 1$$

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Deduce that there exists a sequence $\left( C _ { k } \right) _ { k \in \mathbf { N } }$ of strictly positive reals, independent of $p$, such that
$$\forall k \in \mathbf { N } , \left| \mathbf { E } \left( X ^ { k } \right) - \frac { 1 } { p ^ { k } } \right| \leq \frac { C _ { k } q } { p ^ { k } }$$

We denote by $G_{X_n}$ and $G_Y$ the generating functions of the variables $X_n$ and $Y$ from the previous question, respectively. Express $G_{X_n}(s)$ as a sum, for $s$ real, and verify that $$\forall s \in \mathbf{R} \quad \lim_{n \rightarrow +\infty} G_{X_n}(s) = G_Y(s)$$

Let us randomly place circle stones $\bigcirc$ and square stones $\square$ one by one in a line from left to right. The circle and square stones are placed with probability $1 - q$ and $q$, respectively, according to the independent and identical distribution, where $0 < q < 1$. The placement stops right after $M$ square stones are placed in a row, where $M$ is a positive integer. We show examples of the lines for $M = 4$ as follows.
Let $L$ be a random variable which represents the number of the stones after stopping the placement. For the case of the lines shown above, $L = 5$ and $L = 9$ for lines 1 and 2, respectively.
Here, we consider intermediate states during the placement. Let $k$ be a non-negative integer and let $C_k$ be a state of a line where there are $k$ square stones in a row from the right end. Since we are considering the case of $M = 4$, lines 3 and 4 are not stopped yet. Line 3 is in state $C_2$ since there are 2 square stones in a row from the right end. Line 4 is in state $C_0$ since there is no square stone at the right end. Let $a_{kn}$ be the probability that the stopping condition is met after placing $n$ stones starting from state $C_k$, where $n$ is a non-negative integer. We define the following generating function $A_k(t)$ for $a_{kn}$.
$$A_k(t) = \sum_{n=0}^{\infty} t^n a_{kn}$$
Answer the following questions.
(1) Calculate the mean and variance of $L$ for $M = 1$.
(2) Obtain the recurrence relation that $A_k(t)$ satisfies.
(3) Obtain $A_k(t)$ as a function of $q, M, t$, and $k$.
(4) Calculate the mean of $L$.