Infinitely divisible distributions

Let the random variable $\xi$ have an infinitely divisible distribution. Prove that for any real $a$ and $b$, the distribution of the random variable $a \xi +b$ is also infinitely divisible.

Prove that the weak limit of a sequence of infinitely divisible distributions is infinitely divisible.

Let $\xi$ and $\eta$ be independent random variables with an infinitely divisible distribution. Prove that the random variable $\xi +\eta$ also has an infinitely divisible distribution.

Let $f(t)$ be an infinitely divisible characteristic function. Prove that the characteristic function $\left|f(t)\right|$ is also infinitely divisible.

Prove that an infinitely divisible characteristic function is nowhere equal to zero.

Prove that the exponential distribution is infinitely divisible and find its "$k$-th root".

Let $f(t)$ be an infinitely divisible characteristic function. Prove that for any $\alpha > 0$ the function $(f(t))^{\alpha }$ is also an infinitely divisible characteristic function.

Let $f(t)$ be a characteristic function such that for some sequence of positive integers $n_{1}, n_{2}, \ldots$ satisfying $n_{k} \rightarrow \infty$ as $k \rightarrow \infty$, the functions

are characteristic functions. Prove that $f(t)$ is infinitely divisible.

Prove that the uniform distribution on a segment cannot be infinitely divisible.

Prove that the distribution with density $\frac{2}{\pi x^{2}} \sin^{2} \frac{x}{2}$ is not infinitely divisible.

Prove that:

Let $\xi$ and $\eta$ be independent random variables, where $\xi$ has a uniform distribution on some segment. Prove that the random variable $\xi +\eta$ cannot have an infinitely divisible distribution.

Let $F(x)$ be an arbitrary distribution function. Prove that for no $a \neq 0$ can the distribution function

be infinitely divisible.

Prove that a continuous function that is linear on each segment $[n, n+1], n = 0, \pm 1, \ldots$ cannot be an infinitely divisible distribution function.

A random variable $\xi$ defined on some probability space may be called infinitely divisible if for every positive integer $n$ it can be represented as the sum of $n$ independent, identically distributed random variables defined on the same probability space. Give an example of a variable that is not infinitely divisible in this sense but has an infinitely divisible distribution.

Prove that an infinitely divisible nondegenerate random variable cannot be bounded with probability one.

Let $\varphi (t)$ be any characteristic function and let $p$ be an arbitrary positive number. Prove that the function

is an infinitely divisible characteristic function.

Prove that any infinitely divisible characteristic function $f(t)$ can be represented in the form

where $p_{n}$ are positive numbers and $\varphi_{n}(t)$ are certain characteristic functions.

Let $\varphi (t)$ be an arbitrary characteristic function. Prove that for any $a > 1$ the function

is an infinitely divisible characteristic function.

Prove that the geometric distribution is infinitely divisible.

Let $\psi (t)$ be an arbitrary continuous nonpositive even function, convex for $t \geqslant 0, \psi (0) = 0$. Prove that $e^{\psi (t)}$ is an infinitely divisible characteristic function.

Prove that the normal distribution and the Cauchy distribution are stable.

Can one assert that the sum of independent random variables, each of which has a stable distribution, also has a stable distribution?

Prove that any stable distribution is infinitely divisible.

Is the exponential distribution stable?

Prove that the distribution with characteristic function $\exp \left\{ -c\left|t\right|^{\alpha }\right\}$, where $c > 0$ and $0 < \alpha \leqslant 2$, is stable.

Prove that for a stable random variable $\xi$

for all $r \in (0, \alpha )$, where $\alpha$ is the parameter in the canonical representation of stable distributions.

Prove that the characteristic function of a distribution stable with parameter $\alpha$, for $0 < \alpha \leqslant 1$, is not differentiable at zero.

Can an infinitely divisible distribution be discrete but not lattice?

Let $a_{1}, \ldots , a_{n}, c_{1}, \ldots , c_{n}$ be arbitrary positive numbers such that

Prove that the distribution with density

is infinitely divisible.

Let $f(x)$ be a positive measurable function and $F(x)$ a distribution function. Prove that the distribution with density

is infinitely divisible.

Prove that the function

where $a_{h} \geqslant 0, \sum a_{k} = 1, c_{k} \geqslant 0,0 < \alpha_{k} \leqslant 1$, is the characteristic function of an infinitely divisible distribution.

