Math Problems Vault

Problem 4.1

Let $\xi$ be a nonnegative integer-valued random variable with generating function $\varphi (z)$ . Find the generating functions of the random variables $\xi +n$ and $n \xi$ ( $n$ is a nonnegative integer) . 6 A. V. Prokhorov et al.

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Problem 4.2

Find the distributions corresponding to the following generating functions:

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(a)

$\frac{1}{4}(1+z)^{2}$ ;

(b)

$p(1-q z)^{-1}, p, q > 0, p+q = 1$ ;

(c)

$e^{\lambda (z-1)}, \lambda > 0$ ;

(d)

$(p+q z)^{n}$ ;

(e)

$\frac{2 e}{e^{2}-1} \operatorname {ch} z$ ;

(f)

$\left(1+\frac{1-z}{z} \ln (1-z)\right)$ .

Problem 4.3

Find the distribution corresponding to the generating function $\varphi (z)$ , if $\varphi \left(\frac{1}{2^{n}}\right) = \frac{1}{2^{n}}, n = 1,2, \ldots$

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Problem 4.4

Prove that the function $\varphi (z) = \left|z\right|$ cannot be the generating function of a probability distribution.

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Problem 4.5

Let $\xi$ be a random variable taking nonnegative integer values, and let $\varphi (z)$ be its generating function. Prove that

\mathbb {P}\left(\xi = 0\right) = \lim _{n \rightarrow \infty } \frac{1}{n} \sum _{k = 1}^{n} \varphi \left(a_{k}\right)

where $\left|a_{k}\right| \leqslant 1, \quad a_{k} \rightarrow 0$ as $k \rightarrow \infty$ .

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Problem 4.6

Let $\xi$ be a random variable taking nonnegative integer values, and let $\varphi (z)$ be its generating function. Prove that $\mathbb {P}\left(\xi = 0\right) = 0$ if and only if the series

\sum _{n = 1}^{\infty } \varphi \left(\frac{1}{2^{n}}\right) .

converges.

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Problem 4.7

For which values of the parameters is the fractional-linear function $\varphi (z) = \frac{a+b z}{c+d z}$ the generating function of a probability distribution?

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Problem 4.8

Let $\xi$ and $\eta$ be random variables, where $\xi$ takes the values 0 and 1 with probabilities $1 / 2$ each, and $\eta$ takes the values 0, 1, 2, 3 with probabilities $1 / 8, 1 / 4, 1 / 2$ and $1 / 8$ respectively. Prove that there is no random variable $\zeta$ , independent of $\xi$ , such that $\xi +\zeta = \eta$ .

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Problem 4.9

Let $\xi$ and $\eta$ be random variables, where $\xi$ takes the values $0,1,2$ with probabilities $1 / 2,1 / 4,1 / 4$ , and $\eta$ takes the values 0, 1, 2, 3, 4 with probabilities $6 / 10,1 / 10,1 / 10,1 / 10,1 / 10$ respectively. Prove that there is no random variable $\zeta$ , independent of $\xi$ , such that $\xi +\zeta = \eta$ .

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Problem 4.10

Let $\xi$ and $\eta$ be independent random variables such that $\xi +\eta$ takes the values $0,1,2$ with probabilities $1 / 3$ each. Prove that one of the variables $\xi$ and $\eta$ has a degenerate distribution.

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Problem 4.11

Let $F(x), F_{1}(x), F_{2}(x), \ldots$ be a sequence of distribution functions of nonnegative integer-valued random variables, and let $\varphi (z), \varphi_{1}(z), \varphi_{2}(z), \ldots$ be the corresponding generating functions. Prove that if $\varphi_{n}(z) \rightarrow \varphi (z)$ as $n \rightarrow \infty$ , then $F_{n}(x) \rightarrow \rightarrow F(x)$ uniformly in $x$ .

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Problem 4.12

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, let $F(x)$ be the distribution function of $\xi_{1}$ , and let $v$ be a positive integer-valued random variable, independent of $\xi_{1}, \xi_{2}, \ldots$ and having generating function $\varphi (z)$ . Prove that the distribution function of the random variable $\max \left\{ \xi_{1}, \ldots , \xi_{v}\right\}$ equals $\varphi (F(x))$ .

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Problem 4.13

Let $f(t)$ be the characteristic function of a random variable $\xi$ , and let $a$ and $b$ be real numbers. Find the characteristic function of the random variable $a \xi +b$ .

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Problem 4.14

Let $f(t)$ be the characteristic function of a random variable $\xi$ . Find the characteristic function of the random variable $-\xi$ .

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Problem 4.15

Prove that a characteristic function is even if and only if the corresponding distribution function $F(x)$ satisfies the relation

F(x) = 1-F(-x-0) .

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Problem 4.16

Prove that a characteristic function is real if and only if it is even.

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Problem 4.17

Prove that an even characteristic function $\varphi (t)$ can be represented in the form

\varphi (t) = \int _{-\infty }^{\infty } \cos t x d F(x)

where $F(x)$ is the corresponding distribution function.

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Problem 4.18

Let $\xi$ and $\eta$ be independent, identically distributed random variables with characteristic function $f(t)$ . Find the characteristic function of the random variable $\xi -\eta$ .

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Problem 4.19

(a)(b)

Prove that the following functions cannot be characteristic functions:

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(a)

$e^{-i\left|t\right|}$ ;

(b)

$a_{1} \cos t+\ldots +a_{n} \cos n t+b_{1} \sin t+\ldots +b_{n} \sin n t, b_{1} \cdot \ldots \cdot b_{n} \neq 0$ , where all the $a_{i}$ and $b_{i}$ are real numbers.

Problem 4.20

Prove that the characteristic function of any distribution is uniformly continuous on the whole real line.

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Problem 4.21

Is the function $\cos t^{2}$ a characteristic function?

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Problem 4.22

(a)(b)(c)

On the probability space $(\Omega , \mathscr {A}, P)$ , consisting of the interval $[0,1]$ with the $\sigma$ -algebra of Borel subsets and Lebesgue measure, a random variable $\xi (\omega )$ is defined. Find its characteristic function if:

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(a)

$\xi (\omega ) = \begin{cases} 2 \omega , & \quad 0 \leqslant \omega \leqslant 1 / 2 \\ 2 \omega -1, & \quad 1 / 2 < \omega \leqslant 1 ; \end{cases}$

(b)

$\xi (\omega ) = \ln \omega , \quad \xi (0) = 0$ ;

(c)

$\xi (\omega ) = \begin{cases} 1, & 0 \leqslant \omega \leqslant 1 / 3 \\ 0, & 1 / 3 < \omega < 2 / 3 \\ 1, & 2 / 3 \leqslant \omega \leqslant 1. \end{cases}$

Problem 4.23

Find the characteristic function corresponding to the density $\frac{1}{\pi \operatorname {ch} x},-\infty < x < \infty$ .

