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14. Prove that if $X, Y \in L^{2}(\Omega, \mathscr{F}, \mathrm{P}), \mathrm{E}(X \mid Y)=Y, \mathrm{E}(Y \mid X)=X$, then $X=Y$ a.s. | Solution. By the property of iterated expectations
$$
\mathrm{E}(X-Y)^{2}=\mathrm{E}[X-\mathrm{E}(Y \mid X)] X-\mathrm{E}[\mathrm{E}(X \mid Y)-Y] Y=0
$$ | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,029 |
15. Let $X, Y$ be independent random variables, $\mathrm{E} X^{2}, \mathrm{E} Y^{2}<\infty$. Show that for any such variable $Z$, that $\mathrm{E} Z^{2}<\infty$ and $\mathrm{E} Z=0$, the following inequality holds:
$$
\mathrm{E}|\mathrm{E}(Z \mid X)|^{2}+\mathrm{E}|\mathrm{E}(Z \mid Y)|^{2} \leqslant \mathrm{E} Z^{2}
... | Solution. The random variable $\mathrm{E}(Z \mid X)$ is the projection of $Z$ onto the linear subspace (in the Hilbert space $L^{2}$)
$$
L_{X}^{2}=\left\{f(X): f-\text { Borel function, } \mathrm{E}|f(X)|^{2}<\infty, \mathrm{E} f(X)=0\right\}
$$
(we have taken into account that $\mathrm{EE}(Z \mid X)=\mathrm{E} Z=0$)... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,030 |
16. Given three sequences $\left(\mathscr{E}_{n}\right),\left(\mathscr{F}_{n}\right)$ and $\left(\mathscr{G}_{n}\right)$ of $\sigma$-subalgebras of the algebra $\mathscr{F}$. Let $\xi$ be a random variable, $\mathrm{E} \xi^{2}<\infty$, and assume that for each $n$ the conditions
$$
\mathscr{E}_{n} \subseteq \mathscr{F... | Solution. The desired relation is ensured by the estimate
$$
\mathrm{E}\left[\mathrm{E}\left(\xi \mid \mathscr{\mathscr { F }}_{n}\right)-\mathrm{E}\left(\xi \mid \mathscr{E}_{n}\right)\right]^{2} \leqslant \mathrm{E}\left[\mathrm{E}\left(\xi \mid \mathscr{G}_{n}\right)-\mathrm{E}\left(\xi \mid \mathscr{E}_{n}\right)\... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,031 |
17. Let $f=f(x)$ be a Borel function defined on $\mathbb{R}_{+}$ such that
$$
\int_{\mathbb{R}_{+}} e^{-\lambda x} f(x) d x=0 \quad \text { for all } \lambda \in \mathbb{N} \text {. }
$$
Show that $f=0$ a.e. with respect to Lebesgue measure. | Solution. Let's make a change of variables
$$
\int_{\mathbb{R}_{+}} e^{-\lambda x} f(x) d x=\int_{0}^{1} z^{\lambda-1} g(z) d z, \quad g(z)=f(-\ln z) .
$$
The function $g(z)$ is orthogonal on the interval $[0,1]$ to all polynomials. Polynomials form a dense set in the space $C[0,1]$, which in turn forms a dense set i... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,032 |
18. Let the random variable $\xi$ be uniformly distributed on $[-1,1]$. Show that
(a) the optimal (in the mean-square sense) estimates of $\xi^{2}$ based on $\xi$ and $\xi$ based on $\xi^{2}$ are given respectively by the formulas
$$
\mathrm{E}\left(\xi^{2} \mid \xi\right)=\xi^{2}, \quad \mathrm{E}\left(\xi \mid \xi^... | Solution. (a) Since the quantity $\xi^{2}$ is measurable with respect to $\sigma(\xi)$, we obtain that $\mathrm{E}\left(\xi^{2} \mid \xi\right)=\xi^{2}$. The quantity $\xi$ is symmetric, so $\xi^{2}$ does not depend on sign $\xi, \mathrm{E}(\operatorname{sign} \xi)=0$ and $\mathrm{E}\left(\xi \mid \xi^{2}\right)=|\xi| ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,033 |
19. Let $Y, X_{1}, X_{2}, \ldots$ be random variables in $L^{2}$. Suppose also that $X_{i} (i \geqslant 1)$ does not lie in the closure of the linear span in $L^{2}$ of the variables $X_{j}$, $j \neq i$ (the condition of relevance of the factor $X_{i}$). Define $\beta_{1 p}, \ldots, \beta_{p p}$ as the solution to the ... | Solution. (a) Due to the convexity of the function
$$
f\left(b_{1}, \ldots, b_{p}\right)=\mathrm{E}\left(Y-b_{1} X_{1}-\ldots-b_{p} X_{p}\right)^{2}
$$
the coefficients $\beta_{i p}$ can be found from the first-order conditions ($f^{\prime}=0$):
$$
\left(\beta_{1 p}, \ldots, \beta_{p p}\right)=\left(\mathbb{E} \bar{... | \beta_{i}=(-1)^{i+1} | Algebra | proof | Yes | Yes | olympiads | false | 34,034 |
1. Let $X$ be a random variable such that $\mathrm{E} e^{s X}<\infty, s \in \mathbb{R}$ and
$$
\int_{\mathbb{R}}\left|\mathrm{E} e^{(s+i t) X}\right| d t<\infty, \quad s, t \in \mathbb{R}
$$
The latter for $s=0$ guarantees the existence of a density $f=f(x)$ for $X$. Under these conditions, suggest some method for ob... | Solution. The finiteness of $\mathrm{E} e^{s X}$ is equivalent to the fact that the function $e^{s x} f(x)$ defines the density of some finite measure on $\mathbb{R}$, corresponding to the characteristic function $\mathrm{E} e^{(s+i t) X}$. Therefore, by the inversion formula,
$$
e^{s x} f(x)=\frac{1}{2 \pi} \int_{\ma... | \lim_{|x|arrow\infty}\frac{\lnf(x)}{e^{|x|}}=-\frac{e^{2\gamma}\pi}{4} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,035 |
2. Let $X$ be a random variable with density $f=f(x)$ and characteristic function $\varphi=\varphi(t)$. Prove the Poisson summation formula
$$
\sum_{n=-\infty}^{\infty} f(n)=\sum_{n=-\infty}^{\infty} \varphi(2 \pi n)
$$
assuming for simplicity that $X \stackrel{d}{=}-X$ and the series
$$
g(x)=\sum_{n=-\infty}^{\inft... | Solution. Due to the symmetry of $X$, we have $\varphi(t)=\varphi(-t), t>0$, therefore the limit
$$
\lim _{M, N \rightarrow \infty} \sum_{n=-M}^{N} \varphi(2 \pi n)
$$
exists if and only if the limit
$$
\lim _{N \rightarrow \infty} \sum_{n=-N}^{N} \varphi(2 \pi n)=\lim _{N \rightarrow \infty}\left[1+2 \sum_{n=1}^{N}... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,036 |
3. (See [57].) Prove Marcinkiewicz's theorem: The function $\varphi(t)=e^{P(t)}$ with a polynomial $P(t)$ can be a characteristic function only if $\operatorname{deg} P(t)$ is no greater than two. | Solution. If $e^{P(t)}=\mathrm{E} e^{i t X}, t \in \mathbb{R}$, for some random variable $X$, then $e^{P(-i z)}=\mathrm{E} e^{z X}, z \in \mathbb{C}$. It is not difficult to establish the latter by first verifying that $X$ has moments of all orders (as derivatives of the function $e^{P(t)}$ at zero up to a constant fac... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,037 |
4. If in a neighborhood of zero the characteristic function $\varphi=\varphi(t)$ coincides with some entire function $f=f(t)$, i.e., with a function that can be expanded on $\mathbb{R}$ into the series $\sum_{n \geqslant 0} c_{n} t^{n}, c_{n} \in \mathbb{C}$, then $\varphi \equiv f$ on $\mathbb{R}$. Prove this statemen... | Solution. The coefficients $c_{n}$ necessarily coincide with
$$
\frac{\varphi^{(n)}(0)}{n!}=\frac{i^{n} E X^{n}}{n!}
$$
where $X$ is a random variable with characteristic function $\varphi$. Therefore, the series
$$
\sum_{k \geqslant 0} \frac{i^{k} \mathrm{E} X^{k}}{k!} t^{k}
$$
converges on $\mathbb{R}$. Moreover,... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,038 |
5. Let $X$ and $Y$ be independent random variables, $f(x)=f_{1}(x)+i f_{2}(x), g(x)=g_{1}(x)+i g_{2}(x)$, where $f_{k}(x), g_{k}(x)$ are Borel functions, $k=1,2$. Show that if $\mathrm{E}|f(X)|<\infty, \mathrm{E}|g(Y)|<\infty$, then
$$
\mathrm{E}|f(X) g(Y)|<\infty \quad \text { and } \quad \mathrm{E} f(X) g(Y)=\mathrm... | Solution. The statement follows from the analogous result for Borel functions $f(x)$ and $g(x)$ with values in $\mathbb{R}$ (see problems II.5.4 and II.6.9), as well as from the fact that $(a+i b)(c+i d)=a c-b d+i(a d+b c)$ for all $a, b, c, d \in \mathbb{R}$. | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,039 |
6. Let $X=\left(X_{1}, \ldots, X_{n}\right)$ and $\mathrm{E}\|X\|^{n}<\infty$, where $\|X\|=\sqrt{\sum X_{i}^{2}}$. Show that
$$
\varphi(t)=\mathrm{E} e^{i(t, X)}=\sum_{k=0}^{n} \frac{i^{k} \mathrm{E}(t, X)^{k}}{k!}+o(1) \cdot\|t\|^{n}, \quad\|t\| \rightarrow 0
$$
where $t \in \mathbb{R}^{n}$. | Solution. By expanding $\cos x$ and $\sin x$ into a sum of powers of $x$ with the remainder term in the form of Taylor, one can obtain various upper bounds for the modulus of the difference
$$
\left|e^{i x}-\sum_{k=0}^{n} \frac{i^{k} x^{k}}{k!}\right|
$$
valid for any real $x$. For example, estimate it by $3|x|^{n+1}... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,040 |
7. Let $\mu$ and $\nu$ be two probability measures on ( $\mathbb{R}^{n}, \mathscr{B}\left(\mathbb{R}^{n}\right)$ ), such that
$$
\int_{\mathbb{R}^{n}} e^{i(t, x)} d \mu=\int_{\mathbb{R}^{n}} e^{i(t, x)} d \nu
$$
for all $t \in \mathbb{R}^{n}$. Prove that $\mu=\nu$. | Solution. Measures $\mu$ and $\nu$ on $\mathscr{B}(\mathbb{R})$ are defined by their values on "rectangles" of the form $\left(a_{1}, b_{1}\right] \times \ldots \times\left(a_{n}, b_{n}\right]$. Therefore, it is sufficient to prove the equality $\mu=\nu$ only on these sets.
