Characteristic functions

Find two dependent random variables $X$ and $Y$ such that $\phi_{X+Y}(t) = \phi_{X}(t) \phi_{Y}(t)$ for all $t$.

If $\phi$ is a characteristic function, show that $\operatorname {Re}\left\{ 1-\phi (t)\right\} \geq \frac{1}{4} \operatorname {Re}\left\{ 1-\phi (2 t)\right\}$, and deduce that $1-\left|\phi (2 t)\right| \leq 8\left\{ 1-\left|\phi (t)\right|\right\}$.

The cumulant generating function $K_{X}(\theta )$ of the random variable $X$ is defined by $K_{X}(\theta ) = \log \mathbb {E}\left[e^{\theta X}\right]$, the logarithm of the moment generating function of $X$. If the latter is finite in a neighbourhood of the origin, then $K_{X}$ has a convergent Taylor expansion:

and $k_{n}(X)$ is called the $n$th cumulant (or semi-invariant) of $X$.

Let $X$ be $N(0,1)$, and show that the cumulants of $X$ are $k_{2}(X) = 1, k_{m}(X) = 0$ for $m \neq 2$.

The random variable $X$ is said to have a lattice distribution if there exist $a$ and $b$ such that $X$ takes values in the set $L(a, b) = \left\{ a+b m: m = 0, \pm 1, \ldots \right\}$. The span of such a variable $X$ is the maximal value of $b$ for which there exists $a$ such that $X$ takes values in $L(a, b)$.

Let $X$ be a random variable with density function $f$. Show that $\left|\phi_{X}(t)\right| \rightarrow 0$ as $t \rightarrow \pm \infty$.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent variables, $X_{i}$ being $N\left(\mu_{i}, 1\right)$, and let $Y = X_{1}^{2}+X_{2}^{2}+\cdots +X_{n}^{2}$. Show that the characteristic function of $Y$ is

where $\theta = \mu_{1}^{2}+\mu_{2}^{2}+\cdots +\mu_{n}^{2}$. The random variables $Y$ is said to have the non-central chi-squared distribution with $n$ degrees of freedom and non-centrality parameter $\theta$, written $\chi^{2}(n ; \theta )$.

Let $X$ be $N(\mu , 1)$ and let $Y$ be $\chi^{2}(n)$, and suppose that $X$ and $Y$ are independent. The random variable $T = X / \sqrt{Y / n}$ is said to have the non-central $t$-distribution with $n$ degrees of freedom and non-centrality parameter $\mu$. If $U$ and $V$ are independent, $U$ being $\chi^{2}(m ; \theta )$ and $V$ being $\chi^{2}(n)$, then $F = (U / m) /(V / n)$ is said to have the non-central $F$-distribution with $m$ and $n$ degrees of freedom and non-centrality parameter $\theta$, written $F(m, n ; \theta )$.

Let $X$ be a random variable with density function $f$ and characteristic function $\phi$. Show, subject to an appropriate condition on $f$, that

If $X$ and $Y$ are continuous random variables, show that

Let $X$ and $Y$ be independent with the distributions $N\left(\mu , \sigma^{2}\right)$ and $N\left(0, \sigma^{2}\right)$, where $\sigma^{2} > 0$. Show that $R = \sqrt{X^{2}+Y^{2}}$ has density function

The integral may be expressed in terms of a modified Bessel function.

For each of the following joint density functions of the pair $(X, Y)$, find the joint moment generating function $M(s, t) = \mathbb {E}\left[e^{s X+t Y}\right]$, and hence find $\operatorname {Cov}\left[X, Y\right]$.