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Problem 4.24

Find the characteristic functions of the following distributions:

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(a)

the uniform distribution on the interval $[a, b]$ ;

(b)

the binomial distribution;

(c)

the Poisson distribution;

(d)

the Cauchy distribution;

(e)

the exponential distribution;

(f)

the normal distribution;

(g)

the geometric distribution;

(h)

the negative binomial distribution.

Problem 4.25

The characteristic function of a sum of two random variables equals the product of the characteristic functions of the summands. Can we conclude that the summands are independent?

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Problem 4.26

Let $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ be a random vector in $\mathbb {R}^{n}$ , let $f\left(t_{1}, \ldots , t_{n}\right)$ be its characteristic function, and let $f_{1}(t), \ldots , f_{n}(t)$ be the characteristic functions of the random variables $\xi_{1}, \ldots$ , $\xi_{n}$ . Prove that for $\xi_{i}, \ldots$ , $\xi_{n}$ to be independent, it is necessary and sufficient that for any set of real $t_{1}, \ldots , t_{n}$ the equality

f\left(t_{1}, \ldots , t_{n}\right) = \prod _{i = 1}^{n} f_{i}\left(t_{i}\right)

should hold.

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Problem 4.27

Let $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ be a random vector in $\mathbb {R}^{n}$ , let $f\left(t_{1}, \ldots , t_{n}\right)$ be its characteristic function, and let $f_{1}(t), \ldots , f_{n}(t)$ be the characteristic functions of the random variables $\xi_{1}, \ldots , \xi_{n}$ . Show that the equality

f(t, \ldots , t) = \prod _{i = 1}^{n} f_{i}(t)

for any real $t$ does not imply the independence of the random variables $\xi_{1}, \ldots$ , $\xi_{n}$ .

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Problem 4.28

Let $f_{1}(t), f_{2}(t), \ldots$ be characteristic functions, and let $a_{1}, a_{2}, \ldots$ be nonnegative numbers such that $a_{1}+a_{2}+\ldots = 1$ . Prove that the function

g(t) = \sum _{i = 1}^{\infty } a_{i} f_{i}(t)

is a characteristic function.

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Problem 4.29

Let the function $f(t, a), t, a \in R^{1}$ , satisfy the following conditions:

for each fixed $a$ , $f(t, a)$ is a characteristic function,
for each fixed $t$ , $f(t, a)$ is measurable. Prove that for any distribution function $F(x)$ ,

g(t) = \int _{-\infty }^{\infty } f(t, a) d F(a)

is a characteristic function.

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Problem 4.30

Let $f(t)$ be an arbitrary characteristic function. Prove that the function $2 /(2-f(t))-1$ is also a characteristic function.

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Problem 4.31

Let $F(x)$ be a distribution function with characteristic function $f(t)$ . Prove that $\operatorname {Re} f(t)$ is a characteristic function, and find the corresponding distribution function.

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Problem 4.32

Let $F(x)$ be a distribution function with characteristic function $f(t)$ . Prove that $\left|f(t)\right|^{2}$ is a characteristic function, and find the corresponding distribution function.

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Problem 4.33

Let $\xi$ and $\eta$ be independent random variables, each of which has an asymmetric distribution. Can the random variable $\xi +\eta$ have a symmetric distribution?

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Problem 4.34

Let $\xi$ be a random variable with characteristic function $f(t)$ , and suppose that for some $a$ , $\mathbb {P}\left(\xi = a\right) > 1 / 2$ . Prove that $f(t)$ vanishes nowhere on the real line.

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Problem 4.35

Let $\xi$ be a random variable with real characteristic function $f(t)$ . Prove that if $\mathbb {P}\left(\xi = a\right) > 1 / 4$ for some $a \neq 0$ , then $f(t)$ vanishes infinitely many times.

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Problem 4.36

Give an example of a characteristic function that vanishes, but only finitely many times.

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Problem 4.37

Find the distributions corresponding to the following characteristic functions:

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(a)

$\cos t$ ;

(b)

$\cos^{2} t$ ;

(c)

$e^{-t^{2}}$ ;

(d)

$e^{-\left|t\right|}$ ;

(e)

$\frac{1}{1+t^{2}}$ ;

(f)

$\frac{1}{1-i t}$ ;

(g)

$\frac{\sin t}{t}$ ;

(h)

$e^{-\left|t\right|} \cos t$ .

Problem 4.38

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, let $v$ be a random variable, independent of $\xi_{1}, \xi_{2}, \ldots$ , taking positive integer values, and let $p_{k} = \mathbb {P}\left(v = k\right)$ . Let $f(t)$ be the characteristic function of $\xi_{1}$ . Find the characteristic function of the random variable $\xi_{1}+\ldots +\xi_{\nu }$ .

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Problem 4.39

Prove that the function

f(t) = \begin{cases} 1-\left|t\right|, & \left|t\right| \leqslant 1 \\ 0, & \left|t\right| > 1 \end{cases}

is a characteristic function.

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Problem 4.40

Prove that any even continuous function that is convex for $t \geqslant 0$ and such that $0 \leqslant f(t) \leqslant 1$ and $f(0) = 1$ is a characteristic function.

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Problem 4.41

Do there exist two distinct characteristic functions that coincide on some interval containing the origin?

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Problem 4.42

Prove that for any $0 < \alpha \leqslant 1$ the function

f(t) = \frac{1}{1+\left|t\right|^{\alpha }}

is a characteristic function.

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Problem 4.43

Let $f(t)$ be the characteristic function of a continuous distribution. Prove that $\left|f(t)\right| < 1$ for $t \neq 0$ .

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Problem 4.44

Let $\xi$ be a random variable with a symmetric continuous distribution. Prove that the characteristic function of the random variable $\max \left\{ 0, \xi \right\}$ vanishes nowhere. Can the continuity condition be dropped?

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Problem 4.45

Let $\xi_{1}, \xi_{2}, \ldots$ be independent identically distributed integer-valued random variables, and let $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that if $0 < \mathbb {P}\left(\xi_{1} \text{ is divisible by two}\right) < 1$ , then

\lim _{n \rightarrow \infty } \mathbb {P}\left(\eta _{n} \text{ is divisible by two}\right) = 1 / 2.

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Problem 4.46

Let $f(t)$ be an arbitrary characteristic function. Prove that for any real $t$ the following inequalities hold:

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(a)

$1-\operatorname {Re} f(2 t) \leqslant 4(1-\operatorname {Re} f(t))$ ;

(b)

$1-\left|f(2 t)\right|^{2} \leqslant 4\left(1-\left|f(t)\right|^{2}\right)$ ;

(c)

$1-\operatorname {Re} f(2 t) \leqslant 2\left(1-(\operatorname {Re} f(t))^{2}\right)$ ;

(d)

$\quad 1-\left|f(2 t)\right| \leqslant 2\left(1-\left|f(t)\right|^{2}\right)$ ;

(e)

$1-\left|f(2 t)\right| \leqslant 4(1-\left|f(t)\right|)$ .