Notice now that the measure of any "rectangl... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,041 |
8. Let $\varphi=\varphi(t)$ and $\psi=\psi(t)$ be the characteristic functions of some random variables $X$ and $Y$.
(a) Determine what the equality
$$
f(\varphi(t))=g(\psi(t)), \quad t \in \mathbb{R},
$$
can mean for some functions $f=f(x)$ and $g=g(x)$ that are analytic in the disk $B_{r}=\{x:|x|<r\}$ with $r>1$, ... | Solution. Let for $x \in B_{r}$ the expansions hold
$$
f(x)=\sum_{n=1}^{\infty} f_{n} x^{n}, \quad g(x)=\sum_{n=1}^{\infty} g_{n} x^{n}, \quad h(x)=\sum_{n=1}^{\infty} h_{n} x^{n}
$$
(a) The equality specified in the problem can be rewritten as
$\sum_{n: f_{n} \geqslant 0} f_{n}[\varphi(t)]^{n}-\sum_{n: g_{n}<0} g_{... | proof | Other | math-word-problem | Yes | Yes | olympiads | false | 34,042 |
9. Let $X_{1}, X_{2}, \ldots$ be i.i.d. random variables with characteristic function $\varphi=\varphi(t)$, and $S_{n}=X_{1}+\ldots+X_{n}, n \geqslant 1$. Denote
$$
\tau=\inf \left\{n \geqslant 1: S_{n}>0\right\}
$$
Establish that
$$
1-\mathrm{E} e^{i t S_{\tau}} s^{\tau}=\exp \left\{-\sum_{n=1}^{\infty} \frac{s^{n}... | Solution. Since $\{\tau \geqslant n\} \in \sigma\left(X_{1}, \ldots, X_{n-1}\right)$ for $n>1$ and $\{\tau \geqslant 1\}=\Omega$, the random vector $\left(S_{n-1}, I(\tau \geqslant n)\right)$ is independent of the value $X_{n}$. Therefore,
$$
\begin{aligned}
\psi(s, t)+\phi(s, t) & =\sum_{n=1}^{\infty} s^{n-1} \mathrm... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,043 |
11. Let $\varphi_{k}=\varphi_{k}(t), t \in \mathbb{R}, k \geqslant 1,$ be characteristic functions. Show that for any non-negative numbers $\lambda_{k}, \sum \lambda_{k}=1$, the function $\varphi=\sum \lambda_{k} \varphi_{k}$ is a characteristic function. | Solution. The function $\varphi$ is the characteristic function of a mixture of distributions $\mu_{k}$, corresponding to $\varphi_{k}$. In other words, for all $t \in \mathbb{R}$, the following equality holds:
$$
\varphi(t)=\int_{\mathbb{R}} e^{i t x} d \sum \lambda_{k} \mu_{k}
$$ | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,045 |
12. If $\varphi(t)$ is the characteristic function of some random variable $\xi$, will $\operatorname{Re} \varphi(t), \operatorname{Im} \varphi(t), \overline{\varphi(t)},|\varphi(t)|$ and $|\varphi(t)|^{2}$ also be characteristic functions? | Solution. We have $\operatorname{Re} \varphi(t)=\left(\mathrm{E} e^{i t \xi}+\mathrm{E} e^{-i t \xi}\right) / 2=\mathrm{E} e^{i t \xi X}, t \in \mathbb{R}$, where the random variable $X$ is independent of $\xi$ and $\mathrm{P}(X=1)=\mathrm{P}(X=-1)=1 / 2$. Moreover, $\overline{\varphi(t)}=\mathrm{E} e^{i t(-\eta)}$ and... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,046 |
14. Let $\varphi, \phi, \psi$ be characteristic functions and $\psi \varphi \equiv \psi \phi$. Does it follow that $\varphi \equiv \phi$? | Solution. No, it should not. According to Pólya's theorem, the functions $\psi(t)=(1-|t|)^{+}, \quad \varphi(t)=(1-|t / 2|)^{+}, \quad \phi(t)= \begin{cases}(1-|t / 2|)^{+}, & |t| \leqslant 1, \\ (3-|t|)^{+} / 4, & |t|>1,\end{cases}$
are characteristic and $\psi \varphi \equiv \psi \phi$, but $\varphi(t) \neq \phi(t)$ ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,048 |
15. Let $\xi$ be an integer-valued random variable and $\varphi_{\xi}(t)$ its characteristic function. Show that
$$
\mathrm{P}(\xi=n)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{-i n t} \varphi_{\xi}(t) d t, \quad k \in \mathbb{Z}
$$ | Solution. By Fubini's theorem
$$
\int_{-\pi}^{\pi} e^{-i n t} \varphi_{\xi}(t) d t=\mathrm{E} \int_{-\pi}^{\pi} e^{i(\xi-n) t} d t=2 \pi \mathrm{P}(\xi=n)
$$ | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,049 |
16. Show that in the space
$$
L^{2}=L^{2}([-\pi, \pi], \mathscr{B}([-\pi, \pi]))
$$
with the Lebesgue measure, the system of functions $\left\{\frac{e^{i \lambda n}}{\sqrt{2 \pi}}, n \in \mathbb{Z}\right\}$ forms an orthonormal basis. | Solution. The proof can be carried out according to the following scheme:
(a) for $\varepsilon>0$, find some $c>0$ such that
$$
\|\varphi-f\|_{L^{2}}<\varepsilon, \quad f(x)=\varphi(x) I(|\varphi(x)| \leqslant c)
$$
(b) by the Lusin's theorem (see problem II.10.35), find a continuous function $f_{\varepsilon}(x)$ su... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,050 |
17. Show that any characteristic function $\varphi=\varphi(t)$ for any $s$ and $t$ satisfies the inequality
$$
|\varphi(t-s)| \geqslant|\varphi(s) \varphi(t)|-\left[1-|\varphi(s)|^{2}\right]^{1 / 2}\left[1-|\varphi(t)|^{2}\right]^{1 / 2}
$$ | Solution. According to the Bochner-Hinchin theorem, the matrix
$$
\left(\begin{array}{ccc}
1 & \varphi(-t) & \varphi(-s) \\
\varphi(t) & 1 & \varphi(t-s) \\
\varphi(s) & \varphi(s-t) & 1
\end{array}\right)
$$
is non-negatively defined and, consequently, has a non-negative determinant. By direct verification, taking i... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,051 |
18. (a) In the Bochner-Hinchin theorem, it is assumed that the non-negative definite function \(\varphi(t)\) is continuous. Prove that the continuity of \(\varphi(t)\) at zero already ensures continuity on the entire real line. Derive this by establishing the inequality
\[
|\varphi(t)-\varphi(s)|^{2} \leqslant 2[1-\op... | Solution. (a) To prove the statement, it is sufficient to establish the given inequality. Reasoning as in problem II.12.17, we obtain that
$$
\left(1-|\varphi(t)|^{2}\right) \cdot\left(1-|\varphi(s)|^{2}\right) \geqslant|\varphi(t) \overline{\varphi(s)}-\varphi(t-s)|^{2}
$$
On the other hand, the non-negativity of th... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,052 |
19. (Hinchin's Criterion.) Show that the characteristic function $\varphi=\varphi(t)$ of any absolutely continuous distribution with density $f=f(x)$ can be represented in the form
$$
\varphi(t)=\int_{\mathbb{R}} \phi(t+s) \overline{\phi(s)} d s, \quad t \in \mathbb{R}
$$
for some complex-valued function $\phi$ satis... | Solution. Let
$$
\phi(t)=\frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} e^{i t x} \sqrt{f(x)} d x
$$
By the inversion formula (for finite measures with density $g(x)=\sqrt{f(x)}, x \in \mathbb{R})$, and the Fubini's theorem, we have
$$
\begin{aligned}
\frac{1}{2 \pi} \int_{\mathbb{R}} e^{i t x} d t \int_{\mathbb{R}} \phi(... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,053 |
20. (a) Prove that if $\varphi(t)$ is a characteristic function, then so is the function $\exp [\lambda(\varphi(t)-1)]$ for each $\lambda>0$.
(b) Using the equality
$$
C_{\alpha} \int_{\mathbb{R}_{+}} \frac{1-e^{-x}}{x^{1+\alpha}} d x=1
$$
valid for some positive $C_{\alpha}$, prove the Schoenberg theorem: if the fu... | Solution. (a) Let $\eta, \xi_{1}, \xi_{2}, \ldots$ be independent random variables, where $\eta$ is distributed according to the Poisson law with parameter $\lambda$, and $\xi_{n}$ has the characteristic function $\varphi(t)$. Then, as is not difficult to verify, $\exp [\lambda(\varphi(t)-1)]$ is the characteristic fun... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,054 |
21. Which of the functions
$$
\begin{aligned}
& \varphi(t)=\exp \left(-|t|^{\alpha}\right), \quad \alpha \in(0,2], \\
& \varphi(t)=\exp \left(-|t|^{\alpha}\right), \quad \alpha>2, \\
& \varphi(t)=\frac{1}{\left(1+|t|^{\alpha}\right)^{\beta}}, \quad \alpha \in(0,2], \beta>0 \\
& \varphi(t)=\frac{1}{\left(1+|t|^{\alpha}... | Solution. For the functions $\exp \left(-|t|^{\alpha}\right)$ and $\left(1+|t|^{\alpha}\right)^{-\beta}$ when $\alpha>2$, as well as for the function $1-|t|^{3} \wedge 1$ at the point $t=0$, the second derivative exists and is equal to zero. Since
$$
\mathrm{E} \xi^{2}=-\varphi_{\xi}^{\prime \prime}(0)=0 \Rightarrow \... | proof | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,055 |
22. Prove that the function
$$
\varphi(t)= \begin{cases}\sqrt{1-t^{2}}, & |t| \leqslant 1 \\ 0, & |t|>1\end{cases}
$$
is not a characteristic function.
Is the function $\varphi(t)=\frac{\sin t}{t}$ a characteristic function? | Solution. If $\varphi(t)=\sqrt{1-t^{2}} \cdot I(|t| \leqslant 1)=\mathrm{E} e^{i t \xi}$ is the characteristic function for some random variable $\xi$, then $\xi$ has a second moment $\mathrm{E} \xi^{2}=-\varphi^{\prime \prime}(0)$. This implies that the function $\varphi(t)$ is differentiable for any $t \in \mathbb{R}... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,056 |
23. Let $\varphi=\varphi(t)$ be the characteristic function of a non-degenerate distribution $F=F(x)$. Show that if $\varphi \in C^{2 n}, n \geqslant 1$, then the function $\varphi^{(2 n)}(t) / \varphi^{(2 n)}(0)$ is also a characteristic function. | Solution. From the non-degeneracy of the distribution, it follows that $\varphi^{(2 n)}(0) \neq 0$. Now the statement follows from the equality
$$
\frac{\varphi^{(2 n)}(t)}{\varphi^{(2 n)}(0)}=\frac{\int_{\mathbb{R}} x^{2 n} e^{i t x} d F(x)}{\int_{\mathbb{R}} x^{2 n} d F(x)}=\int_{\mathbb{R}} e^{i t x} d G(x)
$$
whe... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,057 |
24. Let $\varphi(t)$ be a characteristic function. Show that the following functions are also characteristic:
$$
\int_{0}^{1} \varphi(u t) d u, \quad \int_{\mathbb{R}_{+}} e^{-u} \varphi(u t) d u
$$ | Solution. If $\varphi(t)=\mathrm{E} e^{i t \xi}$, then
$$
\int_{0}^{1} \varphi(u t) d u=\mathrm{E} e^{i t \xi \eta}, \quad \int_{\mathbb{R}_{+}} e^{-u} \varphi(u t) d u=\mathrm{E} e^{i t \xi \zeta}
$$
where $\eta, \zeta$ are independent of $\xi$, the random variable $\eta$ has a uniform distribution on the interval $... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,058 |
25. Show that for each $n \geqslant 1$ the functions
$$
\varphi_{n}(t)=\frac{e^{i t}-\sum_{k=0}^{n-1}(i t)^{k} / k!}{(i t)^{n} / n!}
$$
are characteristic. | Solution. The given function is the characteristic function of a random variable with the density
$$
f(x)= \begin{cases}n(1-x)^{n-1}, & x \in[0,1] \\ 0, & x \notin[0,1]\end{cases}
$$ | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,059 |
26. Let $\varphi_{n}(t)$ be the characteristic function of a random variable uniformly distributed on $(-n, n)$. Show that
$$
\lim _{n \rightarrow \infty} \varphi_{n}(t)= \begin{cases}1, & t=0 \\ 0, & t \neq 0\end{cases}
$$ | Solution. The statement follows from the formula
$$
\varphi_{n}(t)=\frac{\sin (n t)}{n t}
$$ | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,060 |
27. Let $X$ and $Y$ be independent random variables with characteristic functions $\varphi=\varphi(t)$ and $\psi=\psi(t)$ and distribution functions $F=F(x)$ and $G=G(x)$. For functions $f=f(t)$ and $g=g(t)$ with values in $\mathbb{C}$, denote by $M(f, g)$ the limit (as $T \rightarrow \infty$) of the integrals
$$
\fra... | Solution. By Fubini's theorem
$$
M\left(\varphi, e^{i t x}\right)=\frac{1}{2 T} \int_{-T}^{T} e^{-i t x} \varphi(t) d t=\mathrm{E} \frac{1}{2 T} \int_{-T}^{T} e^{i t(X-x)} d t=\mathrm{E} \frac{\sin T(X-x)}{T(X-x)}
$$
The last expression, by the Lebesgue dominated convergence theorem, tends to
$$
\mathrm{P}(X-x=0)=F(... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,061 |
28. In this problem, following the work [70], we are asked to establish the inequality
$$
1-\operatorname{Re} \varphi(n t) \leqslant n\left[1-\operatorname{Re}^{n} \varphi(t)\right] \leqslant n^{2}[1-\operatorname{Re} \varphi(t)]
$$
valid for all $n \geqslant 1, t \in \mathbb{R}$ and the characteristic function $\var... | Solution. (a) The given estimate can be easily obtained by induction, noting that
$$
\left|\frac{\sin (n+1) x}{\sin x}\right|=\left|\frac{\sin n x}{\sin x} \cos x+\cos n x\right| \leqslant\left|\frac{\sin n x}{\sin x}\right|+1
$$
(b) We need to show that $f(x)=n \cos ^{n} x-\cos n x \leqslant n-1$ for all $x \in \mat... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,062 |
29. Show that every characteristic function $\varphi=\varphi(t)$ satisfies the following inequalities (for all $t \in \mathbb{R}$):
$$
\begin{gathered}
|\operatorname{Im} \varphi(t)|^{2} \leqslant \frac{1-\operatorname{Re} \varphi(2 t)}{2}, \quad|\operatorname{Re} \varphi(t)|^{2} \leqslant \frac{1+\operatorname{Re} \v... | Solution. Let $\varphi(t)=\mathrm{E} e^{i t X}, t \in \mathbb{R}$. By Lyapunov's inequality, we have
$$
|\operatorname{Im} \varphi(t)|^{2} \leqslant \mathrm{E} \sin ^{2} t X=\frac{\mathrm{E}(1-\cos 2 t X)}{2}=\frac{1-\operatorname{Re} \varphi(2 t)}{2}
$$
Similarly, an inequality for $|\operatorname{Re} \varphi(t)|^{2... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,063 |
30. Let $X$ be a random variable with characteristic function $\varphi=\varphi(t)$. Show that
(a) if $\left|\varphi\left(t_{n}\right)\right|=1+o\left(t_{n}^{2}\right)$ for some $t_{n} \rightarrow 0$, then $X=a$ a.s. for some $a \in \mathbb{R}$;
(b) if $\left|\varphi\left(t_{n}\right)\right|=1+O\left(t_{n}^{2}\right)$... | Solution. (a) We have $\left|\varphi\left(t_{n}\right)\right|^{2}=1+o\left(t_{n}^{2}\right)$. Moreover, $|\varphi(t)|^{2}$ is the characteristic function of the random variable $X-Y$, where $Y$ is an independent copy of the random variable $X$. We also have
$$
\begin{aligned}
0 & =\lim _{n} \frac{1-\left|\varphi\left(... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,064 |
31. Let $\xi$ be a random variable with characteristic function $\varphi=\varphi(t)$. Show that
$$
r^{\alpha} \mathrm{P}(|\xi|>r)=O(1), \quad r \rightarrow \infty
$$
is equivalent to
$$
|\varphi(t)|=1+O\left(|t|^{\alpha}\right), \quad t \rightarrow 0
$$
In the case $\alpha=2$, the restriction on $\varphi(t)$ is (on... | Solution. We will prove the necessity of condition (*). Let $\eta$ be an independent copy of the random variable $\xi$. Then
$|\varphi(t)|^{2}=\mathrm{E} e^{i t(\xi-\eta)}=\mathrm{E} \cos (t(\xi-\eta)), \quad \mathrm{P}(|\xi-\eta|>r) \leqslant 2 \mathrm{P}(|\xi|>r / 2) \leqslant \frac{C}{r^{\alpha}}$ for some constant... | proof | Other | proof | Yes | Yes | olympiads | false | 34,065 |
32. (a) Let $\xi$ and $\eta$ be i.i.d. random variables with zero mean and finite variance. Prove that if
$$
\xi \stackrel{d}{=} \frac{\xi+\eta}{\sqrt{2}}
$$
then the variable $\xi$ is Gaussian.
(b) Show that the result of part (a) remains valid if
$$
\xi \stackrel{d}{=} a \xi + b \eta
$$
for some $a, b > 0$.
(c) ... | Solution. (a) We use the central limit theorem. As $n \rightarrow \infty$, we have
$$
\xi \stackrel{d}{=} \frac{\xi_{1}+\xi_{2}}{\sqrt{2}} \stackrel{d}{=} \ldots \stackrel{d}{=} \frac{\xi_{1}+\ldots+\xi_{2^{2 n}}}{2^{n}} \stackrel{d}{\rightarrow} \mathscr{N}\left(0, \mathrm{E} \xi^{2}\right)
$$
where $\xi_{1}, \xi_{2... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,066 |
33. Show that if the distribution function $F=F(x)$ has a density $f=f(x)$, then its characteristic function $\varphi=\varphi(t)$ has the property that
$$
\lim _{t \rightarrow \infty} \varphi(t)=0
$$ | Solution. We have
$$
\varphi(t)=\int_{\mathbb{R}} e^{i t x} f(x) d x=-\int_{\mathbb{R}} e^{i t x-i \pi} f(x) d x=-\int_{\mathbb{R}} e^{i t y} f(y+\pi / t) d y
$$
and according to problem II. 10.37, we get that
$$
\begin{aligned}
&|\varphi(t)|=\frac{1}{2}\left|\int_{\mathbb{R}} e^{i t x}[f(x+\pi / t)-f(x)] d x\right|... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,067 |
35. In this problem, following the book [40], it is proposed to establish that
$$
\varlimsup_{t \rightarrow \infty}|\varphi(t)|=1
$$
for the characteristic function $\varphi(t)$ of an arbitrary discrete random variable $\xi$.
To prove this, one should first consider the Fejér kernel
$$
F_{n}(t)=\frac{1}{n} \sum_{k=... | Solution. The Fejér kernel admits representations of the form
$$
F_{n}(t)=\frac{1}{n} \sum_{k=0}^{n-1} \frac{\sin (k+1 / 2) t}{\sin t / 2}=\frac{1}{n}\left[\frac{\sin n t / 2}{\sin t / 2}\right]^{2}
$$
since for all $t \in \mathbb{R}$ the equalities hold
$$
\begin{gathered}
\sum_{|j| \leqslant k} e^{i j t}=1+2 \sum_... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,069 |
36. A function $\phi=\phi(\lambda), \lambda>0$, is called completely monotonic if there exist derivatives $\phi^{(n)}(\lambda)$ of all orders $n \geqslant 0$ for each $\lambda>0$ and $(-1)^{n} \phi^{(n)}(\lambda) \geqslant 0$. Prove the following Bernstein criterion.
