Math Problems Vault

Problem 6.1

Bernoulli trials with constant success probability are carried out. Let

\xi _{i} = \begin{cases} 1, & \text{ if the } i \text{-th and }(i+1) \text{-th trials both ended in success } \\ 0, & \text{ otherwise. } \end{cases}

Does the LLN hold for the sequence $\xi_{1}, \xi_{2}, \ldots$ ?

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Problem 6.2

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{n}$ takes the values $\sqrt{n}, 0$ , and $-\sqrt{n}$ with probabilities $1 /(2 n), 1-1 / n, 1 /(2 n)$ respectively. Does the LLN hold for this sequence?

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Problem 6.3

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{n}$ takes the values $-n, 0$ , and $n$ with probabilities $1 /\left(2 n^{2}\right), 1-1 / n^{2}$ , and $1 /\left(2 n^{2}\right)$ respectively. Does the LLN apply to this sequence?

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Problem 6.4

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables,

\mathbb {P}\left(\xi _{n} = \pm 2^{n}\right) = 2^{-(2 n+1)}, \quad \mathbb {P}\left(\xi _{n} = 0\right) = 1-2^{-2 n} .

Does the LLN apply to this sequence?

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Problem 6.5

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{n}$ takes the values $2^{n}$ and $-2^{n}$ with probability 1/2 each. Does the LLN apply to this sequence?

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Problem 6.6

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{n}$ takes the values $-2^{n},-1,1,2^{n}$ with probabilities $2^{-n-1}, \frac{1-2^{-n}}{2}, \frac{1-2^{-n}}{2}, 2^{-n-1}$ respectively. Does the LLN apply to this sequence?

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Problem 6.7

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{n}$ takes the values $-n, 0, n$ with probabilities $1 / 4,1 / 2,1 / 4$ respectively. Does the LLN apply to this sequence?

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Problem 6.8

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{n}$ takes the values $-n, 0, n$ with probabilities $2^{-n}, 1-2^{-n+1}, 2^{-n}$ respectively. Does the LLN apply to this sequence?

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Problem 6.9

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{n}$ takes the values $-\varphi (n), 0, \varphi (n)$ with probabilities $1 / \psi (n), 1-2 / \psi (n), 1 / \psi (n)$ respectively, where $\varphi (n)$ and $\psi (n)$ are such that

\varphi (n) \geqslant 0, \quad \psi (n) \geqslant 2, \quad \frac{\varphi (n)}{\psi (n)} \leqslant C

( $C$ a constant). Does the LLN apply to this sequence?

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Problem 6.10

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. When $n$ is a perfect square, $\xi_{n}$ takes the values $-\sqrt{n}, \sqrt{n}$ with probability $1 / 2$ each; for other $n$ , $\xi_{n}$ takes the values $-2^{-n}, 2^{-n}$ with probability $1 / 2$ each. Does the LLN apply to this sequence?

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Problem 6.11

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{n}$ takes the values $-\sqrt{n}$ and $\sqrt{n}$ with probability $1 / 2$ each. Does the LLN apply to this sequence?

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Problem 6.12

For which values $\alpha > 0$ does the LLN apply to the sequence of independent random variables $\xi_{1}, \xi_{2}, \ldots$ such that

\mathbb {P}\left(\xi _{n} = n^{\alpha }\right) = \mathbb {P}\left(\xi _{n} = -n^{\alpha }\right) = 1 / 2,

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Problem 6.13

(Chebyshev's theorem) Prove that the LLN holds for a sequence of independent random variables having uniformly bounded variances.

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Problem 6.14

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with finite variances $\sigma_{i}^{2}$ . Prove that if

\frac{1}{n^{2}} \sum _{i = 1}^{n} \sigma _{i}^{2} \rightarrow 0 \quad \text{ as } \quad n \rightarrow \infty ,

then the LLN applies to the sequence $\xi_{1}, \xi_{2}, \ldots$

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Problem 6.15

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with variances $\sigma_{i}^{2}$ . Prove that if the covariance of $\xi_{i}$ and $\xi_{j}$ is nonpositive for $i \neq j$ , and

\frac{1}{n^{2}} \sum _{i = 1}^{n} \sigma _{i}^{2} \rightarrow 0 \quad \text{ as } \quad n \rightarrow \infty ,

then the LLN holds for the sequence $\xi_{1}, \xi_{2}, \ldots$

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Problem 6.16

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with uniformly bounded variances, where $\xi_{n}$ depends only on $\xi_{n-1}$ and $\xi_{n+1}$ , but is independent of the remaining $\xi_{i}$ . Prove that the LLN holds for this sequence.

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Problem 6.17

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with uniformly bounded variances, where $\operatorname {Cov}\left[ \xi_{i}, \xi_{j} \right] \leqslant 0$ for $i \neq j$ . Prove that the LLN applies to this sequence.

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Problem 6.18

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with finite variances, and let the correlation coefficient of $\xi_{i}$ and $\xi_{j}$ not exceed $g(\left|j-i\right|)$ , where $g(k) \geqslant 0$ . Prove that if

[g(0)+\ldots +g(n-1)]\left[\sigma _{1}^{2}+\ldots +\sigma _{n}^{2}\right] = o\left(n^{2}\right) \quad \text{ as } \quad n \rightarrow \infty ,

then the LLN applies to the sequence $\xi_{1}, \xi_{2}, \ldots$

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Problem 6.19

(Bernstein's theorem) Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with uniformly bounded variances, where $\operatorname {Cov}\left[ \xi_{j}, \xi_{k} \right] \rightarrow 0$ uniformly as $\left|j-k\right| \rightarrow \infty$ . Prove that the LLN applies to this sequence.

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Problem 6.20

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $\mathbb {E}\left[\xi_{i}\right] = a, \operatorname {Var}\left[\xi_{i}\right] = \sigma^{2}, i = 1$ , $2, \ldots$ Prove that the sequence of random variables $\eta_{1}$ , $\eta_{2}, \ldots$ , where

\eta _{n} = \frac{\xi _{1}+\ldots +\xi _{n}}{\xi _{1}^{2}+\ldots +\xi _{n}^{2}},

converges in probability, and find the limit.

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Problem 6.21

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that if $\left|\eta_{n}\right| < c n$ , while $\operatorname {Var}\left[\eta_{n}\right] > \alpha n^{2}$ , then the LLN does not apply to the sequence $\xi_{1}, \xi_{2}, \ldots$

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Problem 6.22

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite nonzero variance, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that the LLN does not hold for the sequence $\eta_{1}, \eta_{2}, \ldots$ , but does hold for the sequence $a_{1} \eta_{1}, a_{2} \eta_{2}, \ldots$ if $a_{n} \rightarrow 0$ as $n \rightarrow \infty$ .

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Problem 6.23

A sequence of independent random variables $\xi_{1}$ , $\xi_{2}, \ldots$ is called equivalent to a sequence of independent random variables $\eta_{1}, \eta_{2}, \ldots$ if the series

\sum _{n = 1}^{\infty } \mathbb {P}\left(\xi _{n} \neq \eta _{n}\right) .

converges.

Prove that if for every $n$ the random variables $\xi_{n}$ and $\eta_{n}$ have the same expectation and the LLN applies to one of the two equivalent sequences, then it also applies to the other.

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Problem 6.24

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables for which the LLN holds. Must the LLN hold for the sequence $\left|\xi_{1}\right|,\left|\xi_{2}\right|, \ldots$ ?

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Problem 6.25

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with zero expectations. Does the convergence $\xi_{n} \xrightarrow {\mathbb {P}} 0$ imply the convergence $\frac{\xi_{1}+\ldots +\xi_{n}}{n} \xrightarrow {\mathbb {P}} 0$ ?

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Problem 6.26

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables such that

\mathbb {E}\left[\left|\xi _{i}-\mathbb {E}\left[\xi _{i}\right]\right|\right] \leqslant C, \quad i = 1,2, \ldots \quad (C \text{ a constant }),

and let $a_{1}, a_{2}, \ldots$ be a sequence of real numbers, $a_{n} \rightarrow 0$ as $n \rightarrow \infty$ . Prove that the LLN holds for the sequence $a_{1} \xi_{1}, a_{2} \xi_{2}, \ldots$

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Problem 6.27

Is the following statement true: if the LLN holds for a sequence of random variables $\xi_{1}, \xi_{2}, \ldots$ and $a_{1}, a_{2}, \ldots$ is a uniformly bounded sequence of nonnegative numbers, then the LLN also holds for the sequence $\eta_{1}, \eta_{2}, \ldots$ , where $\eta_{n} = a_{n} \xi_{n}$ ?

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Problem 6.28

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, and $a_{1}, a_{2}, \ldots$ a uniformly bounded sequence of nonnegative numbers. Can it be asserted that if the LLN holds for $\xi_{1}, \xi_{2}, \ldots$ , then it also holds for $\eta_{1}, \eta_{2}, \ldots$ , where $\eta_{i} = a_{i} \xi_{i}$ ?

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Problem 6.29

Let $\xi_{1}, \xi_{2}, \ldots$ and $\eta_{1}, \eta_{2}, \ldots$ be two sequences of independent random variables, where

\begin{gathered} \xi _{n} = \begin{cases} \varphi (n), & \text{ with probability } p_{n} \\ 0, & \text{ with probability } 1-2 p_{n} \\ -\varphi (n), & \text{ with probability } p_{n} \end{cases} \\ \eta _{n} = \begin{cases} \varphi (n), & \text{ with probability } q_{n} \\ 0, & \text{ with probability } 1-2 q_{n} \\ -\varphi (n), & \text{ with probability } q_{n} \end{cases} \\ \frac{p_{n}}{q_{n}} \leqslant C < \infty , \quad n = 1,2, \ldots \end{gathered}

Prove that if the LLN holds for the sequence $\left\{ \eta_{n}\right\}$ , then it also holds for the sequence $\left\{ \xi_{n}\right\}$ .

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Problem 6.30

Let $\xi_{1}, \xi_{2}, \ldots$ and $\eta_{1}, \eta_{2}, \ldots$ be two sequences, each internally independent, of random variables, where

\begin{gathered} \xi _{n} = \begin{cases} \varphi (n), & \text{ with probability } p_{n} \\ 0, & \text{ with probability } 1-2 p_{n} \\ -\varphi (n), & \text{ with probability } p_{n} \end{cases} \\ \eta _{n} = \begin{cases} \psi (n), & \text{ with probability } p_{n} \\ 0, & \text{ with probability } 1-2 p_{n} \\ -\psi (n), & \text{ with probability } p_{n} \end{cases} \\ \frac{\varphi (n)}{\psi (n)} \leqslant C < \infty , \quad n = 1,2, \ldots \end{gathered}

Prove that if the LLN holds for the sequence $\left\{ \eta_{n}\right\}$ , then it also holds for the sequence $\left\{ \xi_{n}\right\}$ .

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Problem 6.31

Let $\xi_{1}, \xi_{2}, \ldots$ and $\eta_{1}, \eta_{2}, \ldots$ be two sequences, each internally independent, of random variables, and let

\mathbb {P}\left(\left|\xi _{i}-\mathbb {E}\left[\xi _{i}\right]\right| \geqslant t\right) \leqslant \mathbb {P}\left(\left|\eta _{i}-\mathbb {E}\left[\eta _{i}\right]\right| \geqslant t\right)

for every $t \geqslant 0$ . Is it true that if the LLN holds for the sequence $\eta_{1}$ , $\eta_{2}, \ldots$ , then it also holds for the sequence $\xi_{1}, \xi_{2}, \ldots$ ?

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Problem 6.32

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite variances, and $C_{1}, C_{2}, \ldots$ a nondecreasing sequence of positive numbers. Prove that the LLN applies to the sequence $C_{1} \xi_{1}, C_{2} \xi_{2}, \ldots$ if and only if

C_{n} / \sqrt{n} \rightarrow 0 \quad \text{ as } \quad n \rightarrow \infty

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Problem 6.33

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with zero expectations and uniformly bounded variances. Prove that the series

\sum _{n = 1}^{\infty } \xi _{n} / n

converges almost surely.

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Problem 6.34

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent normally distributed random variables with zero expectations. Prove that the series $\sum \xi_{n}$ converges almost surely if and only if the series $\sum \operatorname {Var}\left[\xi_{n}\right]$ converges.

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Problem 6.35

Prove that if a series of independent random variables converges almost surely to a constant, then each term of the series is almost surely constant.

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Problem 6.36

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Can the series $\sum \xi_{n}$ converge almost surely while the sequence of medians $m \eta_{n}$ does not converge?

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Problem 6.37

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Prove that the series $\sum \xi_{n}^{(s)}$ ( $\xi_{n}^{(s)}$ is the symmetrization of $\xi_{n}$ ) converges almost surely if and only if there exists a sequence of real numbers $a_{1}, a_{2}, \ldots$ such that the series $\sum \left(\xi_{n}-a_{n}\right)$ converges almost surely.

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Problem 6.38

Let $\xi_{1}, \xi_{2}, \ldots , \eta_{1}, \eta_{2}, \ldots$ be jointly independent random variables. Prove that if the series $\sum \left(\xi_{n}+\eta_{n}\right)$ converges almost surely, then for some sequences of real numbers $a_{1}, a_{2}, \ldots$ and $b_{1}, b_{2}, \ldots$ the series

\sum \left(\xi _{n}-a_{n}\right) \quad \text{ and } \quad \sum \left(\eta _{n}-b_{n}\right) .

converge almost surely.

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Problem 6.39

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Prove that the series $\sum_{n = 1}^{\infty } \xi_{n}$ converges almost surely if and only if it converges in probability.

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Problem 6.40

Prove that a series of independent random variables converges almost surely if and only if it converges in distribution.

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Problem 6.41

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, $f_{1}(t), f_{2}(t), \ldots$ the corresponding characteristic functions. Prove that the series $\sum \xi_{n}$ converges almost surely if and only if

\lim _{n \rightarrow \infty } \prod _{k = 1}^{n} f_{k}(t) = f(t),

where $f(t)$ is a function continuous at zero.

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Problem 6.42

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with characteristic functions $f_{1}(t), f_{2}(t), \ldots$ respectively. Prove that if the infinite product $\prod_{n = 1}^{\infty } f_{n}(t)$ converges to a nonzero limit on some set of positive Lebesgue measure, then the series $\sum \xi_{n}$ converges almost surely, and conversely.

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Problem 6.43

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with characteristic functions $f_{1}(t), f_{2}(t), \ldots$ respectively. Prove that the infinite product $\prod_{n = 1}^{\infty }\left|f_{n}(t)\right|$ is strictly positive on a set of positive Lebesgue measure if and only if there exists a sequence of real numbers $a_{1}, a_{2}, \ldots$ such that the series $\sum \left(\xi_{n}-a_{n}\right)$ converges almost surely.

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Problem 6.44

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables,

\mathbb {P}\left(\xi _{1} = -1\right) = \mathbb {P}\left(\xi _{1} = 1\right) = 1 / 2.

Prove that the series $\sum c_{n} \xi_{n}$ , where $c_{1}, c_{2}, \ldots$ is a sequence of real numbers, converges if and only if the series $\sum c_{n}^{2}$ converges.

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Problem 6.45

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with the uniform distribution on the segment $[-1,+1]$ . Prove that the series $\sum c_{n} \xi_{n}$ converges almost surely if and only if the series $\sum c_{n}^{2}$ converges.

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Problem 6.46

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed bounded random variables with zero expectations. Prove that the series $\sum c_{n} \xi_{n}$ , where $c_{1}$ , $c_{2}, \ldots$ is a sequence of real numbers, converges almost surely if and only if the series $\sum c_{n}^{2}$ converges.

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Problem 6.47

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite variances and zero expectations. Prove that the series $\sum_{n = 1}^{\infty } c_{n} \xi_{n}$ , where $c_{1}, c_{2}, \ldots$ is a sequence of real numbers, converges almost surely if and only if the series $\sum c_{n}^{2} < \infty$ converges.

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Problem 6.48

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, each having a Poisson distribution.

Prove that the series $\sum_{n = 1}^{\infty } \xi_{n}$ converges almost surely if and only if the series $\sum_{n = 1}^{\infty } \mathbb {E}\left[\xi_{n}\right]$ converges.

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Problem 6.49

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent nonnegative random variables such that $\mathbb {P}\left(\xi_{n} > c\right) = 0$ . Prove that the series $\sum \xi_{n}$ converges almost surely if and only if the series $\sum \mathbb {E}\left[\xi_{n}\right]$ converges.

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Problem 6.50

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables satisfying the condition $\left|\xi_{n}\right| \leqslant c < \infty , n = 1,2, \ldots$ Prove that the series $\sum \xi_{n}$ converges almost surely if the series $\sum \mathbb {E}\left[\left|\xi_{n}-\mathbb {E}\left[\xi_{n}\right]\right|\right]$ converges. Is the converse true?

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Problem 6.51

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables having the same Cauchy distribution with density

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

Prove that the series $\sum c_{n} \xi_{n}$ , where $c_{1}, c_{2}, \ldots$ is a sequence of real numbers, converges almost surely if and only if the series $\sum \left|c_{n}\right|$ converges.

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Problem 6.52

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with density

\frac{1-\cos x}{\pi x^{2}},

and $c_{1}, c_{2}, \ldots$ a sequence of real numbers. Prove that the series $\sum c_{n} \xi_{n}$ converges almost surely if and only if the series $\sum \left|c_{n}\right|$ converges.

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Problem 6.53

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables having a symmetric stable distribution with exponent $\alpha > 0$ . Prove that the series $\sum c_{n} \xi_{n}$ , where $c_{1}, c_{2}, \ldots$ is a sequence of real numbers, converges almost surely if and only if $\sum \left|c_{n}\right|^{\alpha } < \infty$ .

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Problem 6.54

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with zero expectations and finite third moments. Set $\sigma_{i}^{2} = \operatorname {Var}\left[\xi_{i}\right], \beta_{i} = \mathbb {E}\left[\left|\xi_{i}\right|^{3}\right]$ . Prove that if the quantities $\beta_{i} / \sigma_{i}^{2}$ are uniformly bounded: $\beta_{i} / \sigma_{i}^{2} \leqslant c, i = 1,2, \ldots$ , and the series $\sum \xi_{n}$ converges almost surely, then the series $\sum \sigma_{n}^{2}$ converges.

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Problem 6.55

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with finite third moments. Set $\sigma_{i}^{2} = \operatorname {Var}\left[\xi_{i}\right]$ , $\beta_{i} = \mathbb {E}\left[\left|\xi_{i}\right|^{3}\right]$ . Prove that if the quantities $\beta_{i} / \sigma_{i}^{2}$ are uniformly bounded, then a necessary and sufficient condition for the series $\sum \xi_{n}$ to converge almost surely is that the series $\sum \mathbb {E}\left[\xi_{i}\right]$ and $\sum \sigma_{i}^{2}$ converge.

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Problem 6.56

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $c_{1}, c_{2}, \ldots$ a sequence of real numbers. Prove that if the series $\sum c_{n} \xi_{n}$ converges almost surely, then $\sum c_{n}^{2} < \infty$ .

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Problem 6.57

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables having finite expectation, and $c_{1}, c_{2}, \ldots$ a sequence of real numbers. Prove that if $\sum \left|c_{n}\right| < \infty$ , then the series $\sum c_{n}\left(\xi_{n}-a_{n}\right)$ converges almost surely for some choice of real numbers $a_{1}, a_{2}, \ldots$

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Problem 6.58

Give an example of a sequence of independent random variables $\xi_{1}, \xi_{2}, \ldots$ with zero expectations such that the series $\sum \xi_{n}$ converges almost surely, while the series $\sum \operatorname {Var}\left[\xi_{n}\right]$ diverges.

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Problem 6.59

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent integer-valued random variables. Prove that if the series $\sum \xi_{n}$ converges almost surely, then $\prod_{k = 1}^{\infty } p_{k} > 0$ , where $p_{k}$ is the largest jump of the distribution function of the random variable $\xi_{k}$ .

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Problem 6.60

Prove that the radius of convergence $R$ of the power series $\sum_{n = 0}^{\infty } \xi_{n} z^{n}$ , where $\left\{ \xi_{n}\right\}$ are independent random variables, is almost surely constant.

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Problem 6.61

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Prove that the series $\sum \xi_{n}^{2}$ converges almost surely if and only if

\sum \mathbb {E}\left[\frac{\xi _{n}^{2}}{1+\xi _{n}^{2}}\right] < \infty .

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Problem 6.62

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with zero expectations. Prove that if

\sum \mathbb {E}\left[\frac{\xi _{n}^{2}}{1+\left|\xi _{n}\right|}\right] < \infty

then the series $\sum \xi_{n}$ converges almost surely.

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Problem 6.63

A sequence of random variables $\eta_{1}, \eta_{2}, \ldots$ is called bounded in probability if

\limsup _{c \rightarrow \infty } \mathbb {P}\left(\left|\eta _{n}\right| > c\right) = 0

Prove that if $\xi_{1}, \xi_{2}, \ldots$ are independent symmetric random variables, and the random variables $\eta_{n} = \sum_{i = 1} \xi_{i}$ are bounded in probability, then the series $\sum \xi_{n}$ converges almost surely. Can the symmetry condition be dropped?

9 A. V. Prokhorov et al.

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Problem 6.64

Let $v_{n}$ be the number of successes in $n$ Bernoulli trials with success probability $p$ . Prove that

v_{n} / n \rightarrow p \text{ a.s. as } \quad n \rightarrow \infty .

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Problem 6.65

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables. Prove that if

\frac{\xi _{1}+\ldots +\xi _{n}}{n} \rightarrow c \text{ a.s. as } \quad n \rightarrow \infty ,

where $c$ is some real number, then $\mathbb {E}\left[\left|\xi_{1}\right|\right] < \infty$ and $\mathbb {E}\left[\xi_{1}\right] = c$ .

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Problem 6.66

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables having finite variances, $b_{1}, b_{2}, \ldots$ a nondecreasing sequence of real numbers such that $b_{n} \rightarrow \infty$ as $n \rightarrow \infty$ , $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that if

\sum _{n = 1}^{\infty } \frac{\operatorname {Var}\left[\xi _{n}\right]}{b_{n}^{2}} < \infty ,

then

\frac{\eta _{n}-\mathbb {E}\left[\eta _{n}\right]}{b_{n}} \rightarrow 0 \text{ a.s. } \quad \text{ as } \quad n \rightarrow \infty .

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Problem 6.67

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables such that $\left|\xi_{i}\right| \leqslant C, C > 0, i = 1,2, \ldots$ Set $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that

\frac{\eta _{n}-\mathbb {E}\left[\eta _{n}\right]}{\sqrt{n} \log n} \rightarrow 0 \text{ a.s., } \quad n \rightarrow \infty .

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Problem 6.68

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $0 < r < 2$ . Prove that if

\frac{1}{n^{1 / r}} \sum _{k = 1}^{n}\left(\xi _{k}-a_{k}\right) \rightarrow 0 \text{ a.s.: }

then $\mathbb {E}\left[\left|\xi_{k}\right|\right], r < \infty$ , where $a_{k} = 0$ for $r < 1$ and $a_{k} = \mathbb {E}\left[\xi_{k}\right]$ for $r \geqslant 1$ .

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Problem 6.69

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that if for some $r \geqslant 1$

\sum _{n = 1}^{\infty } \frac{\mathbb {E}\left[\left|\xi _{n}\right|^{2 r}\right]}{n^{r+1}} < \infty ,

then

\frac{\eta _{n}}{n} \rightarrow 0 \text{ a.s., } \quad n \rightarrow \infty .

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Problem 6.70

Show that, whatever the sequence of nonnegative numbers $\sigma_{1}^{2}, \sigma_{2}^{2}, \ldots$ such that

\sum _{n = 1}^{\infty } \frac{\sigma _{n}^{2}}{n^{2}} = \infty ,

there exists a sequence of independent random variables $\xi_{1}, \xi_{2}, \ldots$ such that

\mathbb {E}\left[\xi _{n}\right] = 0, \quad \operatorname {Var}\left[\xi _{n}\right] = \sigma _{n}^{2}, \quad n = 1,2, \ldots ,

and the sequence $\eta_{1}, \eta_{2}, \ldots$ , where $\eta_{n} = n^{-1} \sum_{i = 1}^{n} \xi_{i}$ , does not converge almost surely to zero.

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Problem 6.71

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables such that

\mathbb {E}\left[\xi _{n}\right] = 0, \quad \left|\frac{\xi _{n}}{n}\right| \leqslant C, \quad n = 1,2, \ldots ,

where $C$ is some constant, and let

\frac{1}{n} \sum _{i = 1}^{n} \xi _{i} \rightarrow 0 \text{ a.s., } \quad n \rightarrow \infty .

Prove that for every $\varepsilon > 0$

\sum _{n = 1}^{\infty } \frac{\operatorname {Var}\left[\xi _{n}\right]}{n^{2+\varepsilon }} < \infty .

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Problem 6.72

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables having a normal distribution. Prove that if the sequence $\xi_{1}, \xi_{2}, \ldots$ satisfies the condition $\liminf_{n \rightarrow \infty } \frac{\ln n}{n} \operatorname {Var}\left[\xi_{n}\right] > 0$ , then the SLLN does not hold for it.

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Problem 6.73

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $\mathbb {E}\left[\left|\xi_{j}\right|^{r}\right] < \infty , 1 \leqslant r < 2$ . Prove that

n^{-\frac{1}{r}} \sum _{j = 1}^{n}\left(\xi _{j}-\mathbb {E}\left[\xi _{j}\right]\right) \rightarrow 0 \text{ a.s. } \quad \text{ as } \quad n \rightarrow \infty .

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Problem 6.74

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $\mathbb {E}\left[\left|\xi_{j}\right|^{r}\right] < \infty , 0 < r < 1$ .

Prove that

n^{-\frac{1}{r}} \sum _{j = 1}^{n} \xi _{j} \rightarrow 0 \text{ a.s. } \quad \text{ as } \quad n \rightarrow \infty .

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Problem 6.75

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables. Prove that if $\mathbb {E}\left[\left|\xi_{1}\right|^{p}\right] = \infty$ for some $0 < p < 2$ , then

\limsup _{n \rightarrow \infty } n^{-\frac{1}{p}}\left|\sum _{i = 1}^{n}\left(\xi _{i}-a\right)\right| = +\infty \text{ a.s. }

for every real $a$ .

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Problem 6.76

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with expectation $a$ , $\eta_{n} = \left(\xi_{1}+\ldots +\xi_{n}\right) / n$ . Prove that the SLLN applies to the sequence $\eta_{1}, \eta_{2}, \ldots$

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Problem 6.77

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed nondegenerate random variables with finite variances, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that for any finite real numbers $a$ and $b$

\lim _{n \rightarrow \infty } \mathbb {P}\left(a \leqslant \eta _{n} \leqslant b\right) = 0

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Problem 6.78

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite positive variances. Prove that for every real number $x$ the limit

\lim _{n \rightarrow \infty } \mathbb {P}\left(\xi _{1}+\ldots +\xi _{n} < x\right)

equals either 0, or 1, or $1 / 2$ . Indicate the conditions under which each of these cases occurs.

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Problem 6.79

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero expectations and finite variances. Prove that for every positive $x$ the limit

\lim _{n \rightarrow \infty } \mathbb {P}\left(\left|\frac{\xi _{1}+\ldots +\xi _{n}}{n^{\alpha }}\right| \leqslant x\right)

equals 0 for $\alpha < 1 / 2$ and 1 for $\alpha > 1 / 2$ .

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Problem 6.80

Let $P_{n} = \max_{0 \leqslant h \leqslant n} \mathbb {P}\left(v_{n} = k\right)$ , where $v_{n}$ is the number of successes in a Bernoulli scheme with success probability $p$ . Find $\lim_{n \rightarrow \infty } P_{n} \sqrt{n}$ .

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Problem 6.81

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero expectations and finite variances, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Find $\operatorname {Var}\left[\xi_{i}\right]$ if

\lim _{n \rightarrow \infty } \mathbb {P}\left(\frac{\eta _{n}}{\sqrt{n}} > 1\right) = 1 / 3

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Problem 6.82

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Find

\lim _{n \rightarrow \infty } \mathbb {P}\left(0 < \frac{\eta _{n}}{\sqrt{n}} < 1\right)

if $\xi_{n}$ is uniformly distributed on the segment $\left[a_{n}-1, a_{n}+1\right]$ ( $a_{1}, a_{2}, \ldots$ is a sequence of real numbers, $\sum a_{i} = A < \infty$ ).

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Problem 6.83

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables having the uniform distribution on the segment $[0,1]$ , $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Find a sequence of real numbers $a_{1}, a_{2}, \ldots$ satisfying the condition

\lim _{n \rightarrow \infty } \mathbb {P}\left(\eta _{n} \leqslant a_{n} \sqrt{n}\right) = p, \quad 0 \leqslant p \leqslant 1

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Problem 6.84

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite variance and

0 < \lim _{n \rightarrow \infty } \mathbb {P}\left(\left|\frac{\xi _{1}+\ldots +\xi _{n}}{\sqrt{n}}\right| < a\right) = b < 1.

Find

\lim _{n \rightarrow \infty } \mathbb {P}\left(\left|\frac{\xi _{1}+\ldots +\xi _{n}}{\sqrt{n}}\right| < 2 a\right) .

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Problem 6.85

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with unit variances, $\mathbb {E}\left[\left[\xi_{1}\right]\right] = 0$ ( $[x]$ is the integer part of $x$ ), and let

\lim _{n \rightarrow \infty } \mathbb {P}\left(\frac{\xi _{1}+\ldots +\xi _{n}}{\sqrt{n}} > 0\right) = 1 / 2.

Find $\mathbb {E}\left[\left\{ \xi_{1}\right\} \right]$ ( $\left\{ x\right\}$ is the fractional part of $x$ ).

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Problem 6.86

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables taking the values $0, \pm 1, \pm 2, \ldots , \mathbb {P}\left(\xi_{1} = 0\right) > 0, \mathbb {P}\left(\xi_{1} = 1\right) > 0$ . Find the limit (in the sense of weak convergence) of the sequence of distributions of the random variables $\left\{ \eta_{n} / 2+1 / 2\right\}$ ( $\eta_{n} = \sum_{i = 1}^{n} \xi_{i}$ , $\left\{ x\right\}$ is the fractional part of $x$ ). Will convergence in variation hold?

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Problem 6.87

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero expectations and unit variances. Find the limit (in the sense of weak convergence) of the sequence of distributions of the random variables $\left[\eta_{n} / \sqrt{n}\right]$ ( $\eta_{n} = \sum_{i = 1}^{n} \xi_{i}$ , $[x]$ is the integer part of $x$ ). Will convergence in variation hold?

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Problem 6.88

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero expectations and unit variances. Prove that the quantities

\eta = \sqrt{n} \frac{\xi _{1}+\ldots +\xi _{n}}{\xi _{1}^{2}+\ldots +\xi _{n}^{2}} \quad \text{ and } \quad \zeta = \frac{\xi _{1}+\ldots +\xi _{n}}{\sqrt{\xi _{1}^{2}+\ldots +\xi _{n}^{2}}}

are asymptotically normal.

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Problem 6.89

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, where $\xi_{1}, \xi_{3}, \xi_{5}, \ldots$ are identically distributed and $\xi_{2}, \xi_{4}$ , $\xi_{6}, \ldots$ are identically distributed,

\mathbb {E}\left[\xi _{1}^{2}\right] < \infty , \quad \mathbb {E}\left[\xi _{2}^{2}\right] < \infty , \quad \operatorname {Var}\left[\xi _{1}\right] > 0, \quad \operatorname {Var}\left[\xi _{2}\right] > 0.

Set

\eta _{n} = \xi _{1}+\ldots +\xi _{n}, \quad \eta _{n}^{*} = \frac{\eta _{n}-\mathbb {E}\left[\eta _{n}\right]}{\sqrt{\operatorname {Var}\left[\eta _{n}\right]}} .

Find the limiting (in the sense of weak convergence) distribution for $\eta_{n}^{*}$ .

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Problem 6.90

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables having the same Poisson distribution with parameter $\lambda$ . Find the limit

\lim _{n \rightarrow \infty } \mathbb {P}\left(\frac{\sum _{i = 1}^{n}\left(\xi _{2 i-1}-\xi _{2 i}\right)}{\sqrt{n}} < x\right) .

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Problem 6.91

Prove that the Lindeberg condition is satisfied for a sequence of independent identically distributed random variables with finite variance.

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Problem 6.92

Prove that if the Lyapunov condition holds for a sequence of random variables, then the Lindeberg condition also holds.

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Problem 6.93

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent normally distributed random variables, $\mathbb {E}\left[\xi_{k}\right] = 0, k = 1,2, \ldots$ , $\operatorname {Var}\left[\xi_{1}\right] = 1, \operatorname {Var}\left[\xi_{k}\right] = 2^{k-2}, k \geqslant 2$ . Show that in this case the Lindeberg condition does not hold, but the CLT does hold.

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Problem 6.94

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero expectation and unit variance, and $v_{1}, v_{2}, \ldots$ a sequence of positive integer-valued random variables such that for each $i$ , $v_{i}$ is independent of $\xi_{1}, \xi_{2}, \ldots$ Let $\eta_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that if $v_{n} \xrightarrow {\mathbb {P}} \infty$ as $n \rightarrow \infty$ , then the distribution of $\eta_{v_{n}} / \sqrt{v_{n}}$ converges weakly, as $n \rightarrow \infty$ , to the standard normal law.

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Problem 6.95

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero expectation and unit variance, and $v_{1}, v_{2}, \ldots$ a sequence of positive integer-valued random variables such that $\mathbb {E}\left[v_{k}\right] = \alpha_{k}, \operatorname {Var}\left[v_{k}\right] = \beta_{k}$ , and for each $k$ , $v_{k}$ is independent of $\xi_{1}, \xi_{2}, \ldots$ Let $\alpha_{n} \rightarrow \infty$ and $\beta_{n} = o\left(\alpha_{n}^{2}\right)$ as $n \rightarrow \infty$ . Find the limiting (in the sense of weak convergence) distribution for $\eta_{v_{n}} / \sqrt{\operatorname {Var}\left[\eta_{v_{n}}\right]}$ ( $\eta_{k} = \xi_{1}+\ldots +\xi_{k}$ ).

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Problem 6.96

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with zero expectation, unit variance, and absolutely integrable characteristic function. Let $p^{(n)}(x)$ be the density of the distribution of the random variable

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

Prove that, uniformly in $x$ ,

\lim _{n \rightarrow \infty } p^{(n)}(x) = \frac{1}{\sqrt{2 \pi }} e^{-x^{2} / 2}

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Problem 6.97

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables having the same uniform distribution on the segment $[-a, a]$ , and $F_{n}(x)$ the distribution function of the normalized sum

\frac{\xi _{1}+\ldots +\xi _{n}}{\sqrt{n \operatorname {Var}\left[\xi _{1}\right]}} .

Prove that

\sup _{x}\left|F_{n}(x)-\frac{1}{\sqrt{2 \pi }} \int _{-\infty }^{x} e^{-\frac{u^{2}}{2}} d u\right| \leqslant \frac{1}{4 \pi n}\left(1+\frac{4}{n-1}\right) .

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Problem 6.98

Let $\xi_{1}, \ldots , \xi_{n}$ be independent identically distributed random variables with zero expectation, variance $\sigma^{2}$ , and finite third absolute moment $\mathbb {E}\left[\left|\xi_{1}\right|^{3}\right] = \beta_{3}$ . Let $f_{n}(t)$ be the characteristic function of the random variable

\frac{\xi _{1}+\ldots +\xi _{n}}{\sigma \sqrt{n}} .

Prove that

\left|f_{n}(t)-e^{-t^{2} / 2}\right| \leqslant 16 \frac{\beta _{3}}{\sigma ^{3} \sqrt{n}} e^{-t^{2} / 3}\left|t\right|^{3}

for $\left|t\right| \leqslant \frac{\sigma^{3} \sqrt{n}}{4 \beta_{3}}$ .

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Problem 6.99

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with the same distribution function $F(x)$ , where $\mathbb {E}\left[\xi_{1}\right] = 0, \mathbb {E}\left[\xi_{1}^{2}\right] = \sigma^{2}, \mathbb {E}\left[\left|\xi_{1}\right|^{3}\right] = \beta_{3}$ , $F(x)$ is symmetric, and the corresponding characteristic function $f(t)$ is nonnegative. Let

\sup _{x} F^{\prime }(x) \leqslant A < \infty .

Prove that

\left|\mathbb {P}\left(\frac{\xi _{1}+\ldots +\xi _{n}}{\sigma \sqrt{n}} < x\right)-\frac{1}{\sqrt{2 \pi }} \int _{-\infty }^{x} e^{-\frac{t^{2}}{2}} d t\right| \leqslant \frac{C(A) \beta _{3}}{\sigma ^{3} \sqrt{n}}

where $C(A)$ is a positive constant depending only on $A$ .

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Problem 6.100

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables, $b_{1}, b_{2}, \ldots$ a sequence of real numbers, $m \xi_{i}$ the median of the random variable $\xi_{i}, i = 1,2, \ldots$ Prove that if $\xi_{n}-b_{n} \xrightarrow {\mathbb {P}} 0$ as $n \rightarrow \infty$ , then $\xi_{n}-m \xi_{n} \xrightarrow {\mathbb {P}} 0$ .

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Problem 6.101

Let a sequence of independent random variables $\xi_{1}, \xi_{2}, \ldots$ converge almost surely. Prove that there exists a constant $a$ such that $\mathbb {P}\left(\lim_{n \rightarrow \infty } \xi_{n} = a\right) = 1$ .

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Problem 6.102

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with distribution functions $F_{1}(x), F_{2}(x), \ldots$ respectively. Prove that $\xi_{n} \rightarrow 0$ as $n \rightarrow \infty$ if and only if

\lim _{n \rightarrow \infty } \int _{-\infty }^{\infty } \frac{x^{2}}{1+x^{2}} d F_{n}(x) = 0

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Problem 6.103

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with finite variances. Set $a_{n} = \mathbb {E}\left[\xi_{n}\right], \sigma_{n}^{2} = \operatorname {Var}\left[\xi_{n}\right]$ . Prove that if $a_{n} \rightarrow \infty$ and $\sigma_{n}^{2} = o\left(a_{n}^{2}\right)$ as $n \rightarrow \infty$ , then

\frac{\xi _{n}}{a_{n}} \xrightarrow {\mathbb {P}} 1 \quad \text{ as } \quad n \rightarrow \infty .

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Problem 6.104

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables with finite expectations, $\eta_{n} = \xi_{1} \ldots \xi_{n}$ . Prove that if $\mathbb {E}\left[\left|\xi_{1}\right|\right] = \mathbb {E}\left[\left|\xi_{2}\right|\right] = \ldots < 1$ , then $\eta_{n} \xrightarrow {\mathbb {P}} 0$ as $n \rightarrow \infty$ . Is the converse statement true if $\left\{ \xi_{i}\right\}$ are identically distributed?

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Problem 6.105

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables, $\eta_{n} = \xi_{1} \ldots \xi_{n}$ . Prove that if for some $\alpha > 0$ , $\mathbb {E}\left[\left|\xi_{1}\right|^{\alpha }\right] = \mathbb {E}\left[\left|\xi_{2}\right|^{\alpha }\right] = \ldots < 1$ , then $n^{\beta } \eta_{n} \xrightarrow {\mathbb {P}} 0$ as $n \rightarrow \infty$ for every real $\beta$ .

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Problem 6.106

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $\eta_{n} = \xi_{1} \ldots \xi_{n}$ . Prove that if $\eta_{n} \xrightarrow {\mathbb {P}} a$ as $n \rightarrow \infty$ ( $a$ a finite real number), then either $a = 0$ or $a = 1$ , and in the latter case $\mathbb {P}\left(\xi_{i} = 1\right) = 1$ , $i = 1,2, \ldots$

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Problem 6.107

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed nonnegative random variables with expectation $a$ ,

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

Prove that the sequence of random variables $\zeta_{n} = \sqrt[n]{\eta_{1} \ldots \eta_{n}}$ converges almost surely to $a$ .

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Problem 6.108

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables with finite expectations, and $\xi$ a random variable with finite variance, such that for every natural number $n$ , $\xi_{1}+\ldots +\xi_{n}$ and $\xi -\left(\xi_{1}+\ldots +\xi_{n}\right)$ are independent. Prove that in this case all $\xi_{n}$ have finite variances and the series $\sum_{n = 1}^{\infty }\left(\xi_{n}-\mathbb {E}\left[\xi_{n}\right]\right)$ converges almost surely.

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Problem 6.109

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of identically distributed random variables with finite absolute moment of order $p > 0$ . Set

\eta _{n} = \begin{cases} 0, & \left|\xi _{n}\right| < n^{1 / p} \\ 1, & \left|\xi _{n}\right| \geqslant n^{1 / p} \end{cases}

Prove that

\sum _{n = 1}^{\infty } \mathbb {E}\left[\eta _{n}\right] \leqslant \mathbb {E}\left[\left|\xi _{1}\right|^{p}\right]

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Problem 6.110

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of identically distributed random variables with finite absolute moment of order $p > 0$ . Set

\eta _{n} = \begin{cases} \xi _{n}, & \left|\xi _{n}\right| < n^{1 / p} \\ 0, & \left|\xi _{n}\right| \geqslant n^{1 / p} \end{cases}

Prove that

\sum _{n = 1}^{\infty } \mathbb {P}\left(\xi _{n} \neq \eta _{n}\right) \leqslant \mathbb {E}\left[\left|\xi _{1}\right|^{p}\right] < \infty

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Problem 6.111

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of random variables, $\alpha > \beta > 0$ . Does convergence of the series $\sum \mathbb {E}\left[\left|\xi_{n}\right|^{\alpha }\right]$ imply convergence of the series $\sum \mathbb {E}\left[\left|\xi_{n}\right|^{\beta }\right]$ ?

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Problem 6.112

Let $\xi_{1}, \xi_{2}, \ldots$ be a uniformly integrable sequence of random variables. Prove that

\mathbb {E}\left[\frac{1}{n} \sup _{1 \leqslant m \leqslant n}\left|\xi _{m}\right|\right] \rightarrow 0

as $n \rightarrow \infty$ .

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Problem 6.113

Let $\xi_{1}, \xi_{2}, \ldots$ and $\eta_{1}, \eta_{2}, \ldots$ be two sequences of random variables, where $\xi_{n}$ is asymptotically normal with parameters $\left(a, h / \sqrt{n}\right)$ , and $\eta_{n}$ is asymptotically normal with parameters $(b, k / \sqrt{n}), b \neq 0$ . Prove that the random variable

\zeta _{n} = \sqrt{n} \frac{\xi _{n}-a}{\eta _{n}}

is asymptotically normal with parameters $(0, h / b)$ .

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Problem 6.114

Let $f(x)$ be a continuous bounded function on $[0, \infty )$ . Prove that for $h > 0$

\lim _{n \rightarrow \infty } \sum _{k = 0}^{\infty } f\left(x+\frac{k}{n}\right) \frac{(n h)^{k}}{k!} e^{-h n} = f(x+h) .

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Problem 6.115

Prove that a random variable measurable with respect to the tail $\sigma$ -algebra has a degenerate distribution.

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Problem 6.116

Give an example of a tail $\sigma$ -algebra containing more than two events.

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Problem 6.117

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables. Prove that the random variables $\limsup \xi_{n}$ and $\liminf \xi_{n}$ are degenerate.

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Problem 6.118

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables having the standard normal distribution, and let the event $A_{n}$ consist in

\xi _{1}+\ldots +\xi _{n} > a(2 n \ln n)^{1 / 2},

where $a$ is a fixed constant. Prove that for $a > 1$ , with probability 1 only finitely many of the events $A_{n}$ occur.

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Problem 6.119

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables having the standard normal distribution. Prove that

\mathbb {P}\left(\limsup \frac{\xi _{n}}{\sqrt{2 \ln n}} = 1\right) = 1.

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Problem 6.120

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

\mathbb {P}\left(\limsup \frac{\xi _{n} \ln \ln n}{\ln n} = 1\right) = 1.

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Problem 6.121

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with characteristic function

f(t) = \exp \left\{ -\left|t\right|^{\alpha }\right\} , \quad 0 < \alpha < 2.

Prove that

\mathbb {P}\left(\limsup \left|\frac{\xi _{1}+\ldots +\xi _{n}}{n^{1 / \alpha }}\right|^{\frac{1}{\ln \ln n}} = e^{1 / \alpha }\right) = 1 .

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Problem 6.122

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent random variables,

\mathbb {E}\left[\xi _{k}\right] = m_{k} > 0 ; \quad \sum _{k = 1}^{\infty } m_{k} = \infty , \quad \sum _{k = 1}^{\infty } \frac{\operatorname {Var}\left[\xi _{k}\right]}{\left(\sum _{i = 1}^{k} m_{i}\right)^{2}} < \infty .

Prove that with probability 1

\lim _{n \rightarrow \infty } \frac{\sum _{k = 1}^{n} \xi _{k}}{\sum _{k = 1}^{n} m_{k}} = 1

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Problem 6.123

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $B_{1}, B_{2}, \ldots$ a sequence of real numbers. Let the sequence of distributions of the random variables

\frac{\xi _{1}+\ldots +\xi _{n}}{B_{n}}

converge weakly, as $n \rightarrow \infty$ , to some nondegenerate distribution $F$ . Prove that in this case

\lim _{n \rightarrow \infty } B_{n} = \infty \quad \text{ and } \quad \lim _{n \rightarrow \infty } \frac{B_{n+1}}{B_{n}} = 1

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Problem 6.124

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with finite expectation, $B_{1}, B_{2}, \ldots$ a sequence of positive numbers. Prove that if the sequence of distributions of the random variables

\frac{\xi _{1}+\ldots +\xi _{n}}{B_{n}}

converges weakly, as $n \rightarrow \infty$ , to some nondegenerate distribution, then $\mathbb {E}\left[\xi_{i}\right] = 0$ .

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Problem 6.125

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables having characteristic function

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

Choose a sequence of real numbers $B_{1}, B_{2}, \ldots$ so that the sequence of distributions of the random variables

\frac{\xi _{1}+\ldots +\xi _{n}}{B_{n}}

converges weakly, as $n \rightarrow \infty$ , to some nondegenerate limiting distribution. Find the limiting law.

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Problem 6.126

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables with density

\frac{1-\cos x}{\pi x^{2}}

$\eta_{n} = \frac{\xi_{1}+\ldots +\xi_{n}}{n}, F_{n}(x)$ the distribution function of $\eta_{n}$ . Prove that there exists a distribution function $F(x)$ such that $F_{n}$ converges weakly to $F$ as $n \rightarrow \infty$ . Find the distribution function $F(x)$ .

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Problem 6.127

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $B_{1}, B_{2}, \ldots$ a sequence of positive numbers, where the sequence of distributions of the random variables

\frac{\xi _{1}+\ldots +\xi _{n}}{B_{n}}

converges weakly, as $n \rightarrow \infty$ , to a distribution with characteristic function

e^{-c\left|t\right|^{\alpha }}, \quad c > 0, \quad 0 < \alpha \leqslant 2.

Prove that $B_{n} = n^{1 / \alpha } h(n)$ , where $\lim_{x \rightarrow \infty } h(t x) / h(x) = 1$ for $t > 0$ .

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Problem 6.128

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables, $A_{1}, A_{2}, \ldots$ and $B_{1}, B_{2}, \ldots$ ( $B_{n} > 0$ ) two sequences of real numbers, where the sequence of distributions of the random variables

\frac{\xi _{1}+\ldots +\xi _{n}-A_{n}}{B_{n}}

converges weakly, as $n \rightarrow \infty$ , to some distribution. Prove that the limiting distribution is stable.

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Problem 6.129

Let the random variable $\xi_{N,M,n}$ have the hypergeometric distribution with parameters $N, M, n$ : $\mathbb {P}\left(\xi_{N, M, n} = m\right) = \begin{cases} C_{M}^{m} C_{N-M}^{n-m} / C_{N}^{n}, & \max (0, M+n-N) \leqslant m \leqslant \min (n, M) \\ 0 \quad , & \text{ otherwise } \end{cases}$

(M \leqslant N, n \leqslant N)

Prove that as $N \rightarrow \infty , M \rightarrow \infty , \frac{M}{N} \rightarrow p, 0 < p < 1$ , with $n$ fixed, the hypergeometric distribution converges to the binomial distribution with parameter $p$ :

C_{n}^{m} p^{m}(1-p)^{n-m} .

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Problem 6.130

Let a sequence of random variables $\xi_{1}, \ldots , \xi_{n}$ form a Poisson scheme, i.e., $\xi_{1}, \ldots$ , $\xi_{n}$ are mutually independent and distributed so that

\mathbb {P}\left(\xi _{k} = 1\right) = p_{k}, \quad \mathbb {P}\left(\xi _{k} = 0\right) = 1-p_{k}, \quad k = 1, \ldots , n .

Denote $\mu_{n} = \xi_{1}+\ldots +\xi_{n}$ . Prove that if $p_{1}+\ldots +p_{n} \rightarrow \lambda > 0$ as $n \rightarrow \infty$ , then

\lim _{n \rightarrow \infty } \mathbb {P}\left(\mu _{n} = m\right) = e^{-\lambda } \frac{\lambda ^{m}}{m!} .

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Problem 6.131

(Continuation) In the Poisson scheme, let $\mu_{n} = \xi_{1}+\ldots +\xi_{n}$ , $\lambda = p_{1}+\ldots +p_{n}, \delta = p_{1}^{2}+\ldots +p_{n}^{2}, \quad 0 < p_{j} \leqslant 1 / 2$ . Prove that for $n \geqslant 2$

\left|\mathbb {P}\left(\mu _{n} = m\right)-\frac{\lambda ^{m}}{m!} e^{-\lambda }\right| \leqslant 2 \delta .

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Problem 6.132

Let the random variables $\xi_{n}, n = 1,2, \ldots$ , have distributions concentrated on the interval $[0,1)$ . Prove that if for every integer $k \neq 0$

\mathbb {E}\left[e^{2 \pi i k \xi _{n}}\right] \rightarrow 0

as $n \rightarrow \infty$ , then for any $\alpha$ and $\beta , 0 \leqslant \alpha < \beta < 1$ ,

\mathbb {P}\left(\alpha \leqslant \xi _{n} < \beta \right) \rightarrow \beta -\alpha .

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Problem 6.133

Let the random variables $\xi_{1}, \xi_{2}, \ldots , \xi_{n}$ be mutually independent and identically distributed, where $\mathbb {P}\left(0 \leqslant \xi_{n} < 1\right) = 1$ . Let $\eta_{n} = \left\{ \xi_{1}+\ldots +\xi_{n}\right\}$ be the fractional part of the sum $\xi_{1}+\ldots +\xi_{n}$ . If for $k \neq 0$

\mathbb {E}\left[e^{2 \pi i k \eta _{n}}\right] \rightarrow 0, \quad n \rightarrow \infty ,

then for any $\alpha$ and $\beta , 0 \leqslant \alpha < \beta < 1$ ,

\mathbb {P}\left(\alpha \leqslant \eta _{n} < \beta \right) \rightarrow \beta -\alpha .

(This assertion serves as an analogue of the central limit theorem when random variables are added modulo 1.)

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Problem 6.134

Find an approximate value for the probability that the number of "successes" $S_{n}$ in a scheme of $n = 100$ Bernoulli trials with success probability $p = 0.5$ lies within the bounds 35 and 65; 47 and 53. For which values of $n$ will the probability that $0.35 \leqslant \frac{S_{n}}{n} \leqslant 0.65$ be greater than 0.998?

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Problem 6.135

(Continuation) How large must the number of trials $n$ be so that, with probability $1-\alpha$ , the "success" frequency $\frac{S_{n}}{n}$ differs from the "success" probability $p$ by no more than $\varepsilon > 0$ ? Solve the problem for $\alpha = 0.01, \varepsilon = 0.01$ .