Problem 4.47

(a)(b)

Prove that for any characteristic function $f(t)$ and any nonnegative integer $n$ the following inequalities hold:

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(a)

$1-\operatorname {Re} f(t) \geqslant \frac{1}{4^{n}}\left(1-\operatorname {Re} f\left(2^{n} t\right)\right)$ ;

(b)

$1-\left|f(t)\right|^{2} \geqslant \frac{1}{4^{n}}\left(1-\left|f\left(2^{n} t\right)\right|^{2}\right)$ .

Problem 4.48

(a)(b)

Prove that for any characteristic function $f(t)$ and any nonnegative integer $n$ the following inequalities hold:

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(a)

$1-\operatorname {Re} f(n t) \leqslant n\left(1-(\operatorname {Re} f(t))^{n}\right) \leqslant n^{2}(1-\operatorname {Re} f(t))$ ;

(b)

$1-\left|f(n t)\right|^{2} \leqslant n\left(1-\left|f(t)\right|^{2 n}\right) \leqslant n^{2}\left(1-\left|f(t)\right|^{2}\right)$ .

Problem 4.49

Let $f(t)$ be a characteristic function, and let $c$ and $b$ be positive constants. Prove that if $\left|f(t)\right| \leqslant c$ for $\left|t\right| \geqslant b$ , then

\left|f(t)\right| \leqslant 1-\frac{1-c^{2}}{8 b^{2}} t^{2}

for $\left|t\right| \leqslant b$ .

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Problem 4.50

A characteristic function $f(t)$ is called self-decomposable if for any $0 < c < 1$ there exists a characteristic function $f_{c}(t)$ such that $f(t) = f(c t) f_{c}(t)$ . Prove that a self-decomposable characteristic function vanishes nowhere.

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Problem 4.51

Let $F(x)$ be a distribution function with characteristic function $f(t)$ . Prove that for any $t > 0$

1-\operatorname {Re} f(t) \geqslant \frac{2 t^{2}}{\pi ^{2}} \int _{-\pi / t}^{\pi / t} x^{2} d F(x)

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Problem 4.52

Let $\xi$ and $\eta$ be independent random variables with distribution functions $F(x)$ and $G(x)$ and characteristic functions $f(t)$ and $g(t)$ respectively. Find the characteristic function $\varphi (t)$ of the random variable $\xi \cdot \eta$ .

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Problem 4.53

Let $F(x)$ be a distribution function with characteristic function $\varphi (t)$ . Find the $n$ -dimensional distribution function $G\left(x_{1}, \ldots , x_{n}\right)$ corresponding to the characteristic function $\psi \left(t_{1}, \ldots , t_{n}\right) = \varphi \left(t_{1}+\ldots +t_{n}\right)$ .

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Problem 4.54

Let $\xi$ be a random variable having a Cauchy distribution. Can there exist two independent random variables $\xi_{1}$ and $\xi_{2}$ such that $\xi = \xi_{1}+\xi_{2}$ and one of them has a distribution uniform on some interval?

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Problem 4.55

Let $\xi , \eta$ and $\zeta$ be random variables, where $\zeta$ is independent of $\xi$ and $\eta$ . The distributions of the random variables $\xi +\zeta$ and $\eta +\zeta$ coincide. Can we conclude that the distributions of the random variables $\xi$ and $\eta$ coincide?

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Problem 4.56

Let $\xi , \eta$ and $\zeta$ be random variables, where $\zeta$ is independent of $\xi$ and $\eta$ . Prove that if the distributions of the random variables $\xi +c \zeta$ and $\eta +c \zeta$ coincide for every $c > 0$ , then the distributions of $\xi$ and $\eta$ coincide.

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Problem 4.57

Let $\xi$ and $\eta$ be independent identically distributed symmetric random variables, such that the variables $\xi -\eta$ and $\left|\xi \right|-\left|\eta \right|$ are identically distributed. Find the distribution of $\xi$ .

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Problem 4.58

Let $F(x)$ and $G(x)$ be arbitrary distribution functions, and let $f(t)$ and $g(t)$ be the corresponding characteristic functions. Prove that

\int _{-\infty }^{\infty } f(t) d G(t) = \int _{-\infty }^{\infty } g(u) d F(u) .

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Problem 4.59

Let $\varphi (t)$ be a characteristic function. Prove that the function

f(t) = \frac{1}{t} \int _{0}^{t} \varphi (u) d u

is also a characteristic function.

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Problem 4.60

Let $\varphi (t)$ be a characteristic function. Prove that for any $p > 0$ the function

f(t) = \frac{p}{t^{p}} \int _{0}^{t} \varphi (u) u^{p-1} d u

is a characteristic function.

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Problem 4.61

Let $g(t)$ be a characteristic function and let $p$ be an arbitrary positive number. Prove that the function

f(t) = e^{p(g(t)-1)}

is a characteristic function.

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Problem 4.62

Suppose the sequence of distribution functions $F_{1}(x)$ , $F_{2}(x)$ ,... converges uniformly on the whole line to a distribution function $F(x)$ . Can we conclude that the corresponding sequence of characteristic functions $f_{1}(t), f_{2}(t), \ldots$ converges uniformly on the whole line to the characteristic function of the distribution $F$ ?

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Problem 4.63

Let $F_{1}(x), F_{2}(x), \ldots$ be a sequence of distribution functions, and let $f_{1}(t), f_{2}(t), \ldots$ be the corresponding sequence of characteristic functions. Prove that if $\lim_{n \rightarrow \infty } f_{n}(t) = 1$ for all $-\varepsilon \leqslant t \leqslant \varepsilon$ , where $\varepsilon > 0$ , then $F_{n}(x)$ converges weakly as $n \rightarrow \infty$ to the distribution degenerate at zero.

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Problem 4.64

Let $F_{1}(x), F_{2}(x), \ldots$ and $G_{1}(x), G_{2}(x), \ldots$ be two sequences of distribution functions, and suppose there exists a distribution function $F(x)$ such that both sequences $F_{n}$ and $F_{n} * G_{n}$ converge weakly to $F$ as $n \rightarrow \infty$ . Prove that the sequence $G_{n}$ converges weakly to the distribution degenerate at zero.

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Problem 4.65

Prove that a sequence of distribution functions $F_{1}(x), F_{2}(x), \ldots$ converges weakly to some distribution function $F(x)$ if and only if the corresponding sequence of characteristic functions $f_{1}(t), f_{2}(t), \ldots$ converges to some function $f(t)$ and this convergence is uniform in some neighborhood of zero.

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Problem 4.66

Let $F(x), F_{1}(x), F_{2}(x), \ldots$ be a sequence of distribution functions, and let $f(t), f_{1}(t), f_{2}(t), \ldots$ be the corresponding sequence of characteristic functions. Prove that if for any $a \leqslant b$

\lim _{n \rightarrow \infty } \int _{a}^{b} f_{n}(t) d t = \int _{a}^{b} f(t) d t,

then $F_{n}$ converges weakly to $F$ .

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Problem 4.67

Let $f(t)$ be a characteristic function, and suppose that for an infinitely increasing sequence $h_{1}, h_{2}, \ldots$ the functions $f(t) g\left(h_{n} t\right)$ are also characteristic functions. Prove that in this case $g(t)$ is a characteristic function.