Can the generating function of an integer-valued nonnegative infinitely divisible distribution vanish?

Let $\xi$ be an integer-valued random variable with an infinitely divisible distribution. Prove that $\mathbb {P}\left(\xi \text{ is divisible by two}\right) \neq \mathbb {P}\left(\xi \text{ is not divisible by two}\right)$.

Let $\xi$ be an integer-valued random variable with a symmetric infinitely divisible distribution. Prove that \mathbb {P}\left(\xi is divisible by two \right) > \mathbb {P}\left(\xi is not divisible by two.

Let $\xi$ be a nonnegative integer-valued random variable with an infinitely divisible distribution, $\mathbb {P}\left(\xi = 0\right) > 0$. Prove that $\mathbb {P}$ ($\xi$ is divisible by two) $> \mathbb {P}$ ($\xi$ is not divisible by two).

Let $\xi$ be a nonnegative integer-valued random variable with an infinitely divisible distribution. Show that if $\mathbb {P}\left(\xi = 0\right) > 0$ and $\mathbb {P}\left(\xi = 1\right) > 0$, then $\mathbb {P}\left(\xi = k\right) > 0$ for every positive integer $k$.

Show that the density of an infinitely divisible distribution, symmetric about some point $a$, attains its maximum at the point $a$.

Let $F(x)$ be an integer-valued infinitely divisible distribution function, symmetric about the point $k$. Prove that $F(x)$ has its largest jump at the point $k$.

Show that the characteristic function

is not infinitely divisible.

Can an infinitely divisible characteristic function be the product of two characteristic functions that are not infinitely divisible?

Let $\xi$ be a nonnegative integer-valued random variable with an infinitely divisible distribution, $p_{k} = \mathbb {P}\left(\xi = k\right)$, $p_{0} > 0, p_{1} > 0$. Prove that $p_{1}^{2} \leqslant 2 p_{0} p_{2}$.

Let $\xi_{n k}, n \geqslant 1,1 \leqslant k \leqslant n$ be a sequence of random variables, where for each $n$ the random variables

are independent and identically distributed. Set

Prove that if the distributions of $\eta_{n}$ converge weakly as $n \rightarrow \infty$ to some limiting distribution, then the latter is infinitely divisible.

Let $\xi$ be a nonnegative integer-valued random variable, $\mathbb {P}\left(\xi = 0\right) > 0$. Prove that for $\xi$ to have an infinitely divisible distribution, it is necessary and sufficient that its generating function admit the representation

where $Q(z)$ is the generating function of some integer-valued nonnegative random variable, and $\lambda$ is a positive number.

Let $\psi (t)$ be an arbitrary characteristic function. Prove that the function

is the characteristic function of an infinitely divisible distribution with finite variance.

Prove that the function

is the characteristic function of an infinitely divisible distribution with finite variance.

Let $\xi$ be an integer-valued random variable having an infinitely divisible distribution. Prove that its characteristic function $\varphi (z)$ can be represented in the form

where $m$ is an integer, $q_{k} \geqslant 0$.

Let $\xi$ be an infinitely divisible random variable with a lattice distribution ($h$ is the lattice step). Prove that

where $v_{1}, v_{2}, \ldots$ are independent random variables having a Poisson distribution.

Prove that for a sequence of infinitely divisible characteristic functions $\varphi_{1}(z), \varphi_{2}(z), \ldots$ to converge to a characteristic function $\varphi (z)$, it is necessary and sufficient that for almost all $x$, $G_{n}(x) \rightarrow G(x)$ and $\gamma_{n} \rightarrow \gamma$ as $n \rightarrow \infty$, where $G_{n}, G, \gamma_{n}, \gamma$ are the functions and constants from the Lévy–Khinchin canonical representation of the characteristic functions $\varphi_{n}(z)$ and $\varphi (z)$.

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, and let $v$ be a random variable, independent of $\xi_{1}, \xi_{2}, \ldots$, having a Poisson distribution. Prove that the random variable

has an infinitely divisible distribution.

Prove that for all $\alpha > 0$ the function

is an infinitely divisible characteristic function. Find its representation in Lévy–Khinchin form.

Find the Lévy–Khinchin representation of the characteristic functions of the normal and Poisson distributions.

Let $\xi = \xi_{1}+\xi_{2}$, where $\xi_{1}$ and $\xi_{2}$ are independent and have infinitely divisible distributions. Prove that: