Topics from the Theory of Characteristic Functions

Let $g$ and $h$ be real-valued functions defined on $\mathbb {R}$ for which $\int_{\mathbb {R}} g(x) d \mu$ and $\int_{\mathbb {R}} h(x) d \mu$ are finite, where $\mu$ is a ($\sigma$-finite) measure in $\mathbb {R}$. Then show that:

In particular, $\left|\mathbb {E}\left[Z\right]\right| \leq \mathbb {E}\left[\left|Z\right|\right]$, where $Z$ is a complex-valued r.v.; i.e., $Z = X+i Y$ with $X$ and $Y$ real-valued r.v.s.

In reference to the proof of Theorem 2, show that $\lim_{t \rightarrow 0} g(t) = b-a$ as claimed there.

In reference to the proof of Proposition 1 (see also proof of Proposition 3), provide the details of the convergence

Show that $\left|e^{i x}-1\right| \leq \left|x\right|$ for all $x \in \mathbb {R}$.

Show that $\left(1+\frac{c_{n}}{n}\right)^{n} \underset {n \rightarrow \infty }{\rightarrow } e^{c}$, when $c_{n} \rightarrow \infty$, where $c_{n} \in \mathbb {C}$, the complex plane, $n \geq 1$.

Let $X$ be a r.v. having the Cauchy distribution with parameters $\mu = 0$ and $\sigma = 1$ (i.e., the p.d.f. of $X$ is given by $p(x) = \frac{1}{\pi } \times \frac{1}{1+x^{2}}, x \in \mathbb {R}$). Then show that:

Show that: $\frac{[(2 n)!]^{1 / 2 n}}{[(2 n-1)!]^{1 /(2 n-1)}} \xrightarrow [n \rightarrow \infty ]{\rightarrow } 1$.

Recall the following lemma (Lemma 6). Let $X$ be a r.v. such that $\mu^{(n)} = \mathbb {E}\left[\left|X\right|^{n}\right] < \infty$ for all $n = 1, 2, \ldots$ (so that $m^{(n)} = \mathbb {E}\left[X^{n}\right]$ are also finite). Then the series

have the same radii of convergence.

Establish the validity of the relations claimed in the following corollary (the Corollary to Lemma 6). A sufficient condition for the series $S(t)$ to converge for some $t_{0} \neq 0$ is that

If $X_{1}, \ldots , X_{n}$ are i.i.d. r.v.s with $\mathbb {E}\left[X_{1}\right] = \mu \in \mathbb {R}$ and $\sigma^{2}\left(X_{1}\right) = \sigma^{2} \in (0, \infty )$, then (by Application 2 to Theorem 9 in this chapter) it follows that $\frac{S_{n}-n \mu }{\sigma \sqrt{n}} \underset {n \rightarrow \infty }{\Rightarrow } Z \sim N(0,1)$, where $S_{n} = \sum_{j = 1}^{n} X_{j}$. Show that $\left\{ \frac{S_{n}-n \mu }{\sigma \sqrt{n}}\right\}$ does not converge in probability as $n \rightarrow \infty$.

According to the WLLN (Application 1 to Theorem 9), if the r.v.s $X_{1}, \ldots , X_{n}$ are i.i.d. with finite $\mathbb {E}\left[X_{1}\right]$, then

The following example shows that it is possible for $\left\{ \frac{S_{n}}{n}\right\}$ to converge in probability to a finite constant, as $n \rightarrow \infty$, even if the $\mathbb {E}\left[X_{1}\right]$ does not exist. To this effect, for $j = 1,2, \ldots$, let $X_{j}$ be i.i.d. r.v.s such that $\mathbb {P}\left(X_{j} = -n\right) = \mathbb {P}\left(X_{j} = n\right) = c / n^{2} \log n, n \geq 3$, where $c = \frac{1}{2} \sum_{n = 3}^{\infty } \frac{1}{n^{2}} \log n$. Then show that $\mathbb {E}\left[X_{1}\right]$ does not exist, but $\frac{S_{n}}{n} \xrightarrow [n \rightarrow \infty ]{P} 0$, where $S_{n} = \sum_{j = 1}^{n} X_{j}$.

If $X \sim B(n, p)$, show that

If $X \sim \operatorname {Pois}\left(\lambda \right)$, show that

If $X$ has the Gamma distribution with parameters $\alpha$ and $\beta$; i.e., if its p.d.f. is given by

where the Gamma function $\Gamma (\alpha )$ is given by $\Gamma (\alpha ) = \int_{0}^{\infty } y^{\alpha -1} e^{-y} d y(\alpha , \beta > 0)$, then show that

In particular, for $\alpha = 1$ and $\beta = 1 / \lambda$, we get the ch.f. of the Negative Exponential distribution with parameter $\lambda$; i.e., $f_{X}(t) = 1 /\left(1-\frac{i t}{\lambda }\right)$; and for $\alpha = \frac{r}{2}(r > 0$ integer) and $\beta = 2$, we get the ch.f. of the chi-square distribution with $r$ degrees of freedom; i.e., $f_{X}(t) = 1 /(1-2 i t)^{r / 2}$.

If the r.v.s $X$ and $Y$ have the Bivariate Normal distribution with parameters $\mu_{1}, \mu_{2}$ in $\mathbb {R}, 0 < \sigma_{1}, \sigma_{2} < \infty$, and $\rho \in [-1,1]$, show that their joint ch.f. is given by

For this purpose, do the following:

Let the r.v.s $X$ and $Y$ have the Bivariate Normal distribution with parameters $\mu_{1}, \mu_{2} \in \mathbb {R}, 0 < \sigma_{1}, \sigma_{2} < \infty$, and $\rho \in [-1,1]$, and set $U = X+Y, V = X-Y$.

In this exercise, the r.v.s $X_{1}, \ldots , X_{k}$ are independent with distributions as indicated, and $X = X_{1}+\cdots +X_{k}$. Then use Exercise 13 in Chapter 10, the inversion formula (Theorem 2), and the appropriate ch.f.s in order to show that:

Show that the r.v.s $X_{1}, \ldots , X_{k}$ are independent if and only if, for all $t_{1}, \ldots , t_{k}$ in $\mathbb {R}, f_{X_{1}, \ldots , X_{k}}\left(t_{1}, \ldots , t_{k}\right) = f_{X_{1}}\left(t_{1}\right) \times \cdots \times f_{X_{k}}\left(t_{k}\right)$.

If $f$ is a ch.f., then show that $f$ is positive definite; i.e., for all $t_{k}, t_{l}$ reals and all complex numbers $z_{k}, z_{l}$, we have

for every integer $n \geq 1$.

For $n = 1,2, \ldots$, let $X_{n}$ be a r.v. distributed as Poisson with parameter $n, X_{n} \sim \operatorname {Pois}\left(n\right)$ (so that $\mathbb {E}\left[X_{n}\right] = \operatorname {Var}\left[X_{n}\right] = n$), and set $Y_{n} = \left(X_{n}-n\right) / \sqrt{n}$. Then show that $Y_{n} \xrightarrow [n \rightarrow \infty ]{d} Z \sim N(0,1)$.

Let $F_{1}$ and $F_{2}$ be d.f.s, and let $G$ be their convolution, $G = F_{1} * F_{2}$. Then:

If the r.v.s $X$ and $Y$ are i.i.d. with ch.f. $f$, then the ch.f. of the r.v. $X-Y$ is $\left|f(t)\right|^{2}$.

Let $X_{1}, \ldots , X_{n}$ be independent r.v.s, each of which is symmetric about 0. Then the r.v. $X_{1}+\cdots +X_{n}$ is also symmetric about 0.

For $n \geq 1$, let $g_{n}$ and $g$ be functions defined on $E \subseteq \mathbb {R}^{k}$ into $\mathbb {R}$, and recall that $\left\{ g_{n}\right\}$ is said to converge continuously to $g$ on $E$, if for every $x \in E, g_{n}\left(x_{n}\right) \rightarrow g(x)$ whenever $x_{n} \rightarrow x$, as $n \rightarrow \infty$. Then show that if $\left\{ g_{n}\right\}$ converges continuously to $g$ on $E$, it follows that $g$ is continuous on $E$.

For $n \geq 1$, let $g_{n}, g$ and $E$ be as in Exercise 24. Then:

For $n \geq 1$, let $X_{n}, Y_{n}$ and $X, Y$ be r.v.s defined on the probability space $(\Omega , \mathcal{A}, P)$, let $d_{1}, d_{2}$ and $c_{n}$ be constants with $0 \neq c_{n} \rightarrow \infty$ as $n \rightarrow \infty$, and suppose that $c_{n}\left(X_{n}-d_{1}, Y_{n}-d_{2}\right) \xrightarrow {d}(X, Y)$ as $n \rightarrow \infty$. Let $g: \mathbb {R}^{2} \rightarrow \mathbb {R}$ be differentiable, and suppose its (first-order) partial derivatives $\dot{g}_{x}, \dot{g}_{y}$ are continuous at ($d_{1}, d_{2}$). Then show that, as $n \rightarrow \infty$,