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Problem 4.68

Let $F(x)$ and $G(x)$ be distribution functions, and let $f(t)$ and $g(t)$ be the corresponding characteristic functions. Prove that

\sup _{t}\left|f(t)-g(t)\right| \leqslant 2 \operatorname {Var}(F, G) .

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Problem 4.69

Let $f_{1}(t), f_{2}(t), \ldots$ be a sequence of characteristic functions. Prove that the following two conditions are equivalent:

the products
$\prod _{i = 1}^{n} f_{i}(t)$
converge as $n \rightarrow \infty$ uniformly on every compact subset of the line,
the products
$\prod _{i = 1}^{n} f_{i}(t)$
converge as $n \rightarrow \infty$ to a nonzero limit on some set of positive Lebesgue measure.

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Problem 4.70

Prove that a characteristic function $f(t)$ corresponds to a lattice distribution if and only if there exists a real number $t_{0} \neq 0$ such that $\left|f\left(t_{0}\right)\right| = 1$ .

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Problem 4.71

Let $f(t)$ be a characteristic function. Prove that if there exist $t_{1}$ and $t_{2}$ such that $t_{1} / t_{2}$ is irrational and $\left|f\left(t_{1}\right)\right| = \left|f\left(t_{2}\right)\right| = 1$ , then $\left|f(t)\right| \equiv 1$ .

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Problem 4.72

Let the characteristic function $f(t)$ equal unity at some point $t_{0} > 0$ . Prove that $t_{0}$ is a period of the function $f(t)$ .

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Problem 4.73

Give an example of a lattice distribution whose characteristic function is not periodic.

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Problem 4.74

Let a lattice distribution be concentrated on some subset of the set of points $a k, k = 0, \pm 1, \pm 2, \ldots$ Prove that the characteristic function of such a distribution is periodic.

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Problem 4.75

Prove that the characteristic function of any lattice distribution can be represented in the form $e^{i t a} f(t)$ , where $f(t)$ is a periodic function and $a$ is a real number.

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Problem 4.76

Can the real part of a characteristic function be a periodic function while the imaginary part is not?

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Problem 4.77

Let $u(t)$ be the real part of some characteristic function, and let $v(t)$ be the imaginary part of some other characteristic function. Is the function $u(t) + i v(t)$ a characteristic function in general?

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Problem 4.78

Let $f(t)$ be the characteristic function of a purely discrete distribution. Prove that

\limsup _{\left|t\right| \rightarrow \infty }\left|f(t)\right| = 1

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Problem 4.79

Prove that the density corresponding to an absolutely integrable characteristic function is continuous.

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Problem 4.80

Prove that a characteristic function differentiable at zero is continuously differentiable on the whole line.

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Problem 4.81

Prove that the function

f(t) = \begin{cases} 1-t^{2}, & \left|t\right| < 1 \\ 0, & \left|t\right| \geqslant 1 \end{cases}

cannot be a characteristic function.

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Problem 4.82

Let $\xi$ be a random variable with characteristic function $f(t)$ . Prove that if for some positive integer $n$ , $\mathbb {E}\left[\left|\xi \right|^{n}\right] < \infty$ , then $f(t)$ is $n$ times differentiable and

\left|f^{(n)}(t)\right| \leqslant \mathbb {E}\left[\left|\xi \right|^{n}\right]

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Problem 4.83

Prove that the characteristic function of the random variable $\xi$ with distribution

\mathbb {P}\left(\xi = 2 k\right) = \mathbb {P}\left(\xi = -2 k\right) = \frac{c}{k^{2} \ln k}, \quad k = 2,3, \ldots

is differentiable at zero, but $\mathbb {E}\left[\xi \right]$ does not exist.

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Problem 4.84

Let $\xi$ be a random variable with characteristic function $f(t)$ and finite expectation. Prove that

\mathbb {E}\left[\left|\xi \right|\right] = \frac{1}{\pi } \int _{-\infty }^{\infty } \frac{1-\operatorname {Re} f(t)}{t^{2}} d t

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Problem 4.85

For which real $\alpha$ is the function

f(t) = \begin{cases} 1-\left|t\right|^{\alpha }, & \left|t\right| \leqslant 1 \\ 0, & \left|t\right| > 1 \end{cases}

a characteristic function, and for which is it not?

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Problem 4.86

Is the function $e^{-t^{4}}$ a characteristic function?

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Problem 4.87

Let $f(t)$ be the characteristic function of a random variable having finite variance. Prove that for sufficiently small $\varepsilon > 0$ , the function $\left|f(t)\right|$ is nonincreasing for $0 < t < \varepsilon$ and nondecreasing for $-\varepsilon < t < 0$ .

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Problem 4.88

(a)(b)

Let $f(t)$ be the characteristic function of a symmetric distribution, and let $a_{k}$ be the corresponding moment of order $k$ . Prove that:

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(a)

$\lim_{t \rightarrow 0} \frac{f^{(2 q-1)}(t)}{t} = (-1)^{q} a_{2 q}$ ;

(b)

$\lim_{t \rightarrow 0} \frac{f^{(2 q)}(t)-f^{(2 q)}(0)}{t^{2}} = \frac{(-1)^{q+1}}{2} a_{2 q+2}$ .

Problem 4.89

Let $f_{1}(t)$ and $f_{2}(t)$ be characteristic functions of symmetric distributions, and suppose that for some positive $\alpha_{1}$ and $\alpha_{2}$ , in some neighborhood of zero, the equality

f_{1}^{\alpha _{1}}(t) f_{2}^{\alpha _{2}}(t) = e^{-t^{2}}

holds.

Prove that the distributions corresponding to the characteristic functions $f_{1}(t)$ and $f_{2}(t)$ have finite variances.

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Problem 4.90

Suppose the conditions of the previous problem hold. Prove that the distributions corresponding to the characteristic functions $f_{1}(t)$ and $f_{2}(t)$ have moments of all orders.

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Problem 4.91

Express the first four semiinvariants of a random variable in terms of the mean and the central moments.

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Problem 4.92

Prove that under the linear transformation $\xi^{\prime } = a \xi +b$ of a random variable $\xi$ , its semiinvariants change according to the law $x_{1}^{\prime } = a x_{1}+b, x_{n}^{\prime } = a^{n} x_{n} \quad (n > 1)$ .

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Problem 4.93

Prove that for any random variable $\xi$ and any $u > 0$ ,

\mathbb {P}\left(\left|\xi \right| > 2 / u\right) \leqslant \frac{1}{u} \int _{-u}^{u}(1-\varphi (t)) d t

where $\varphi (t)$ is the characteristic function of $\xi$ .

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Problem 4.94

Let $\xi$ be a random variable with characteristic function $f(t)$ . Prove that if $1-f(t) = o\left(\left|t\right|^{\alpha }\right)$ as $t \rightarrow 0$ , then $\mathbb {P}\left(\left|\xi \right| > x\right) = o\left(x^{-\alpha }\right)$ as $\quad x \rightarrow \infty$ .

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Problem 4.95

Let $\xi$ be a random variable with real characteristic function $f(t)$ and variance $\sigma^{2}$ . Prove that

f(t) \geqslant 1-\frac{t^{2} \sigma ^{2}}{2}

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Problem 4.96

Let $\xi$ be a random variable with characteristic function $f(t)$ and variance $\sigma^{2}$ . Prove that if $t_{0}$ is the smallest positive root of $f(t)$ , then $t_{0} \geqslant 1 / \sigma$ .

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Problem 4.97

Let $f(t)$ be the characteristic function of a distribution having finite variance. Prove that there exist positive constants $c$ and $\varepsilon$ such that

\left|f(t)\right| \geqslant e^{-c t^{2}}

for $\left|t\right| \leqslant \varepsilon$ .

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Problem 4.98

Let $f(t)$ be the characteristic function of a random variable having finite expectation. Prove that there exist positive constants $c$ and $\varepsilon$ such that

\left|f(t)\right| \geqslant e^{-c\left|t\right|}

for $\quad \left|t\right| \leqslant \varepsilon$ .

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Problem 4.99

Prove that for the characteristic function $f(t)$ of any nondegenerate random variable $\xi$ , there exist positive constants $\delta$ and $\varepsilon$ such that

\left|f(t)\right| \leqslant 1-\varepsilon t^{2}

for $\left|t\right| \leqslant \delta$ .

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Problem 4.100

Let $f(t)$ be a characteristic function. Prove that if $f(t) = 1+w(t)+o\left(t^{2}\right)$ as $t \rightarrow 0$ , where $w(t) = -w(-t)$ , then $f(t) = 1$ .

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Problem 4.101

Let $\xi$ be a bounded random variable, $\left|\xi \right| \leqslant c$ , having a symmetric distribution with characteristic function $f(t)$ and variance $\sigma^{2}$ . Prove that

0 \leqslant f(t) \leqslant e^{-\frac{4}{\pi ^{2}} \sigma ^{2} t^{2}}

for $\left|t\right| \leqslant \frac{\pi }{2 c}$ .

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Problem 4.102

Let $\xi$ be a bounded random variable, $\left|\xi \right| \leqslant c$ , with characteristic function $f(t)$ and variance $\sigma^{2}$ . Prove that

\left|f(t)\right| \leqslant e^{-\frac{4}{\pi ^{2}} \sigma ^{2} t^{2}}

for $\left|t\right| \leqslant \frac{\pi }{4 c}$ .

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Problem 4.103

Let $\xi$ be a random variable with characteristic function $f(t)$ and variance $\sigma^{2}$ . Prove that for any $c < 2 \sigma^{2}$ there exists $\varepsilon > 0$ such that

\left|f(t)\right| \leqslant e^{-c t^{2}}

for $\quad \left|t\right| \leqslant \varepsilon$ .

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Problem 4.104

Let $\xi$ be a random variable with characteristic function $f(t)$ . Prove that if $E\left|\xi \right|^{2} = \infty$ , then for any $c > 0$ there exists $\varepsilon > 0$ such that

\left|f(t)\right| \leqslant e^{-c t^{2}}

for $\left|t\right| \leqslant \varepsilon$ .

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Problem 4.105

Let $p(x)$ be a symmetric unimodal density, and let $f(t)$ be the corresponding characteristic function. Prove that if

p(0) \leqslant A < \infty ,

then

\left|f(t)\right| \leqslant 2 A /\left|t\right|

for any real $t$ .

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Problem 4.106

Suppose the conditions of the previous problem hold. Prove that

\left|f(t)\right| \leqslant \frac{\sin (t / 2 A)}{t / 2 A}

for $\left|t\right| \leqslant \pi A$ .

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Problem 4.107

Let $F(x)$ be a distribution function, and let $f(t)$ be the corresponding characteristic function. Prove that for any $r > 0$ ,

t^{2} \int _{-r}^{r} x^{2} d F(x) \leqslant 3\left|1-f(t)\right|

for $\left|t\right| \leqslant 1 / r$ .

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Problem 4.108

(a)(b)(c)

Let $\xi_{1}$ and $\xi_{2}$ be random variables with characteristic functions $f_{1}(t)$ and $f_{2}(t)$ and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ respectively, and suppose there exists $\varepsilon > 0$ such that $\left|f_{1}(t)\right| > \left|f_{2}(t)\right|$ for $0 < \left|t\right| \leqslant \varepsilon$ . Can we have:

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(a)

$\sigma_{1} > \sigma_{2}$ ,

(b)

$\sigma_{1} < \sigma_{2}$ ,

(c)

$\sigma_{1} = \sigma_{2}$ ?

Problem 4.109

Prove that a distribution with differentiable characteristic function $f(t)$ has finite variance if and only if there exists $\varepsilon > 0$ such that the function

\frac{f^{\prime }(t)-f^{\prime }(0)}{t}

is bounded for $0 < \left|t\right| \leqslant \varepsilon$ .

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Problem 4.110

Let $F(x)$ be a distribution function with characteristic function $f(t)$ . Prove that if for some positive $\lambda < 2$ and some sequence $t_{1}, t_{2}, \ldots$ such that $t_{n} \rightarrow 0$ as $n \rightarrow \infty$ , the inequalities

\left|f\left(t_{n}\right)\right| \leqslant e^{-c\left|t_{n}\right|^{\lambda }}

hold ( $c$ a positive constant), then

\int _{-\infty }^{\infty }\left|x\right|^{\delta } d F(x) = \infty

for all $\delta > \lambda$ .

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Problem 4.111

Let $F(x)$ be a distribution function with characteristic function $f(t)$ . Prove that $F(x)$ has finite variance if and only if there exist $c > 0$ and a sequence $t_{1}, t_{2}, \ldots$ such that $t_{n} \rightarrow 0$ as $n \rightarrow \infty$ and

\left|f\left(t_{n}\right)\right| \geqslant e^{-c t_{n}^{2}}, \quad n = 1,2, \ldots

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Problem 4.112

Let $F(x)$ be a distribution function with characteristic function $f(t)$ . Prove that if there exists a sequence $t_{1}, t_{2}, \ldots$ such that $t_{n} \rightarrow 0$ as $n \rightarrow \infty$ and

\frac{\ln \left|f\left(t_{n}\right)\right|}{t_{n}^{2}} \rightarrow 0

as $n \rightarrow \infty$ , then $F(x)$ is a degenerate distribution function.

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Problem 4.113

Let $\xi$ be a nonnegative random variable with density $p(x)$ and characteristic function $f(t)$ . Prove that if $p(x)$ is monotonically decreasing for $x > 0$ , then $f(t)$ vanishes nowhere.

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Problem 4.114

Let $\xi$ be a nonnegative random variable with density $p(x)$ and characteristic function $f(t)$ . Prove that if $p(x)$ is monotonically nonincreasing for $x \geqslant 0$ , then for $t \geqslant 0$ ,

0 \leqslant \operatorname {Im} f(t) \leqslant \frac{2 p(0)}{t}

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Problem 4.115

Let $F(x)$ be a distribution function, and let $f(t)$ be the corresponding characteristic function. Prove that if $f(t)$ is absolutely integrable, then $F(x)$ is absolutely continuous and the corresponding density is expressed by the formula

p(x) = \frac{1}{2 \pi } \int _{-\infty }^{\infty } e^{-i t x} f(t) d t

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Problem 4.116

Prove that the density $p(x)$ corresponding to a real integrable characteristic function $f(t)$ is expressed by the formula

p(x) = \frac{1}{2 \pi } \int _{-\infty }^{\infty } \cos t x f(t) d t

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Problem 4.117

Let $p(x)$ be a density with characteristic function $f(t)$ . Prove that if $f(t)$ is symmetric, positive, and integrable, then $p(x)$ has a unique maximum. At what point is it attained?

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Problem 4.118

Suppose the conditions of the previous problem hold, and moreover that the second derivative $p^{\prime \prime }(x)$ exists. Prove that for any $x \neq 0$ ,

p(0) > p(x) > p(0)-\frac{x^{2}}{2} p^{\prime \prime }(0) .

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Problem 4.119

Prove that the distribution with characteristic function

e^{-\left|t\right|^{\alpha }} \quad (0 < \alpha \leqslant 2)

has a bounded density.

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Problem 4.120

Let the characteristic function $f(t)$ be such that $f(t) = 0$ for $\left|t\right| \geqslant c$ ( $c > 0$ ). Prove that the corresponding distribution has a bounded density $p(x)$ , and that

\sup _{x} p(x) \leqslant c / \pi .

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Problem 4.121

(Parseval's equality) Let $p(x)$ be a density with characteristic function $f(t)$ . Prove that if the function $p^{2}(x)$ is integrable, then

\int _{-\infty }^{\infty } p^{2}(x) d x = \frac{1}{2 \pi } \int _{-\infty }^{\infty }\left|f(t)\right|^{2} d t

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Problem 4.122

Let $\xi$ be an integer-valued random variable with characteristic function $f(t)$ . Prove that

\mathbb {P}\left(\xi = k\right) = \frac{1}{2 \pi } \int _{-\pi }^{\pi } e^{-i t k} f(t) d t, \quad k = 0, \pm 1, \ldots

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Problem 4.123

Let $F(x)$ be a distribution function with characteristic function $f(t)$ . Prove that for any $x$ ,

\lim _{T \rightarrow \infty } \frac{1}{2 T} \int _{-T}^{T} f(t) e^{-i t x} d t = F(x+0)-F(x-0)

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Problem 4.124

Let $F(x)$ be a distribution function with characteristic function $f(t)$ . Prove that $F(x)$ is continuous if and only if

\lim _{T \rightarrow \infty } \frac{1}{2 T} \int _{-T}^{T}\left|f(t)\right|^{2} d t = 0

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Problem 4.125

Let $\xi$ be a discrete random variable taking the values $x_{1}, x_{2}, \ldots$ with probabilities $p_{1}, p_{2}, \ldots$ , and let $f(t)$ be the corresponding characteristic function. Prove that

\lim _{T \rightarrow \infty } \frac{1}{2 T} \int _{-T}^{T}\left|f(t)\right|^{2} d t = \sum _{i = 1}^{\infty } p_{i}^{2}

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Problem 4.126

Let $f(t)$ be a characteristic function such that

\liminf _{t \rightarrow \infty } \operatorname {Re} f(t) \geqslant a > 0

Prove that the function

g(t) = \frac{f(t)-a}{1-a}

is a characteristic function.

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Problem 4.127

Prove that for any real $a$ such that $0 \leqslant a \leqslant 1$ , there exists a characteristic function $f(t)$ satisfying the relation

\lim _{t \rightarrow \infty } f(t) = a .

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Problem 4.128

Let $a$ be a negative number, $\left|a\right| \leqslant 1$ . Does there exist a characteristic function $f(t)$ such that

\lim _{t \rightarrow \infty } f(t) = a ?

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Problem 4.129

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed integer-valued random variables, and let $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that if $\xi_{1}$ is not constant with probability 1, then

\sup _{k} \mathbb {P}\left(\eta _{n} = k\right) \rightarrow 0

as $n \rightarrow \infty$ .

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Problem 4.130

Let $F(x)$ and $G(x)$ be distribution functions, and let $f(t)$ and $g(t)$ be the corresponding characteristic functions. Prove that

\sup _{x}\left|F(x)-G(x)\right| \leqslant \frac{1}{\pi } \int _{-\infty }^{\infty }\left|\frac{f(t)-g(t)}{t}\right| d t .

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Problem 4.131

Let $f(t)$ and $g(t)$ be absolutely integrable characteristic functions, and let $p(x)$ and $q(x)$ be the corresponding densities. Prove that

\sup _{x}\left|p(x)-q(x)\right| \leqslant \frac{1}{2 \pi } \int _{-\infty }^{\infty }\left|f(t)-g(t)\right| d t .

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Problem 4.132

Let $F(x)$ and $G(x)$ be distribution functions with symmetric densities $p(x)$ and $q(x)$ and nonnegative characteristic functions $f(t)$ and $g(t)$ , such that

\sup _{x} p(x) \leqslant A, \quad \sup _{x} q(x) \leqslant B .

Prove that for any positive $T$ ,

\sup _{x}\left|F(x)-G(x)\right| \leqslant \frac{1}{\pi } \int _{-T}^{T}\left|\frac{f(t)-g(t)}{t}\right| d t+\frac{2(A+B)}{T} .

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Problem 4.133

Let $\xi$ and $\eta$ be integer-valued random variables with characteristic functions $f(t)$ and $g(t)$ respectively. Prove that

\sup _{n}\left|\mathbb {P}\left(\xi = n\right)-\mathbb {P}\left(\eta = n\right)\right| \leqslant \frac{1}{2 \pi } \int _{-\pi }^{\pi }\left|f(t)-g(t)\right| d t .

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Problem 4.134

Let $\xi$ be a nonnegative random variable, and let $\varphi (u)$ be the corresponding Laplace transform. Find the Laplace transform of the random variable $a \xi +b(a, b \geqslant 0)$ .

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Problem 4.135

Find the Laplace transform of the distribution uniform on the interval $[0,1]$ .

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Problem 4.136

Let $\varphi_{1}(u), \varphi_{2}(u), \ldots$ be Laplace transforms of some probability distributions, and let $a_{1}, a_{2}, \ldots$ be nonnegative numbers such that

\sum _{i = 1}^{\infty } a_{i} = 1.

Prove that the function

\varphi (u) = \sum _{k = 1}^{\infty } a_{k} \varphi _{k}(u)

is the Laplace transform of some probability distribution.

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Problem 4.137

Let $\varphi (u)$ be the Laplace transform of some probability distribution. Prove that the function

\frac{2}{2-\varphi (u)}-1

is also a Laplace transform.

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Problem 4.138

Find the Laplace transform of the distribution whose characteristic function equals

\left(\frac{\sin (t / 2)}{t / 2}\right)^{n} e^{\frac{i t n}{2}} .

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Problem 4.139

Prove that the Laplace transform of any nondegenerate distribution is a monotonically decreasing function.

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Problem 4.140

Prove that the Laplace transform of any distribution is a convex function.

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Problem 4.141

A function $\varphi (u)$ , defined on the positive half-line, is called completely monotone if it has derivatives of all orders, with all even derivatives nonnegative and all odd derivatives nonpositive:

(-1)^{n} \varphi ^{(n)}(u) \geqslant 0, \quad u > 0

Prove that the Laplace transform of any distribution is a completely monotone function.

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Problem 4.142

Show that each of the functions listed below cannot be the Laplace transform of a probability distribution:

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(a)

$f(t) = \begin{cases} 1, & 0 \leqslant t \leqslant 1, \\ 0, & t > 1 ; \end{cases}$

(b)

$f(t) = \begin{cases} 1-t, & 0 \leqslant t \leqslant 1 ; \\ 0, & t > 1 ; \end{cases}$

(c)

$f(t) = t e^{-t}$ ;

(d)

$f(t) = e^{-t}+\frac{\sin t}{100}$ ;

(e)

$f(t) = e^{-t^{2}}$ .

Problem 4.143

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed nonnegative random variables, and let $\nu$ be a positive integer-valued random variable, independent of $\xi_{1}, \xi_{2}, \ldots$ Denote by $\varphi (u)$ the Laplace transform of $\xi_{1}$ , and by $p(s)$ the generating function of $\nu$ . Find the Laplace transform of the random variable $\xi_{1}+\ldots +\xi_{\nu }$ .

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Problem 4.144

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables having a Cauchy distribution with density

p(x) = \frac{1}{\pi \left(1+x^{2}\right)}

Find the density of the random variable

\eta _{n} = \frac{\xi _{1}+\ldots +\xi _{n}}{n}

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Problem 4.145

Let $\varphi (t)$ be a characteristic function. Prove that the function

f(t) = \frac{\sin t}{t^{2}} \int _{0}^{t} \varphi (u) d u

is the characteristic function of an absolutely continuous distribution.

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Problem 4.146

Does there exist a probability distribution concentrated on a finite interval such that its characteristic function is nonzero everywhere on the real line?

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Problem 4.147

(a)(b)

Let the functions $f_{1}(x), f_{2}(x), f_{3}(x)$ satisfy the relation

f_{3}(x) = \int _{-\infty }^{\infty } f_{1}(x-u) f_{2}(u) d u

Find $f_{1}(x)$ if:

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(a)

$f_{2}(x) = \frac{1}{\pi \left(1+x^{2}\right)}, \quad f_{3}(x) = \frac{2}{\pi \left(4+x^{2}\right)} ;$

(b)

$f_{2}(x) = e^{-x^{2}}, \quad f_{3}(x) = e^{-x^{2} / 2}$ .

Problem 4.148

Prove that the integral equation for the unknown function $x(t)$

\int _{-\infty }^{\infty } \frac{x(t) d t}{(y-t)^{2}+1} = e^{-y^{2}}

has no solutions in the class of nonnegative functions.

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Problem 4.149

Prove that the integral equation

\int _{-\infty }^{\infty } \frac{2(1-\cos (y-t))}{(y-t)^{2}} x(t) d t = \frac{1}{2 \pi } \int _{-\infty }^{\infty } e^{-i u y} \max \left\{ 0,(1-\left|u\right|),\left(1-\frac{\left|u\right|}{2}\right)\right\} d u

for the unknown function $x(t)$ has infinitely many solutions in the class of nonnegative functions.

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Problem 4.150

Do there exist nondegenerate random variables $\xi_{1}, \xi_{2}, \eta_{1}, \eta_{2}$ such that $\xi_{1}$ is independent of $\xi_{2}$ , $\eta_{1}$ is independent of $\eta_{2}$ , $\mathbb {E}\left[\xi_{i}^{4}\right] < \mathbb {E}\left[\eta_{i}^{4}\right] < \infty , \quad i = 1,2$ , but $\mathbb {E}\left[(\xi_{1}+\xi_{2})^{4}\right] > \mathbb {E}\left[(\eta_{1}+\eta_{2})^{4}\right]$ ?

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Problem 4.151

Let $f(t)$ be a characteristic function that has a derivative of order $2 n(n \geqslant 0)$ . Prove that the function

f^{(2 n)}(t) / f^{(2 n)}(0)

is the characteristic function of a probability distribution.

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Problem 4.152

Let $f(t)$ be a characteristic function that has a derivative of order $2 n-1$ everywhere on the real line. Prove that the distribution corresponding to the characteristic function $f(t)$ has a finite moment of order $2 n$ if and only if there exists $\varepsilon > 0$ such that the function

\frac{f^{(2 n-1)}(t)-f^{(2 n-1)}(0)}{t}

is bounded for $0 < \left|t\right| < \varepsilon$ .

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Problem 4.153

Prove that the function $f(t)$ , equal to $1-\frac{\left|t\right|}{a}$ for $\left|t\right| \leqslant 2 a (a > 0)$ and periodic with period $4 a$ , is a characteristic function.

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Problem 4.154

Prove that the function $f(t)$ , equal to $1-\frac{\left|t\right|}{a}$ for $\left|t\right| \leqslant a (a > 0)$ and periodic with period $2 a$ , is a characteristic function.

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Problem 4.155

Give an example of two distinct characteristic functions $\varphi (t)$ and $\psi (t)$ such that $\varphi^{2}(t) = \psi^{2}(t)$ .

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Problem 4.156

A cart standing on an absolutely rigid, smooth, and level surface is acted upon, constantly in time, by two forces: $F_{1}$ from the left and $F_{2}$ from the right, whose magnitudes are random variables uniformly distributed on the intervals $\left[0, a_{1}\right]$ and $\left[0, a_{2}\right]$ respectively. Can the forces be considered to act independently, if the distance traveled by the cart in time $t$ is a random variable having a uniform distribution on the interval $\left[-\frac{a_{2} t^{2}}{2 m}, \frac{a_{1} t^{2}}{2 m}\right]$ , where $m$ is the mass of the cart? Motion to the right is taken as positive, to the left as negative.

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Problem 4.157

Suppose the conditions of the previous problem hold, but in addition to the forces $F_{1}$ and $F_{2}$ , the cart is also acted upon by another force $F_{0}$ , whose magnitude is a random variable with unknown distribution. Can the forces $F_{0}, F_{1}$ and $F_{2}$ be considered to act independently?

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Problem 4.158

Let $\xi$ be a random variable with a symmetric distribution. Let

\eta = \begin{cases} \xi , & \text{ for }\left|\xi \right| \leqslant c, \\ 0 \text{ for }\left|\xi \right| > c, & \quad c > 0. \end{cases}

Denote by $f(t)$ and $g(t)$ the characteristic functions of $\xi$ and $\eta$ respectively. Prove that there exists $\varepsilon > 0$ such that

f(t) \leqslant g(t)

for $\left|t\right| \leqslant \varepsilon$ .