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Problem 6.136

(Continuation) Suppose that in a Bernoulli scheme with success probability $p = \mathbb {P}\left(\xi_{k} = 1\right), k = 1, \ldots , n$ , the value of $p$ is unknown and must be determined from the values taken by the random variables $\xi_{1}, \ldots , \xi_{n}$ . It is most natural to take as an estimate of $p$ the "success" frequency $\hat{p}_{n} = \frac{S_{n}}{n}, S_{n} = \xi_{1}+\ldots +\xi_{n}$ , since $\mathbb {E}\left[\hat{p}_{n}\right] = p$ and $\mathbb {P}\left\{ \left|\hat{p}_{n}-p\right| < \varepsilon \right\} > 1-\frac{p(1-p)}{n \varepsilon^{2}}$ for every $n$ . As an estimate one can instead give an interval $\left[\bar{p}_{n}\left(\xi_{1}, \ldots , \xi_{n}\right), \overline{\bar{p}}_{n}\left(\xi_{1}, \ldots , \xi_{n}\right)\right], 0 < \bar{p}_{n} < \overline{\overline{p_{n}}} < 1$ , such that

\mathbb {P}\left(\bar{p}_{n} \leqslant p \leqslant \overline{\bar{p}}_{n}\right) \geqslant 1-\alpha

for any preassigned $0 < \alpha < 1$ . Such an interval is called a confidence interval for $p$ of level $1-\alpha$ . Using the De Moivre–Laplace theorem, find an approximate confidence interval for $p$ .

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Problem 6.137

(a)(b)

(experimental estimation of $\pi$ ). Buffon's needle experiment, tossing a needle onto a plane ruled with parallel lines (see problem 1.75), has been used to compute the number $\pi$ . In the 19th and 20th centuries many experiments were carried out (see more on them in the book: Kendall M. and Moran P. Geometric Probabilities: translated from English. — Moscow: Nauka, 1972). In R. Wolf's experiment in Zurich, the needle length was $l = 36$ mm, the distance between the lines $a = 45$ mm, the needle was tossed $n = 5000$ times and crossed the lines $m = 2532$ times. If we assume that the sequence of needle tosses corresponds to a Bernoulli scheme with "success" probability (the needle crosses a line) $p = \frac{2 l}{a \pi }$ , then we can estimate the error of the experimental estimate of $\pi$ . 1. In Wolf's experiment the deviation $\left|\frac{m}{n}-p\right|$ does not exceed 0.0029. Find the number of needle tosses for which such a deviation occurs with probability greater than 0.5. 2. From the results of Wolf's experiment, find a confidence interval for the number $\pi$ .

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(a)

(b)

Problem 6.138

In January 1935 in Sweden (according to data of H. Cramér), out of a total of 7280 newborns, 3743 boys were born. The hypothesis about a specific value of the probability $p$ of a boy being born can be tested as follows. Suppose that a Bernoulli scheme can be used. From the given data, for a certain $0 < \alpha < 1$ , one needs to construct a confidence interval $\left[\bar{p}_{n}, \overline{\bar{p}}_{n}\right]$ for $p$ of level $1-\alpha$ , and then compare the hypothetical value $p_{0}$ with the bounds of the confidence interval: if $p_{0} \in \left[\bar{p}_{n}, \overline{\bar{p}}_{n}\right]$ , then the hypothesis that $p = p_{0}$ is considered consistent with the data, and in the case $p_{0} < \bar{p}_{n}$ or $p_{0} > \overline{\bar{p}}_{n}$ — the hypothesis is rejected, keeping in mind that the probability of an erroneous conclusion does not exceed $\alpha$ . Test the hypotheses $p_{0} = 0.5 ; p_{0} = 0.515 ; p_{0} = 0.55$ .

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Problem 6.139

An urn contains white and black balls. The only thing known about the composition of the urn is that the proportion of white balls is either 0.5 or 0.4. From the urn, 100 balls are drawn with replacement, and it is found that white balls make up the larger part of the sample. On the basis of this observation it is concluded that the proportion of white balls in the urn is 0.5. What is the probability that an erroneous conclusion has been made?

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Problem 6.140

Suppose $n$ Bernoulli trials with success probability $p$ have been carried out. Suppose the number of successes $S_{n}$ turns out to equal $m$ , $0 < m < n$ , and we need to check whether this is consistent with some hypothesis about the unknown value of $p$ . We can use the following criterion. Fix a number $\alpha , 0 < \alpha < 1$ , and, assuming the hypothesis $p = p_{0}$ is true, compute the probability $\mathbb {P}\left(S_{n} \geqslant m\right)$ . If this probability is less than $\alpha$ , we reject the hypothesis. Otherwise, we consider the hypothesis to be consistent with the statistical data.

In 1000 independent coin tosses, heads came up 540 times. Test the hypothesis that the coin is fair at $\alpha = 0.05$ .

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Problem 6.141

(Continuation) Let $\alpha$ be given. Assuming that some hypothesis $p = p_{0}$ is true, find the smallest integer $m_{\alpha }$ such that $\mathbb {P}\left(S_{n} \geqslant m_{\alpha }\right) \leqslant \alpha$ . A criterion for testing the hypothesis $p = p_{0}$ can be as follows: if $S_{n} \geqslant m_{\alpha }$ , we reject the hypothesis $p = p_{0}$ ; if $S_{n} < m_{\alpha }$ , we consider the hypothesis to be consistent with the data. The probability of an erroneous rejection of the hypothesis does not exceed $\alpha$ .

Test the hypothesis about the probability of a boy being born, from problem 6.138.

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Problem 6.142

Suppose that in a scheme of Bernoulli trials there are two hypotheses about the "success" probability, $p_{1}$ and $p_{2}, 0 < p_{1} < p_{2} < 1$ . To distinguish between these hypotheses, $n$ trials are carried out and $\left\{ S_{n} = m\right\}$ is obtained as a result. Let numbers $\alpha$ and $\beta , 0 < \alpha , \beta < 1$ , be given, and let $n$ be such that there exists a positive integer $m^{*}$ such that

\mathbf{Q}_{1} = \mathbb {P}_{p_{1}}\left(S_{n} > m^{*}\right) \leqslant \alpha , \quad \mathbf{Q}_{2} = \mathbb {P}_{p_{2}}\left(S_{n} \leqslant m^{*}\right) \leqslant \beta

where the first probability is computed under the assumption that $p = p_{1}$ , and the second under the assumption $p = p_{2}$ . Then a criterion for testing the hypotheses is constructed as follows: if $m > m^{*}$ , the hypothesis $p = p_{1}$ is rejected, and the hypothesis $p = p_{2}$ is considered acceptable; if $m \leqslant m^{*}$ , the reverse holds: the hypothesis $p = p_{1}$ is accepted, and $p = p_{2}$ is rejected. The probabilities $\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$ above are interpreted as the probabilities of erroneous conclusions.

Apply the proposed testing procedure to the de Méré problem (see problem 1.26), namely, answer the question: how many trials must be carried out to distinguish the two success probabilities $p_{1} = 1-\left(\frac{35}{36}\right)^{24}$ and $p_{2} = 1-\left(\frac{5}{6}\right)^{4}$ for given $\alpha = \beta = 0.05$ .

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Problem 6.143

Prove that the hypergeometric distribution, as $N \rightarrow \infty , M \rightarrow \infty , n \rightarrow \infty , \frac{n M}{N} \rightarrow \lambda >0$ , converges to the Poisson distribution with parameter $\lambda$ .

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Problem 6.144

Prove that the hypergeometric distribution, as $N \rightarrow \infty , M \rightarrow \infty , n \rightarrow \infty , \frac{n M}{N} \rightarrow \infty$ , converges to the normal distribution.

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Problem 6.145

A random variable $\chi_{n}^{2}$ has the chi-square distribution with $n$ degrees of freedom (this is the name given to the distribution with density

p(x) = \frac{1}{2^{\frac{n}{2}-1} \Gamma \left(\frac{n}{2}\right)} x^{n-1} e^{-x^{2} / 2}, x \geqslant 0,

see problem 3.167). Prove that the distribution of the normalized random variable $\frac{\chi_{n}^{2}-\mathbb {E}\left[\chi_{n}^{2}\right]}{\sqrt{\operatorname {Var}\left[\chi_{n}^{2}\right]}}$ is asymptotically normal with parameters (0,1).

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Problem 6.146

A random variable $\xi_{\lambda }$ has the Poisson distribution with parameter $\lambda > 0$ . Prove that as $\lambda \rightarrow \infty$ the random variable $\frac{\xi_{\lambda }-\lambda }{\sqrt{\lambda }}$ is asymptotically normal with parameters (0,1).

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Problem 6.147

Let the random variables $\xi_{1}, \ldots , \xi_{n}$ be mutually independent and have the same Poisson distribution with parameter $\lambda$ . Denote $\bar{\xi }_{n} = \frac{1}{n}\left(\xi_{1}+\ldots +\xi_{n}\right)$ . 1. Prove that $\bar{\xi }_{n} \xrightarrow {\mathbb {P}} \lambda$ as $n \rightarrow \infty$ . 2. Construct a confidence interval for $\lambda$ , i.e., indicate $\lambda_{1}\left(\bar{\xi }_{n}\right)$ and $\lambda_{2}\left(\bar{\xi }_{n}\right)$ such that for a given $0 < \alpha < 1$ ,

\mathbb {P}\left(\lambda _{1}\left(\bar{\xi }_{n}\right) \leqslant \lambda \leqslant \lambda _{2}\left(\bar{\xi }_{n}\right)\right) \geqslant 1-\alpha .

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Problem 6.148

The random variables $\xi_{1}, \ldots , \xi_{n}$ are mutually independent and have the same uniform distribution on the segment $[\theta -1 / 2$ , $\theta +1 / 2]$ . It is assumed that the value of $\theta$ is unknown. Suppose that for $n = 100$ the variables $\xi_{1}, \ldots , \xi_{n}$ have taken some specific values, so that the arithmetic mean $\bar{\xi } = \frac{1}{n}\left(\xi_{1}+\ldots +\xi_{n}\right)$ equals $a$ . Using the central limit theorem, find a confidence interval that contains the unknown value $\theta$ with probability approximately equal to 0.95.

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Problem 6.149

One needs to add a million numbers, each rounded to the fifth decimal place. Assuming that the rounding errors of all the numbers are mutually independent and have a uniform distribution on the corresponding interval, find the bounds within which, with probability 0.95, the total rounding error lies.

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Problem 6.150

Let $\alpha$ be an arbitrary irrational number. Consider the multiples of $\alpha$ , $n \alpha$ , for $n = 1, \ldots , N$ , and their fractional parts $\left\{ n \alpha \right\} = n \alpha -[n \alpha ]$ . Prove that the fractional parts $\left\{ n \alpha \right\}$ are distributed on the interval $[0,1)$ almost uniformly in the following sense: for any $\alpha$ and $\beta , 0 \leqslant \alpha < \beta < 1$ ,

\frac{N(\alpha , \beta )}{N} \rightarrow \beta -\alpha , \quad N \rightarrow \infty ,

where $N(\alpha , \beta )$ is the number of values $\left\{ n \alpha \right\}$ , for $n = 1, \ldots , N$ , contained in the interval $(\alpha , \beta )$ .

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Problem 6.151

Let a real random variable $\xi$ be distributed with density $p(x)$ having a derivative $p^{\prime }(x)$ absolutely integrable over the whole line. Consider the decimal expansion of $\xi$ :

\xi = [\xi ]+\frac{a_{1}(\xi )}{10}+\frac{a_{2}(\xi )}{10^{2}}+\ldots +\frac{a_{n}(\xi )}{10^{n}}+\delta ^{(n)}(\xi ),

where $a_{1}, \ldots , a_{n}$ are the first $n$ decimal digits, and $\delta^{(n)}(\xi )$ is the rounding error. Prove that the random variable $\Delta^{(n)} = 10^{n} \delta^{(n)}(\xi )$ has, as $n \rightarrow \infty$ , an asymptotically uniform distribution on the interval $[0,1)$ , i.e., as $n \rightarrow \infty$ and $0 \leqslant \alpha < \beta < 1$ ,

\mathbb {P}\left(\alpha \leqslant \Delta ^{(n)} \leqslant \beta \right) \rightarrow \beta -\alpha .

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Problem 6.152

Let the random variables $\xi_{1}, \ldots , \xi_{n}$ , for each $n$ , be mutually independent and identically distributed with distribution function $F(x)$ . Define, for each real $x$ , the random variable $\quad \widehat{F}_{n}(x) = \frac{1}{n} \mu_{n}(x)$ , where $\mu_{n}(x)$ is the number of $\xi_{1}, \ldots , \xi_{n}$ satisfying the inequality $\xi_{k} < x$ . Prove that for every $x$

\widehat{F}_{n}(x) \xrightarrow {\mathbb {P}} F(x) .

Will this convergence hold with probability 1?

$\widehat{F}_{n}(x)$ is called the empirical distribution function for $\xi_{1}, \ldots , \xi_{n}$ .

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Problem 6.153

(Continuation) Denote by $\alpha_{k}$ and $\mu_{k}$ the corresponding moments of the distribution of $\xi_{i}$ :

\begin{aligned} & \alpha _{k} = \mathbb {E}\left[\xi ^{k}\right] = \int _{-\infty }^{+\infty } x^{k} d F(x) \\ & \mu _{k} = \mathbb {E}\left[\left(\xi -\mathbb {E}\left[\xi \right]\right)^{k}\right] = \int _{-\infty }^{+\infty }\left(x-\alpha _{1}\right)^{k} d F(x) \end{aligned}

Consider the random variables

a_{1}^{(n)} = \frac{1}{n} \sum _{i = 1}^{n} \xi _{i}, \quad m_{2}^{(n)} = \frac{1}{n} \sum _{i = 1}^{n}\left(\xi _{i}-a_{1}^{(n)}\right)^{2}

Show that as $n \rightarrow \infty$ and $\varepsilon > 0$

\mathbb {P}\left(\left|a_{1}^{(n)}-\alpha _{1}\right| > \varepsilon \right) \rightarrow 0, \quad \mathbb {P}\left(\left|m_{2}^{(n)}-\mu _{2}\right| > \varepsilon \right) \rightarrow 0.

The random variables $a_{1}^{(n)}$ and $m_{2}^{(n)}$ are called, respectively, the sample mean and the sample variance, and the property asserted for them is called consistency.

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Problem 6.154

(Continuation) Prove the following statements. If $\alpha_{2} < \infty$ , then as $n \rightarrow \infty$ the quantity $a_{1}$ is asymptotically normal with parameters $\left(\alpha_{1}, \frac{\alpha_{2}-\alpha_{1}^{2}}{n}\right)$ .

If $\mu_{4} < \infty$ , then as $n \rightarrow \infty$ the quantity $m_{2}$ is asymptotically normal with parameters $\left(\mu_{2}, \frac{\mu_{4}-\mu_{2}^{2}}{n}\right)$ .

If $\alpha_{2 k} < \infty$ , then as $n \rightarrow \infty$ the quantity $a_{k}$ is asymptotically normal with parameters $\left(\alpha_{k}, \frac{\alpha_{2 k}-\alpha_{k}^{2}}{n}\right)$ .

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Problem 6.155

(the Monte Carlo method of statistical trials) The computation of the integral $I = \int_{0}^{1} f(x) d x$ can be described as follows. Let a random variable $\xi$ have the uniform distribution on the segment $[0,1]$ . Then

\mathbb {E}\left[f(\xi )\right] = \int _{0}^{1} f(x) d x = I

Let $\xi_{1}, \ldots , \xi_{n}$ be mutually independent and uniformly distributed on $[0,1]$ . Consider $\bar{f}_{n} = \frac{1}{n}\left[f\left(\xi_{1}\right)+\ldots +f\left(\xi_{n}\right)\right]$ and suppose that $\sigma^{2} = \operatorname {Var}\left[\bar{f}_{n}\right] \leqslant C$ . Show that $\mathbb {E}\left[\bar{f}_{n}\right] = I$ and $\bar{f}_{n} \xrightarrow {\mathbb {P}} I, n \rightarrow \infty$ . Estimate $\mathbb {P}\left(\left|\bar{f}_{n}-I\right| < \varepsilon \right)$ for arbitrary $\varepsilon > 0$ using the central limit theorem.

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Problem 6.156

(the Weierstrass theorem) Let $f(x)$ be a continuous function on the segment $[0,1]$ . Let a sequence of random variables $\xi_{1}, \ldots , \xi_{n}$ correspond to a Bernoulli scheme with success probability $\mathbb {P}\left(\xi_{i} = 1\right) = x, 0 < x < 1$ , and $S_{n} = \xi_{1}+\ldots +\xi_{n}$ . Introduce the polynomials

B_{n}(x) = \mathbb {E}\left[f\left(\frac{S_{n}}{n}\right)\right] = \sum _{m = 0}^{n} f\left(\frac{m}{n}\right) C_{n}^{m} x^{m}(1-x)^{n-m} .

Prove that as $n \rightarrow \infty$

\sup _{0 \leqslant x \leqslant 1}\left|f(x)-B_{n}(x)\right| \rightarrow 0.

(The polynomials $B_{n}(x)$ are called Bernstein polynomials.)

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Problem 6.157

Consider the binary expansion of a number $\xi \in [0,1)$

\xi = \frac{\xi _{1}}{2}+\frac{\xi _{2}}{2^{2}}+\ldots +\frac{\xi _{n}}{2^{n}}+\ldots

(with infinitely many zeros). Then, with respect to Lebesgue measure, the digits of the binary expansion $\xi_{1}, \xi_{2}, \ldots , \xi_{n}$ are independent identically distributed random variables with $\mathbb {P}\left(\xi_{i} = 1\right) = \mathbb {P}\left(\xi_{i} = 0\right) = 1 / 2$ .

Prove that for almost all numbers $\xi \in [0,1)$ the proportion of ones and zeros in the binary expansion of $\xi$ tends, with probability 1, to $1 / 2$ :

\frac{1}{n} \sum _{k = 1}^{n} I_{\left(\xi _{k} = 1\right)} \rightarrow \frac{1}{2} .

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