A function $\phi=\phi(\lambda)$ on $(0, \infty)$ is... | Solution. To prove necessity, it is necessary to note that if $\phi(\lambda)=\mathrm{E} e^{-\lambda x}, \lambda>0$, then for all $n \geqslant 0$ (and $\lambda>0$) the relation holds
$$
(-1)^{n} \phi^{(n)}(\lambda)=\mathrm{E} X^{n} e^{-\lambda X} \geqslant 0
$$
The existence of all derivatives can easily be derived fr... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,070 |
38. If $\xi \sim \mathscr{N}(0, v), v>0$, then
$$
\mathrm{E} e^{-\xi}=\int_{-\infty}^{\infty} e^{-u} \frac{e^{-\frac{u^{2}}{2 v}}}{\sqrt{2 \pi v}} d u=e^{\frac{v}{2}}
$$
Compute the integral $(u>0)$
$$
\int_{0}^{\infty} e^{-v} \frac{e^{-\frac{u^{2}}{2 v}}}{\sqrt{2 \pi v}} d v
$$ | Solution. The Boolean transformation $x \rightarrow x-\lambda / x, x \neq 0(\lambda>0)$, preserves the Lebesgue measure on $\mathbb{R}$ (see problem II.6.103). Therefore,
$$
\begin{aligned}
& \int_{0}^{\infty} e^{-v} \frac{e^{-\frac{u^{2}}{2 v}}}{\sqrt{2 \pi v}} d v=\sqrt{2} \int_{0}^{\infty} e^{-z^{2}} \frac{e^{-\fra... | \frac{e^{-\sqrt{2}u}}{\sqrt{2}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,072 |
39. Let $X$ be a random variable with characteristic function $\varphi=\varphi(t)$ such that $\int_{\mathbb{R}}|\varphi(t)| d t<\infty$. Using the fact that $\psi(t)=(1-|t|)^{+}$ is the characteristic function of a distribution with density
$$
g(x)=\frac{1-\cos x}{\pi x^{2}}, \quad x \in \mathbb{R}
$$
prove that $X$ ... | Solution. Let $Y$ be a random variable independent of $X$ and having density $g(x)$. Then the density $f_{\sigma}=f_{\sigma}(x)$ of the variable $X+\sigma Y (\sigma>0)$ is
$$
f_{\sigma}(x)=\mathrm{E} \frac{1}{\sigma} g\left(\frac{x-X}{\sigma}\right), \quad x \in \mathbb{R}
$$
Obviously, $X+\sigma Y \xrightarrow{d} X,... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,073 |
40. Let $X$ be a random variable with density $f=f(x)$ and characteristic function $\varphi=\varphi(t)$. Suppose that either of the following conditions is satisfied:
(a) $\int_{\mathbb{R}}|\varphi(t)|^{2} d t<\infty \quad$ or
(b) $\int_{\mathbb{R}} f^{2}(x) d x<\infty$.
Using the inversion formula from problem II.12.... | Solution. Let condition (a) be satisfied. Note that $\psi(t)=$ $=|\varphi(t)|^{2}, t \in \mathbb{R}$, is the characteristic function of the variable $X-Y$, where $Y$ is an independent copy of the variable $X$. In turn, the density of the variable $X-Y$ is
$$
g(z)=\int_{\mathbb{R}} f(x-z) f(x) d x
$$
where the conside... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,074 |
41. Let $X$ and $Y$ be independent random variables, $\varphi=\varphi(t)$ and $\phi=\phi(t)$ be their characteristic functions. Prove the validity of the Parseval's relation:
$$
\mathrm{E} \phi(X-t)=\mathrm{E} e^{-i t Y} \varphi(Y)
$$
for each $t \in \mathbb{R}$.
From this, derive that for any $\sigma>0$ the followi... | Solution. All integrals exist, so by Fubini's theorem
$$
\mathrm{E} \phi(X-t)=\mathrm{E} e^{i(X-t) Y}=\mathrm{E} e^{-i t Y} \varphi(Y)
$$
Assuming now that $Y \sim \mathscr{N}\left(0, \sigma^{-2}\right)$, we obtain the second relation. | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,075 |
42. Derive from the equality $(**)$, established in the previous problem, that if the distribution functions $F$ and $G$ have the same characteristic function, then $F=G$. | Solution. Let $X$ and $Y$ be random variables with distribution functions $F$ and $G$. From equality (**) it follows that for all $t \in \mathbb{R}$ and $\sigma>0$ the relation
$$
\mathrm{E} e^{-\frac{(t-X)^{2}}{2 \sigma^{2}}}=\mathrm{E} e^{-\frac{(t-Y)^{2}}{2 \sigma^{2}}}
$$
holds, and thus, for any $x \in \mathbb{R... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,076 |
43. Show that if $\varphi(t)$ is the characteristic function of a random variable $\xi$, then the Laplace transform of the magnitude $|\xi|$ is given by the formula
$$
\mathrm{E} e^{-\lambda|\xi|}=\frac{1}{\pi} \int_{\mathbb{R}} \frac{\lambda \varphi(t)}{\lambda^{2}+t^{2}} d t, \quad \lambda>0
$$ | Solution. Consider the quantity $\eta$, independent of $\xi$ and having a Cauchy distribution with density $\frac{1}{\pi\left(1+x^{2}\right)}$ and characteristic function $e^{-|t|}$. By Fubini's theorem, we obtain
$$
\mathrm{E} e^{-\lambda|\xi|}=\mathrm{E} e^{i \lambda \xi \eta}=\mathrm{E} \varphi(\lambda \eta)=\int_{... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,077 |
44. Let $F=F(x)$ be a distribution function and $\varphi(t)=\int_{\mathbb{R}} e^{i t x} d F(x)$ its characteristic function. According to statement b) of Theorem 3 from V1.II. 12, the property $\int_{\mathbb{R}}|\varphi(t)| d t<\infty$ ensures the existence of a continuous density $f(x)$.
Provide an example where the ... | Solution. We can take a function $\varphi(t)$ that satisfies the conditions of Pólya's theorem, and such that
$$
\int_{\mathbb{R}}|\varphi(t)| d t=\infty \quad \text { and } \quad(\varphi(t))_{t \geqslant 0} \in C^{1}
$$
and then use the remark following problem II.12.61. The condition $(\varphi(t))_{t \geqslant 0} \... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 34,078 |
45. Let $\varphi=\varphi(t)$ be a characteristic function. Prove that
(a) $\varphi$ corresponds to a distribution with a bounded density $f=f(x)$, if
$$
\int_{\mathbb{R}}|\operatorname{Re} \varphi(t)| d t<\infty$ and there exists a singular distribution for which $\int_{\mathbb{R}}|\operatorname{Re} \varphi(t)|^{p} d... | Solution. Let $X$ be a random variable with characteristic function $\varphi$ and $\xi$ be a random variable independent of $X$, $\mathrm{P}(\xi=1)=\mathrm{P}(\xi=-1)=1 / 2$. Then
$$
\operatorname{Re} \varphi(t)=\mathrm{E} \cos t X=\mathrm{E} \frac{e^{-i t X}+e^{i t X}}{2}=\mathrm{E} e^{i t \xi X}
$$
is the character... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,079 |
46. (a) Show that the function
$$
f(x)=\frac{1-\cos x}{\pi x^{2}}=\frac{1}{2 \pi}\left[\frac{\sin (x / 2)}{x / 2}\right]^{2}, \quad x \in \mathbb{R}
$$
is the density of some probability distribution. The characteristic function corresponding to $f$ is
$$
\varphi(t)= \begin{cases}1-|t|, & |t| \leqslant 1 \\ 0, & |t|... | Solution. (a) The function $\varphi$ is absolutely integrable on $\mathbb{R}$, so the density corresponding to $\varphi$ can be found using the inversion formula
$$
\begin{aligned}
f(x)=\frac{1}{2 \pi} \int_{\mathbb{R}} e^{-i t x} \varphi(t) d t=\frac{1}{\pi} \int_{0}^{1} \cos (t x)(1 & -t) d t \\
= & \frac{1}{\pi} \i... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,080 |
47. Let $f=f(x)$ be the density of some probability distribution, $\varphi=\varphi(t)$ its characteristic function. Using problem II.12.46(a), verify that for all $\sigma>0$ the function
$$
f_{\sigma}(x)=\frac{1}{2 \pi} \int_{\sigma|t| \leqslant 1} e^{-i t x} \varphi(t)(1-\sigma|t|) d t
$$
is a density, and that
$$
... | Solution. According to problem II.12.46(a), we have
$$
\varphi(t)(1-\sigma|t|)^{+}=\mathrm{E} e^{i t(X+\sigma Y)}
$$
where the quantities $X, Y$ are independent, $X$ has density $f(x)$, and the density of the quantity $Y$ is
$$
g(x)=\frac{1}{2 \pi}\left[\frac{\sin (x / 2)}{x / 2}\right]^{2}
$$
Therefore, due to the... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,081 |
48. (a) Show that the series
$$
\sum_{n \geqslant 1} a_{n} \sin n t
$$
converges uniformly on $\mathbb{R}$ when $n a_{n} \rightarrow 0$ and $a_{n} \geqslant a_{n+1} \geqslant 0$ for all $n$.
(b) Verify by example that the smoothness on $\mathbb{R}$ of the characteristic function $\varphi(t)$ of a random variable $\x... | Solution. (a) Due to the periodicity of the function $f(x)=\sin n x$, it is sufficient to prove uniform convergence for $t \in(0, \pi]$. We will need auxiliary relations of the form
$$
s_{n}=\sum_{l=1}^{n} \sin l t=\frac{\sin \frac{(n+1) t}{2} \cdot \sin \frac{n t}{2}}{\sin \frac{t}{2}}
$$
$$
\left|s_{n}\right| \leqs... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,082 |
49. Let $\varphi=\varphi(t)$ be the characteristic function of a random variable $X$.
(a) Show that $\mathrm{E}|X|^{p}<\infty$ for a given $p, 0<p<2$, if and only if
$$
\int_{\mathbb{R}} \frac{1-\operatorname{Re} \varphi(t)}{|t|^{1+p}} d t<\infty
$$
and in this case
$$
\mathrm{E}|X|^{p}=C_{p} \int_{\mathbb{R}} \fra... | Solution. (a) The statement follows from Fubini's theorem and the equality
$$
|x|^{p}=C_{p} \int_{\mathbb{R}} \frac{1-\cos x t}{|t|^{1+p}} d t, \quad 0<p<2,
$$
valid for the specified $C_{p}$.
(b) First, note that the derivative $[\operatorname{Re} \varphi(t)]^{\prime}=\operatorname{Re} \varphi^{\prime}(t)=$ $=\math... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,083 |
50. Moments $\mathrm{E} X^{n}$ and absolute moments $\mathrm{E}|X|^{n}$ of a random variable $X$ can be expressed for integers $n \geqslant 1$ through the $n$-th order derivatives of the characteristic function $\varphi(t)=\mathrm{E} e^{i t X}, t \in \mathbb{R}$. For fractional $\alpha>0$, to obtain the corresponding r... | Solution. If condition 1 is satisfied, then $\varphi^{(n)}(t)$ exist for all $t \in \mathbb{R}$. If, in addition, condition 2 holds, then
$$
\left.\frac{d^{n+a} \operatorname{Re} \varphi(t)}{d t^{n+a}}\right|_{t=0}=\frac{(-1)^{n / 2} a}{\Gamma(1-a)} \int_{\mathbb{R}_{+}} \mathrm{E} X^{n}(1-\cos s X) \frac{d s}{s^{1+a}... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,084 |
51. Let $X$ be a non-negative random variable with Laplace transform $\phi(\lambda)=\mathrm{E} e^{-\lambda X}, \lambda \geqslant 0$.
(a) Show that for any $0 < p$ and $\lambda > 0$, the following equality holds:
$$
\mathrm{E} X^{-p}=\frac{1}{\Gamma(p)} \int_{\mathbb{R}_{+}} \phi(\lambda) \lambda^{p-1} d \lambda
$$ | Solution. The desired representations follow (by Fubini's theorem) from the following formulas, valid for $x \geqslant 0$:
$$
\begin{aligned}
x^{p} & =\frac{p}{\Gamma(1-p)} \int_{\mathbb{R}_{+}} \frac{1-e^{-\lambda x}}{\lambda^{p+1}} d \lambda, & & p \in(0,1) \\
x^{-p} & =\frac{1}{\Gamma(p)} \int_{\mathbb{R}_{+}} e^{-... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,085 |
52. Let $X$ be a non-negative random variable.
(a) The double Laplace transform or Stieltjes transform of the variable $X$ is the function $\bar{\psi}(\lambda)=$ $=\mathrm{E}(X+\lambda)^{-1}, \lambda>0$. Prove that for $0<\lambda<\infty$, $\bar{\psi}(\lambda)$ is a completely monotone function. Show that for any $0<p<... | Solution. The required relations follow from the equalities
$$
\begin{aligned}
x^{-p} & =\frac{1}{\Gamma(p) \Gamma(1-p)} \int_{\mathbb{R}_{+}} \frac{\lambda^{-p}}{x+\lambda} d \lambda \\
x^{p} & =\frac{p}{\Gamma(p) \Gamma(1-p)} \int_{\mathbb{R}_{+}} \ln (1+x / \lambda) \lambda^{p-1} d \lambda
\end{aligned}
$$
valid f... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,086 |
53. Let $X, Y$ be independent random variables with distribution $\mathscr{N}(0,1)$, and let the variable $C$ have a Cauchy distribution with density $\frac{1}{\pi\left(1+x^{2}\right)}, x \in \mathbb{R}$.
Prove by direct calculation of the characteristic function of the variable $X / Y$ that $X / Y \stackrel{d}{=} C$. | Solution. Using the independence of the variables $X$ and $Y$, we find that
$$
\begin{aligned}
& \mathrm{E} e^{i t X / Y}=\frac{1}{2 \pi} \int_{\mathbb{R}^{2}} e^{i t x / y-\frac{x^{2}+y^{2}}{2}} d x d y=\frac{1}{2 \pi} \int_{0}^{2 \pi} \int_{0}^{\infty} e^{i t \operatorname{tg} \theta-\frac{r^{2}}{2}} r d \theta d r=... | proof | Other | proof | Yes | Yes | olympiads | false | 34,087 |
54. Let $\xi_{1}, \xi_{2}, \ldots$ be independent random variables distributed according to $\mathscr{N}(0,1)$. For $n \geqslant 1$ we set
$$
X_{n}=\xi_{1}+\ldots+\xi_{n} \quad \text { and } \quad S_{n}=X_{1}+\ldots+X_{n}
$$
Using the method of characteristic functions, show that
$$
\frac{S_{n}}{n^{3 / 2}} \xrightar... | Solution. The statement follows from the relation
$$
\ln \varphi_{S_{n} / n^{3 / 2}}(t)=-\frac{t^{2}}{2 n^{3}} \sum_{k=0}^{n-1}(n-k)^{2} \rightarrow-\frac{t^{2}}{6}, \quad n \rightarrow \infty
$$ | proof | Other | proof | Yes | Yes | olympiads | false | 34,088 |
55. Let $\xi$ and $\eta$ be independent random variables, and $\eta \sim \mathscr{N}(0,1)$. Let also $f=f(x)$ be a bounded Borel function with compact support. Show that for any $\sigma>0$ the following equality holds:
$$
\mathrm{E} f(\xi+\sigma \eta)=\frac{1}{2 \pi} \int_{\mathbb{R}} e^{-\frac{\sigma^{2} t^{2}}{2}} \... | Solution. The characteristic function of the quantity $\zeta=\xi+\sigma \eta$ is absolutely integrable on $\mathbb{R}$ and equals $e^{-\frac{\sigma^{2} t^{2}}{2}} \varphi(t)$. This means that the density of the quantity $\zeta$ is
$$
\frac{1}{2 \pi} \int_{\mathbb{R}} e^{-i t x-\frac{\sigma^{2} t^{2}}{2}} \varphi(t) d ... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,089 |
56. Using the equality (*) from the previous problem, prove that the characteristic function $\varphi(t)$ of any random variable $\xi$ determines its distribution | Solution. Since $\xi+\sigma \eta \rightarrow \xi$ a.s. as $\sigma \rightarrow 0$, we obtain that $f(\xi+\sigma \eta) \rightarrow f(\xi)$ with probability one, when $\mathrm{P}\left(\xi \in \Delta_{f}\right)=0\left(\Delta_{f}-\right.$ the set of discontinuity points of the function $f$ ). Therefore, by the Lebesgue domi... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,090 |
57. Let $\varphi=\varphi(t)$ be a characteristic function such that for some $b>0$ the condition
$$
|\varphi(t)| \leqslant a \quad \text { for } \quad|t| \geqslant b
$$
holds, where $0<a<1$. Show that then for $|t|<b$ the Cramér inequality
$$
|\varphi(t)| \leqslant 1-\left(1-a^{2}\right) \frac{t^{2}}{8 b^{2}}
$$
is... | Solution. For $t=0$ the inequality is trivial. Let $|t|>0$ and $n \in \mathbb{N}$ such that $2^{n-1}|t|<b \leqslant 2^{n}|t|$. It remains to apply the inequality $1-|\varphi(2 t)|^{2} \leqslant 4\left(1-|\varphi(t)|^{2}\right)$ from II.12.29, which provides the desired estimate
$$
1-a^{2} \leqslant 1-\left|\varphi\lef... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,091 |
58. (a) (Barr, Esseen.) Let $\varphi=\varphi(t)$ be the characteristic function of a random variable $\xi$. Show that for any $t \neq 0$ the following inequalities hold:
\[
\begin{aligned}
& \left|\frac{1-\varphi(t)}{t^{p}}\right| \leqslant C_{p} \mathrm{E}|\xi|^{p}, \quad 0 < p \leqslant 2, \\
& \left|\frac{1-\varphi... | Solution. (a) For $p \in(0,1]$ we have
$$
|1-\varphi(t)| \leqslant \mathrm{E}\left|e^{i t \xi}-1\right| \leqslant \mathrm{E}|t \xi| I(|t \xi| \leqslant 1)+2 \mathrm{P}(|t \xi|>1) \leqslant 3 \mathrm{E}|t \xi|^{p}
$$
Now let $p \in(1,2]$. Then
$|1-\operatorname{Re} \varphi(t)| \leqslant \mathrm{E}[1-\cos (t \xi)] \le... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,092 |
59. (Drier.) Let $(\xi, \eta)$ be a random vector uniformly distributed on the set
$$
A=\left\{(x, y) \in[-1,1]^{2}: \frac{x}{y}>1 \text { or } \frac{y}{x}<-1\right\}
$$
Show that the random variables $\xi$ and $\eta$ are dependent, uniformly distributed on $[-1,1]$, but
$$
\varphi_{\xi+\eta}(t)=\varphi_{\xi}(t) \va... | Solution. Analyzing the graphical representation of the set $A$ on the plane, it is easy to verify that $\xi$ and $\eta$ are uniformly distributed on $[-1,1]$ and $\xi+\eta \stackrel{d}{=} \xi-\eta$. Therefore, $\xi$ and $\eta$ are dependent (otherwise the vector $(\xi, \eta)$ would be uniformly distributed on $\left.[... | \frac{1-\cos2}{2^{2}} | Algebra | proof | Yes | Yes | olympiads | false | 34,093 |
60. Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of independent identically distributed random variables taking values $0,1, \ldots, 9$ with probability $1 / 10$. Show that the series
$$
\sum_{n \geqslant 1} \frac{\xi_{n}}{10^{n}}
$$
converges almost surely to a random variable that has a uniform distribution on $[0,... | Solution. Since $\xi_{n} \geqslant 0$ with probability one and
$$
\mathrm{E} \sum_{n \geqslant 1} \frac{\xi_{n}}{10^{n}} \leqslant \frac{1}{1-1 / 10}<\infty
$$
the series converges almost surely. Moreover,
$$
\begin{aligned}
\varphi_{X_{n}}(t)=\prod_{k=1}^{n} \varphi_{\xi_{k}}\left(\frac{t}{10^{k}}\right)=\frac{1}{1... | proof | Other | proof | Yes | Yes | olympiads | false | 34,094 |
61. Prove Pólya's theorem, stating that
any such even convex function on $\mathbb{R}_{+}$, $\varphi=\varphi(t), t \in \mathbb{R}$,
that $\varphi(0)=1$ and $\varphi(\infty)=0$, is a characteristic function.
What random variable corresponds to the characteristic function $\varphi$? (See problems II.6.104, II.12.46(a). | Solution. It is not difficult to show that from problem II.6.104 follows the relation
$$
\varphi(t)=\int_{\mathbb{R}_{+}}(u-|t|)^{+} \mu(d u),
$$
where $x^{+}=x \vee 0, \mu$ - some finite measure. Let the random variables $\xi$ and $\eta$ be independent, $\xi$ has a density from problem II.12.46(a), and the distribut... | proof | Other | proof | Yes | Yes | olympiads | false | 34,095 |
62. Let $\xi$ and $\eta$ be independent random variables such that the distribution of the variable $\xi + \eta$ coincides with the distribution of the variable $\xi$. Prove that $\eta = 0$ almost surely. | Solution. The characteristic function $\varphi_{\eta}(t)$ is necessarily equal to 1 in a small neighborhood of 0, so $\eta$ has a second moment and $\mathrm{E} \eta^{2}=-\varphi_{\eta}^{\prime \prime}(0)=0$. The latter means that $\eta=0$ a.s.
## § 13. Gaussian systems | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,096 |
1. Let $\left(a_{n}\right)_{n \geqslant 1}$ be a numerical sequence such that for all $t \in[c, d], c<d$, there exists $\lim _{n} e^{i t a_{n}}$. Show that then there exists a finite limit $\lim _{n} a_{n}=a$. | Solution. Let $\left|a_{n_{k}}\right| \rightarrow \infty$ for some $\left(n_{k}\right) \subset(n)$. Then there exists a subsequence $\left(m_{l}\right) \subset\left(n_{k}\right)$ such that $\left|b_{l}\right| \rightarrow \infty$, where $b_{l}=a_{m_{l+1}}-a_{m_{l}}$. According to the condition,
$$
f_{l}(t)=\exp \left(i... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,097 |
2. Let $\xi_{n} \sim \mathscr{N}\left(a_{n}, \sigma_{n}^{2}\right)$ and $\xi_{n} \xrightarrow{d} \xi$. Show that
(a) the quantity $\xi$ has a Gaussian distribution $\mathscr{N}\left(a, \sigma^{2}\right)$ when
$$
a=\lim _{n} a_{n}, \quad \sigma^{2}=\lim _{n} \sigma_{n}^{2} \geqslant 0
$$
where all the limits consider... | Solution. (a) By the condition
$$
\mathrm{E} e^{i t \xi_{n}}=e^{i t a_{n}-\sigma_{n}^{2} t^{2} / 2} \rightarrow \varphi(t)=\mathrm{E} e^{i t \xi}, \quad t \in \mathbb{R}
$$
so $e^{-\sigma_{n}^{2} / 2} \rightarrow|\varphi(1)|$ and, therefore,
$$
\sigma_{n}^{2} \rightarrow \sigma^{2}=2 \ln |\varphi(1)|
$$
From this, ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,098 |
3. Let $\left(X_{1}, \ldots, X_{n}\right)$ be a Gaussian vector consisting of independent $\mathscr{N}\left(0, \sigma^{2}\right)$-distributed random variables. Consider the spherical coordinates $\left(R, \Phi_{1}, \ldots, \Phi_{n-1}\right)$ of the vector $\left(X_{1}, \ldots, X_{n}\right)$, i.e., let $R \geqslant 0, \... | Solution. The statement follows from the formula for the transformation of density under a smooth change of coordinates, as well as from the fact that the absolute value of the Jacobian determinant for the spherical change of coordinates is
$$
r^{n-1} \sin ^{n-2} \varphi_{1} \sin ^{n-3} \varphi_{2} \ldots \sin \varphi... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,099 |
4. Let $(\xi, \zeta)$ be a two-dimensional Gaussian vector, such that $\xi, \zeta \sim$ $\sim \mathscr{N}(0,1)$, and $\mathrm{E} \xi \zeta=\rho$. Show that the density of the ratio $\xi / \zeta$ is given by
$$
f_{\xi / \zeta}(z)=\frac{\sqrt{1-\rho^{2}}}{\pi\left(z^{2}-2 \rho z+1\right)}
$$ | Solution. To find the density $f_{\xi / \zeta}(z)$, we represent the distribution function $F_{\xi / \zeta}(z)$ as $\int_{-\infty}^{z} f_{\xi / \zeta}(u) d u$:
$$
\begin{aligned}
& F_{\xi / \zeta}(z)=\int_{x / y \leqslant z} \frac{e^{-\frac{x^{2}-2 \rho x y+y^{2}}{2\left(1-\rho^{2}\right)}}}{2 \pi \sqrt{1-\rho^{2}}} d... | f_{\xi/\zeta}(z)=\frac{\sqrt{1-\rho^{2}}}{\pi(z^{2}-2\rhoz+1)} | Algebra | proof | Yes | Yes | olympiads | false | 34,100 |
5. Let $A$ be some matrix of order $m \times n$. We call a matrix $A^{+}$ of order $n \times m$ the pseudoinverse of matrix $A$ if
$$
A A^{+} A=A, \quad A^{+} A A^{+}=A^{+}, \quad\left(A A^{+}\right)^{*}=A A^{+}, \quad\left(A^{+} A\right)^{*}=A^{+} A
$$
Show that a matrix $A^{+}$ with these properties exists and is u... | Solution. For an arbitrary matrix $A$, the following decomposition holds:
$$
A=U\left(\begin{array}{ll}
D & 0 \\
0 & 0
\end{array}\right) V^{*}
$$
where $U$ and $V$ are unitary matrices of sizes $m \times m$ and $n \times n$ respectively, 0 are zero matrices, and $D$ is a diagonal matrix with positive numbers on the ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,101 |
6. Let $X$ and $Y$ be random matrices of the same size, and suppose that the matrix $\mathrm{E}^{\top}{ }^{\top}$ is invertible. Prove the matrix Cauchy-Bunyakovsky inequality:
$$
\mathrm{E} X Y^{\top} \cdot\left[\mathrm{E} Y Y^{\top}\right]^{+} \cdot \mathrm{E} Y X^{\top} \leqslant \mathrm{E} X X^{\top}
$$
where “s”... | Solution. The desired inequality is equivalent to the following:
$$
\mathrm{E}(X-C Y)(X-C Y)^{\top} \geqslant 0
$$
where $C=\mathrm{E} X Y^{\top} \cdot\left[\mathrm{E} Y Y^{\top}\right]^{+}$. The last inequality is obviously true. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,102 |
7. Let $X$ be a Gaussian vector with values in $\mathbb{R}^{n}$. Prove that the matrix $\mathrm{D} X$ is singular if and only if there exists a non-zero vector $b \in \mathbb{R}^{n}$ such that $\mathrm{P}\left(X^{*} b=\mathrm{E} X^{*} b\right)=1$, where, as usual, $X^{*}=X^{\top}$. | Solution. The matrix $\mathrm{D} X$ is degenerate only in the case when $\mathrm{D} X \cdot b=0$ for some $b \in \mathbb{R}^{n}$. In this case, it is obvious that,
$$
0=b^{*} \cdot \mathrm{D} X \cdot b=\mathrm{E}\left|X^{*} b-\mathrm{E} X^{*} b\right|^{2}=0
$$
or $X^{*} b=\mathrm{E} X^{*} b$ with probability one. Mor... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,103 |
8. For an arbitrary Gaussian vector $X$, show that
$$
[X-\mathrm{E} X]^{*}(\mathrm{D} X)^{+}[X-\mathrm{E} X] \sim \chi_{\operatorname{rank}(\mathrm{D} X)}^{2}
$$
where $\operatorname{rank}(\mathrm{D} X)$ is the rank of the matrix $\mathrm{D} X$ and the symbol $\sim$ means "has the distribution". | Solution. According to the remark to problem II.13.5, we have
$$
\mathrm{D} X=U\left(\begin{array}{ll}
D & 0 \\
0 & 0
\end{array}\right) U^{*}, \quad(\mathrm{D} X)^{+}=U\left(\begin{array}{cc}
D^{-1} & 0 \\
0 & 0
\end{array}\right) U^{*}
$$
for some unitary matrix $U$ and diagonal matrix $D$ (with positive numbers on... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,104 |
9. Let $(X, Y)$ be a Gaussian system, where $Y, X$ are random vectors (possibly of different dimensions). Prove that
$$
Z=Y-\operatorname{cov}(Y, X) \cdot(\mathrm{D} X)^{+} X \quad \text { and } \quad X \text { are independent. }
$$
Here, as usual, $\operatorname{cov}(Y, X)=\mathrm{E}[Y-\mathrm{E} Y][X-\mathrm{E} X]^... | Solution. Due to the Gaussian nature of the pair $(X, Z)$, it is sufficient to verify that
$$
\operatorname{cov}(Z, X)=\operatorname{cov}(Y, X)-\operatorname{cov}(Y, X)(\mathrm{D} X)^{+} \mathrm{D} X=0
$$
By the remark following problem II. 13.5, we have
$$
\mathrm{D} X=U\left(\begin{array}{cc}
D & 0 \\
0 & 0
\end{a... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,105 |
11. Let $(X, Y)$ be a Gaussian system, where $Y$ is a random variable and $X$ is a vector. Prove that the conditional expectation $\mathrm{E}(Y \mid X)$ coincides almost surely with the optimal linear estimates $\widehat{\mathrm{E}}(Y \mid X)=\alpha+\beta^{*} X$ for
$$
(\alpha, \beta) \in \operatorname{Arg} \min _{a, ... | Solution. Due to the finiteness of $\mathrm{E}|Y|^{2}$, we have
$$
\mathrm{E}(Y \mid X=\cdot)=\arg \min _{f} \mathrm{E}|Y-f(X)|^{2}
$$
where the minimum is taken over all Borel functions $f$, and in particular, over all linear functions. In this case, as follows from II.13.10, $\mathrm{E}(Y \mid X)$ is a pointwise li... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,107 |
12. Let $(X, Y)$ be a Gaussian system, where $Y$ is a random variable in $\mathbb{R}$, and $X$ is a random vector in $\mathbb{R}^{d}$. Determine the structure of the conditional expectations $\mathrm{E}\left(Y^{n} \mid X=x\right), n \geqslant 1$. | Solution. We will assume that $\mathrm{E} X=0$ (otherwise, we subtract $\mathrm{E} X$ from $X$). Due to problem II.13.9, the quantity $Z=Y-\mathrm{E} Y X^{*} \cdot(\mathrm{D} X)^{+} X$ is independent of the vector $X$. Therefore,
$$
\begin{aligned}
& \mathrm{E}\left(Y^{n} \mid X=x\right)=\mathrm{E}\left(Z+\mathrm{E} Y... | \mathrm{E}(Y^{n}\midX=x)=\sum_{k\leqslantn/2}C_{n}^{2k}(\mathrm{E}Z+\mathrm{E}YX^{*}\cdot(\mathrm{D}X)^{+}x)^{n-2k}\frac{(2k)!}{2^{k}\cdotk!}(\mathrm{D} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 34,108 |
13. Let $X=\left(X_{1}, \ldots, X_{n}\right)$ and $Y=\left(Y_{1}, \ldots, Y_{n}\right)$ be two centered Gaussian vectors
$$
\mathrm{D} X_{k}=\mathrm{D} Y_{k} \quad \text { and } \quad \operatorname{cov}\left(Y_{k}, Y_{l}\right) \leqslant \operatorname{cov}\left(X_{k}, X_{l}\right), \quad 1 \leqslant k, l \leqslant n .... | Solution. The probabilities mentioned in the condition are approximately equal to $\mathrm{E} f(X)$ and $\mathrm{E} f(Y)$ for a suitably chosen smooth function $f=f(z), z \in \mathbb{R}^{n}$, so the problem reduces to considering the difference $\mathrm{E} f(X)-\mathrm{E} f(Y)$. Assuming further that $X$ and $Y$ are in... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,109 |
14. Let $\xi_{1}, \xi_{2}, \xi_{3}$ be independent random variables distributed according to the law $\mathscr{N}(0,1)$. Show that the quantities
$$
\frac{\xi_{1}+\xi_{2} \xi_{3}}{\sqrt{1+\xi_{3}^{2}}}, \quad \Phi^{-1}\left(\frac{\left|\xi_{1}\right|}{\sqrt{\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2}}}\right), \quad \frac{\le... | Solution. It is sufficient to note that for any fixed $a \in \mathbb{R}$, the relation
$$
\frac{\xi_{1}+a \xi_{2}}{\sqrt{1+a^{2}}} \sim \mathscr{N}(0,1)
$$
holds, and to show that conditionally on $\xi_{3}$, the quantity $\frac{\xi_{1}+\xi_{2} \xi_{3}}{\sqrt{1+\xi_{3}^{2}}}$ has the distribution $\mathscr{N}(0,1)$. T... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,110 |
15. Prove that the functions $s \wedge t, s \wedge t - s t$ and $e^{-|t-s|}$ are non-negative definite on $\mathbb{R}_{+}^{2}, [0,1]^{2}$ and $\mathbb{R}^{2}$ respectively. | Solution. For the first two functions, we have the representations
$$
\begin{gathered}
s \wedge t=\int_{\mathbb{R}_{+}} \mathbf{1}(x \leqslant s) \mathbf{1}(x \leqslant t) d x \\
s \wedge t-s t=\int_{0}^{1}[\mathbf{1}(x \leqslant s)-s][\mathbf{1}(x \leqslant t)-t] d x
\end{gathered}
$$
from which it is easy to deduce... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,111 |
16. Let $\xi$ and $\eta$ be independent random variables with distribution $\mathscr{N}(0,1)$.
(a) Show that the variables $\xi + a \eta$ and $a \xi - \eta$ are independent Gaussian variables for any $a \in \mathbb{R}$.
(b) Using part (a), show that for any $a \in \mathbb{R}$, the following equality holds:
$$
C \sta... | Solution. (a) The vector $(\xi, \eta)$ is a standard Gaussian with zero mean and identity covariance matrix, so the quantities $\xi + a \eta$ and $a \xi - \eta$ are jointly Gaussian, and
$$
\operatorname{cov}(\xi + a \eta, a \xi - \eta) = a - a = 0
$$
therefore, they are independent.
(b) By part (a), the quantities ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,112 |
17. (Bernstein.) Let $\xi$ and $\eta$ be independent random variables with finite variance. Prove that if $\xi+\eta$ and $\xi-\eta$ are independent, then $\xi$ and $\eta$ are Gaussian variables. Can the condition of the identical distribution of the variables $\xi$ and $\eta$ be dropped so that this property still hold... | Solution. Without loss of generality, we assume that $\mathrm{E} \xi=0$. Let $\varphi_{\zeta}(t)$ denote the characteristic function of the random variable $\zeta$. Then
$$
\varphi_{\xi}(t)=\varphi_{\frac{\xi+\eta}{2}}(t) \varphi_{\frac{\xi-\eta}{2}}(t)=\varphi_{\xi / 2}(t)^{3} \varphi_{-\xi / 2}(t)
$$
hence,
$$
\xi... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,113 |
18. Let $X=X_{t}(\omega),(\omega, t) \in \Omega \times[0,1],-\mathscr{F} \times \mathscr{B}[0,1]$-measurable function, and $\int_{0}^{1}\left|X_{t}\right| d t<\infty$ a.s. Suppose that $\left\{X_{t}\right\}_{0 \leqslant t \leqslant 1}$ is a
Gaussian system. Prove that the random variable $\int_{0}^{1} X_{t} d t$ is als... | Solution. First, we can consider that
$$
\mathrm{E}\left[\int_{0}^{1} X_{t} d t\right]^{2} \leqslant \mathrm{E} \int_{0}^{1} X_{t}^{2} d t=\int_{0}^{1} \mathrm{E} X_{t}^{2} d t<\infty
$$
Alternatively, we can first consider
$$
\frac{X_{t}}{\sqrt{1+\varepsilon \mathrm{E} X_{t}^{2}}}
$$
and then take the limit as $\v... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,114 |
20. (See [87], [106].) Let $\xi_{1}, \xi_{2}, \ldots \sim \mathscr{N}(0,1)$. Then
$$
\begin{gathered}
\mathrm{E} \xi_{n: n} \leqslant \sqrt{2 \ln n}, \quad n \geqslant 1, \\
\frac{\lim _{n}}{}\left[\xi_{n: n}-\sqrt{2 \ln n}\right] \leqslant 0 \quad \text { a.s., }
\end{gathered}
$$
where $\xi_{n: n}=\max \left\{\xi_{... | Solution. By Jensen's inequality for all $s>0$ we have
$$
\mathrm{E} \xi_{n: n} \leqslant \frac{1}{s} \ln \mathrm{E} e^{s \xi \xi_{n n}} \leqslant \frac{1}{s} \ln \sum_{i=1}^{n} \mathrm{E} e^{s \xi_{i}}=\frac{1}{s} \ln n e^{\frac{s^{2}}{2}}=\frac{\ln n}{s}+\frac{s}{2}
$$
Minimizing over $s>0$, we find that $\mathrm{E... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,116 |
22. Let $\xi_{0}, \xi_{1}, \xi_{2}, \ldots$ be i.i.d. random variables, $\xi_{1} \sim \mathscr{N}(0,1)$. Show that the series
$$
B_{t}^{\circ}=\sum_{n=1}^{\infty} \xi_{n} \frac{\sqrt{2} \sin n \pi t}{n \pi}, \quad 0 \leqslant t \leqslant 1
$$
indeed defines a Brownian bridge, and the series
$$
B_{t}=\xi_{0} t+\sum_{... | Solution. Since $B_{t}^{\circ}=B_{t}-t B_{1}, t \in[0,1]$, it suffices to establish that $(B_{t})$ is a Brownian motion. The latter, as well as the fact that $\left(W_{t}\right)$ is a Brownian motion, follows from the remark to problem II.13.21. Indeed, it is easy to verify that for all $t,|t| \leqslant 1$, the equalit... | proof | Other | proof | Yes | Yes | olympiads | false | 34,118 |
23. Establish that the process $\left(B_{t}\right)_{0 \leqslant t \leqslant 1}$, given by the formula
$$
B_{t}=\sum_{n=1}^{\infty} \frac{2 \sqrt{2} \xi_{n}}{(2 n-1) \pi} \sin \frac{(2 n-1) \pi t}{2}
$$
with independent random variables $\xi_{1}, \xi_{2}, \ldots \sim \mathscr{N}(0,1)$, is a Brownian motion (the given ... | Solution. This representation can be derived from Mercer's Theorem II.13.19. However, we will obtain it here by modifying the construction described in the remark to Problem II.13.21. Specifically, we have
$$
B_{t}=\sum_{n=1}^{\infty}\left(e_{n}, J_{t}\right) \zeta_{n}
$$
where $J_{t}=J_{t}(x)(0 \leqslant t \leqslant... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,119 |
24. Check that for Brownian motion $B=\left(B_{t}\right)_{t \geqslant 0}$ the following processes are also Brownian motions:
$$
\begin{gathered}
\left(-B_{t}\right)_{t \geqslant 0}, \quad\left(B_{t+a}-B_{a}\right)_{t \geqslant 0}, \quad\left(B_{a}-B_{a-t}\right)_{0 \leqslant t \leqslant a}, \\
\left(a^{-1} B_{a^{2} t}... | Solution. All processes form Gaussian systems as linear transformations of the Gaussian system $\left(B_{t}\right)_{t \geqslant 0}$. All processes have zero mean and a covariance function that coincides with the covariance function of Brownian motion (established by direct calculation).
The continuity of trajectories ... | proof | Other | proof | Yes | Yes | olympiads | false | 34,120 |
26. Let $B^{\circ}=\left(B_{t}^{\circ}\right)_{0 \leqslant t \leqslant 1}$ be a Brownian bridge. Show that the process $B=\left(B_{t}\right)_{t \geqslant 0}, B_{t}=(1+t) B_{t /(1+t)}^{\circ}$, is a Brownian motion. | Solution. The process $B=\left(B_{t}\right)_{t \geqslant 0}$ forms a Gaussian system as a linear transformation of the Gaussian system $\left(B_{t}^{\circ}\right)_{0 \leqslant t \leqslant 1}$. By direct verification, we obtain that $\mathrm{E} B_{t}=0$ and $\mathrm{E} B_{s} B_{t}=s \wedge t$ for all $s, t \geqslant 0$.... | proof | Other | proof | Yes | Yes | olympiads | false | 34,122 |
28. Let $\left(X_{1}^{(n)}, \ldots, X_{n}^{(n)}\right)$ be a random vector uniformly distributed on the sphere $S_{n}(\sqrt{n})$ of radius $\sqrt{n}$ centered at the origin $(n \geqslant 1)$. Prove the validity of the following "Poincaré observation":
$$
\left(X_{1}^{(n)}, X_{2}^{(n)}, \ldots, X_{m}^{(n)}\right) \xrig... | Solution. The entire proof essentially boils down to establishing the equality
$$
\left(X_{1}^{(n)}, \ldots, X_{n}^{(n)}\right) \stackrel{d}{=} \sqrt{n} \frac{\left(\xi_{1}, \ldots, \xi_{n}\right)}{\left[\sum_{i=1}^{n} \xi_{i}^{2}\right]^{1 / 2}}
$$
The latter is ensured by the fact that the distribution of the vecto... | proof | Other | proof | Yes | Yes | olympiads | false | 34,124 |
29. Let $\mu$ be the uniform distribution on the unit sphere $S_{n}$ in $\mathbb{R}^{n} (n \geqslant 1)$ centered at the origin. Establish the isoperimetric inequality of P. Lévy.
For any closed set $F \subseteq S_{n}, \mu(F)>0$, and all $\varepsilon>0$, the inequality
$$
\mu\left(F^{\varepsilon}\right) \geqslant \mu... | Solution. (a) Let $B_{k} \in \mathscr{B}, k \geqslant 1$, be such that
$$
\lim _{k} \mu\left(B_{k} C\right)=\sup _{B \in \mathscr{B}} \mu(B C)
$$
The metric space $\left(S_{n}, \rho_{n}\right)$ is compact and separable, so by points (d) and (f) of problem II.1.30, there exist sets $\left(B_{l}\right) \subseteq\left(B... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,125 |
30. (Continuing from problem II.13.29.) (a) Establish that from the isoperimetric inequality of Lévy on the sphere, in particular, follows the classical isoperimetric inequality in $\mathbb{R}^{d}$, which states that
among all bodies in $\mathbb{R}^{d}$ with a smooth boundary and any fixed volume, the closed ball has ... | Solution. (a) It is sufficient to prove that among all bodies $V$ in $\mathbb{R}^{d}$ with a smooth boundary and an arbitrary fixed volume, the minimum of the quantity
$$
\lambda\left(V^{\varepsilon}\right)=\int_{0}^{\varepsilon} \lambda\left(V^{\delta}\right)^{\prime} d \delta
$$
is realized on a closed ball for eac... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 34,126 |
31. Let $(X, Y)$ be a two-dimensional Gaussian vector with $\mathrm{E} X=\mathrm{E} Y=0$, $\mathrm{E} X^{2}=\mathrm{E} Y^{2}=1$ and correlation coefficient $\rho=\mathrm{E} X Y$.
(a) Show that the quantities $X$ and $Z=(Y-\rho X) / \sqrt{1-\rho^{2}}$ are independent $\mathscr{N}(0,1)$-distributed.
(b) Prove that
$$
\... | Solution. (a) The quantity $Z$ is Gaussian as a linear combination of jointly Gaussian. Moreover, $\mathrm{E} Z=0, \mathrm{D} Z=\mathrm{E} Z^{2}=1$ and $\operatorname{cov}(X, Z)=0$, so $X$ and $Z$ are independent $\mathscr{N}(0,1)$-variables.
(b) Let $\rho=\sin \theta$ and $\theta \in[-\pi / 2, \pi / 2]$, then $Y=Z \c... | proof | Other | proof | Yes | Yes | olympiads | false | 34,127 |
32. Let $Z=X Y$, where $X$ and $Y$ are independent random variables, $X \sim \mathscr{N}(0,1)$ and $\mathrm{P}(Y=1)=\mathrm{P}(Y=-1)=1 / 2$. Show that $Z \sim \mathscr{N}(0,1)$, and find the distributions of the vectors $(X, Z)$, $(Y, Z)$, as well as the distribution of the random variable $X+Z$. Verify that $X$ and $Z... | Solution. We have $\operatorname{Law}(Z \mid Y=1)=\operatorname{Law}(X)=\mathscr{N}(0,1)$ and $\operatorname{Law}(Z \mid Y=$ $=-1)=\operatorname{Law}(-X)=\mathscr{N}(0,1)$, therefore $Z$ is independent of $Y$ and $Z \sim \mathscr{N}(0,1)$. Moreover,
$$
\begin{aligned}
\mathrm{P}(X \leqslant x, Z \leqslant z) & =\frac{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,128 |
33. Let $\xi \sim \mathscr{N}(0,1)$ and
$$
\eta_{\alpha}= \begin{cases}\xi, & \text { if }|\xi| \leqslant \alpha \\ -\xi, & \text { if }|\xi|>\alpha\end{cases}
$$
Show that $\eta_{\alpha} \sim \mathscr{N}(0,1)$ and for such $\alpha$ that
$$
\int_{0}^{\alpha} x^{2} \varphi(x) d x=\frac{1}{4}, \quad \varphi(x)=\frac{1... | Solution. By the definition of $\eta_{\alpha}$, taking into account the equality $\xi \stackrel{d}{=}-\xi$, we obtain
$$
\begin{aligned}
\mathrm{P}\left(\eta_{\alpha} \leqslant x\right) & =\mathrm{P}(\xi \leqslant x,|\xi| \leqslant \alpha)+\mathrm{P}(-\xi \leqslant x,|\xi|>\alpha)= \\
& =\mathrm{P}(\xi \leqslant x,|\x... | proof | Calculus | proof | Yes | Yes | olympiads | false | 34,129 |
34. Let $\xi$ and $\eta$ be normally distributed random variables, $\mathrm{E} \xi=\mathrm{E} \eta=0, \mathrm{E} \xi^{2}=\mathrm{E} \eta^{2}=1$ and $\mathrm{E} \xi \eta=\rho$. Show that
(a) $\mathrm{E}(\xi \vee \eta)=\sqrt{(1-\rho) / \pi}, \quad \mathrm{E}(\xi \vee \eta)^{2}=1$,
(b) $\mathrm{E}(\xi \mid \eta)=\rho \e... | Solution. (a) Since $(\xi, \eta) \stackrel{d}{=}(-\xi,-\eta)$, we obtain that $\xi \vee \eta \stackrel{d}{=}$ $\stackrel{d}{=}-\xi \wedge \eta$ and
$$
\mathrm{E}(\xi \vee \eta)=\mathrm{E} \frac{\xi \vee \eta-\xi \wedge \eta}{2}=\frac{\mathrm{E}|\xi-\eta|}{2}=\sqrt{\frac{\mathrm{D}(\xi-\eta)}{2 \pi}}=\sqrt{\frac{1-\rho... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,130 |
35. Let $\xi=\left(\xi_{1}, \ldots, \xi_{n}\right)$ be a non-degenerate Gaussian vector with zero means and covariance matrix $R=\left\|\mathrm{E} \xi_{i} \xi_{j}\right\|$. Let $\lambda_{1}, \ldots, \lambda_{n}$ be the eigenvalues of the matrix $R$. Show that the characteristic function $\varphi(t)$ of the random varia... | Solution. Matrix $R$ is symmetric and non-negative definite. According to a well-known fact from linear algebra, there exists an orthogonal matrix $C$ such that the matrix $D = C R C^{\top}$ is diagonal with non-negative numbers $\lambda_{1}, \ldots, \lambda_{n}$ on the diagonal. Therefore, we can take
$$
\zeta = (\ze... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,131 |
36. Let $\xi_{1}, \ldots, \xi_{n}$ be i.i.d. random variables, $n \geqslant 2$. Show that the distribution of the vector $\xi=\left(\xi_{1}, \ldots, \xi_{n}\right)$ is spherically symmetric if and only if each of the variables $\xi_{1}, \ldots, \xi_{n}$ is normally distributed with zero mean. | Solution. Suppose initially that the distribution of $\xi$ is spherically symmetric. In terms of characteristic functions $\varphi_{\xi}=$ $\varphi_{\xi}(\lambda)$, this condition means that for any orthogonal matrix $C$ such that $\operatorname{det} C=1$, the equality
$$
\varphi_{\xi}(\lambda)=\varphi_{C \xi}(\lambda... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,132 |
37. (See [62].) Let $X=\left(X_{1}, X_{2}, \ldots\right)$ be a Gaussian random sequence, and $L$ be an arbitrary linear subspace of $\mathbb{R}^{\infty}$.
Prove that then the zero-one law for Gaussian systems holds:
$$
\mathrm{P}(X \in L)=0 \text{ or } 1.
$$
In particular, derive that $\mathrm{P}\left(\sup _{n}\left... | Solution. Let $Y=\left(Y_{1}, Y_{2}, \ldots\right)$ be an independent copy of $X$. Then for all $0<\theta<\pi / 2$ the random sequences
$$
X \cos \theta+Y \sin \theta \text { and }-X \sin \theta+Y \cos \theta
$$
are independent and distributed as $X$. Therefore,
$$
\mathrm{P}\left(B_{\theta}\right)=\mathrm{P}(X \in ... | proof | Other | proof | Yes | Yes | olympiads | false | 34,133 |
38. (See [78].) Under the conditions of problem II. 13.37, show that if
$$
\mathrm{P}\left(\sup _{n}\left|X_{n}\right|<\infty\right)=1 \text {, }
$$
then
$$
\mathrm{E} \sup _{n}\left|X_{n}\right|<\infty
$$
$\langle$ See problem II.6.40. $\rangle$ | Solution. Let $Y=\left(Y_{1}, Y_{2}, \ldots\right)$ be an independent copy of the random variable $X$. Then the random sequences
$$
\frac{X+Y}{\sqrt{2}} \quad \text { and } \quad \frac{X-Y}{\sqrt{2}}
$$
are independent and distributed the same as $X$. Let, further, $\left|\left(x_{1}, x_{2}, \ldots\right)\right|=$ $=... | proof | Other | proof | Yes | Yes | olympiads | false | 34,134 |
39. Let $\xi \sim \mathscr{N}\left(a, \sigma^{2} I_{n}\right)$ be a Gaussian vector in $\mathbb{R}^{n}$ with mean $\mathrm{E} \xi=a$ and covariance matrix $\sigma^{2} I_{n}=\mathrm{D} \xi=\mathrm{E}(\xi-\mathrm{E} \xi)(\xi-\mathrm{E} \xi)^{\top}$, where $I_{n}$ is the identity matrix of size $n \times n$ and $\sigma>0$... | Solution. Since the vector formed by the components of vectors $P \xi$ and $Q \xi$ is Gaussian, to prove the independence of $P \xi$ and $Q \xi$, it is sufficient to check their uncorrelatedness:
$$
\mathrm{E}(P \xi-P a)(Q \xi-Q a)^{\top}=P \cdot \mathrm{E}(\xi-a)(\xi-a)^{\top} \cdot Q=P \sigma^{2} I_{n} Q=O_{n}
$$
M... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,135 |
40. Let $\xi_{1}, \ldots \xi_{n}, n \geqslant 2,$ be independent random variables distributed according to $\mathscr{N}\left(a, \sigma^{2}\right)$. Show that the quantities
$$
\bar{\xi}=\frac{1}{n} \sum_{i=1}^{n} \xi_{i}, \quad s_{1}^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(\xi_{i}-\bar{\xi}\right)^{2}
$$
are independen... | Solution. Applying the statement of problem II. 13.39 for orthogonal projectors $P$ and $Q=I_{n}-P$ onto the linear space spanned by the vector $e=(1, \ldots, 1)$ and its orthogonal complement, respectively, we find that $P \xi=\bar{\xi}$ and $s_{1}^{2}=\frac{\|Q \xi\|^{2}}{n-1}$ are independent, and $\frac{(n-1) s_{1}... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,136 |
41. (On the statistics of the normal distribution $\mathscr{N}\left(a, \sigma^{2}\right)$: main statistics.) Let $\xi_{1}, \ldots \xi_{n}$ be i.i.d. random variables, $\xi_{1} \sim \mathscr{N}\left(a, \sigma^{2}\right)$ and $x=\left(x_{1}, \ldots, x_{n}\right)$ be a sample obtained from observations of $\xi=\left(\xi_{... | Solution. The statement follows from the following factorization criterion.
If $\xi$ is a random vector in $\mathbb{R}^{n}$ with density $f=f(x, \theta)$, $(x, \theta) \in \mathbb{R}^{n} \times \Theta$, then the vector-function $T=T(x)$ will be a sufficient statistic if and only if $f$ admits the representation
$$
f(... | proof | Algebra | proof | Yes | Yes | olympiads | false | 34,137 |
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