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Problem 4.159

Let $\xi$ be a random variable with characteristic function $f(t)$ , $\mathbb {E}\left[\xi \right] = 0, \operatorname {Var}\left[\xi \right] = \sigma^{2}, \mathbb {E}\left[\left|\xi \right|^{3}\right] = \beta$ . Prove that there exists an absolute (universal) constant $K$ such that

\left|f(t)\right| \leqslant \exp \left\{ -\frac{t^{2} \sigma ^{2}}{2}\left(1-K \frac{\beta }{\sigma ^{2}}\left|t\right|\right)\right\}

for all real $t$ .

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Problem 4.160

Let $v, \xi_{1}, \xi_{2}, \ldots$ be independent random variables, where $\xi_{1}, \xi_{2}, \ldots$ are identically distributed and $v$ has a Poisson distribution with parameter $\lambda$ . Prove that if $F(x)$ is the distribution function of $\xi_{1}$ , then the characteristic function of the random variable $\xi_{1}+\ldots +\xi_{\nu }$ equals

\exp \left\{ \lambda \int _{-\infty }^{\infty }\left(e^{i t u}-1\right) d F(u)\right\}

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Problem 4.161

Let $F(x), F_{1}(x), F_{2}(x), \ldots$ be a sequence of distribution functions, and let $f(t), f_{1}(t), f_{2}(t), \ldots$ be the corresponding sequence of characteristic functions. Prove that if $F_{n}$ converges to $F$ in variation as $n \rightarrow \infty$ , then $f_{n}$ converges to $f$ uniformly on the whole line.

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Problem 4.162

Let $\xi$ and $\eta$ be random variables with characteristic functions $f(t)$ and $g(t)$ respectively. Prove that

\sup _{t}\left|f(t)-g(t)\right| \leqslant 2 \mathbb {P}\left(\xi \neq \eta \right) .

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Problem 4.163

Let $\xi$ be a random variable with zero mean, variance $\sigma^{2}$ , and characteristic function $f(t)$ . Let

\eta = \begin{cases} \xi , & \text{ for } \left|\xi \right| \leqslant c, \\ 0, & \text{ for } \left|\xi \right| > c, \end{cases} \quad c > 0.

Denote by $g(t)$ the characteristic function of $\eta$ . Prove that

\sup _{t}\left|f(t)-g(t)\right| \leqslant \frac{2 \sigma ^{2}}{c^{2}} .

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Problem 4.164

Let $\xi$ be a random variable with characteristic function $f(t)$ . The degree of dispersion $\delta_{\xi }$ of the random variable $\xi$ is the quantity

\delta _{\xi } = -\ln \frac{1}{\pi } \int _{-\infty }^{\infty } \frac{\left|f(u)\right|^{2}}{1+u^{2}} d u

Prove that $\delta_{\xi } = 0$ if and only if $\xi$ is a constant with probability 1.

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Problem 4.165

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables. Prove that $\delta_{\xi_{n}} \rightarrow 0$ as $n \rightarrow \infty$ (for the definition of $\delta_{\xi_{n}}$ , see the previous problem) if and only if there exists a sequence of real numbers $a_{1}, a_{2}, \ldots$ such that $\quad \xi_{n}-a_{n} \xrightarrow {P} 0$ as $n \rightarrow \infty$ .

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Problem 4.166

Prove that when independent random variables are added, the degree of dispersion does not decrease (for the definition of degree of dispersion, see problem 4.164), i.e., if $\xi$ and $\eta$ are independent random variables, then

\delta _{\xi +\eta } \geqslant \max \left\{ \delta _{\xi }, \delta _{\eta }\right\} .

Show that equality can hold if and only if one of the summands is a constant with probability 1.

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Problem 4.167

Let $\psi (t)$ be a complex function of a real variable such that

A_{\psi } = \int _{-\infty }^{\infty }\left|\psi (t)\right|^{2} d t < \infty

Prove that the function

\varphi (u) = \frac{1}{A_{\psi }} \int _{-\infty }^{\infty } \psi (t+u) \bar{\psi }(t) d t

is the characteristic function of an absolutely continuous distribution.

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Problem 4.168

Let $f(t)$ be a characteristic function such that for some $\varepsilon > 0$ , $f(t) = e^{-t^{2} / 2}$ for all $\left|t\right| \leqslant \varepsilon$ . Prove that $f(t) = e^{-t^{2} / 2}$ .

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Problem 4.169

Prove that for any positive integers $k$ and $n$ , $k \leqslant n$ , the inequality

\mathrm{C}_{n}^{k} \leqslant 2^{n} \sqrt{\frac{2}{\pi n}}

holds.

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Problem 4.170

Let $P$ be the uniform distribution on the unit circle in $\mathbb {R}^{2}$ ( $P$ is evidently singular). Prove that the convolution $P * P$ is absolutely continuous and has a bounded density.

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Problem 4.171

Let the $n$ -dimensional random vector $\xi$ and the $m$ -dimensional random vector $\eta$ be related by the linear dependence $\eta = A \xi +a$ , where $A$ is a rectangular matrix of size $m \times n$ and $a$ is a nonrandom $n$ -dimensional vector. Express the characteristic function of the random vector $\eta$ in terms of the characteristic function of the random vector $\xi$ .

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Problem 4.172

(A. Ya. Khinchin's theorem) Prove that in order for a function $f(t)$ to be the characteristic function of a unimodal distribution, it is necessary and sufficient that it be representable in the form

f(t) = \frac{e^{i t a}}{t} \int _{0}^{t} \varphi (u) d u

where $\varphi (u)$ is a characteristic function and $a$ is a real number.

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Problem 4.173

Prove that the function

\frac{1}{1+\left|t\right|^{\alpha }} \quad (0 < \alpha \leqslant 1)

is the characteristic function of a unimodal distribution.

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Problem 4.174

Let $f(t)$ be the characteristic function of a unimodal distribution. Prove that the function

\varphi (t) = f(t)+t f^{\prime }(t)

is a characteristic function.

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Problem 4.175

Does there exist a characteristic function that is nowhere differentiable?

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Problem 4.176

Give an example of two distinct characteristic functions $f(t)$ and $g(t) \quad (f(t) \neq g(-t))$ such that

\left|f(t)\right|^{2} = \left|g(t)\right|^{2}

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Problem 4.177

Give an example of two distinct characteristic functions $f(t)$ and $g(t) \quad (f(t) \neq g(-t))$ , corresponding to bounded random variables, such that

\left|f(t)\right|^{2} = \left|g(t)\right|^{2}

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Problem 4.178

Let the random vector $\xi = \left(\xi_{1}, \ldots , \xi_{n}\right)$ have a spherically symmetric distribution (see the definition in problem 3.300). Prove that if the random variables $\xi_{1}, \ldots , \xi_{n}$ are independent, then the vector $\xi$ has an $n$ -dimensional normal distribution.

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