Problem set 5: Limit Theorems
You have invited 64 guests to a party. You need to make sandwiches for the guests. You believe that a guest might need \(0\), \(1\), or \(2\) sandwiches with probabilities
\[ \frac14,\qquad \frac12,\qquad \frac14, \]
respectively.
You assume that the number of sandwiches each guest needs is independent of the other guests.
How many sandwiches should you make so that you are \(95\%\) sure that there is no shortage?
When typing, a stenographer makes a mistake on a character with probability \(0.0005\).
Find an approximate value for the probability that, when typing \(10\,000\) characters, the stenographer will make a mistake no more than three times.
Let \(\{\xi_n\}\) be a sequence of independent identically distributed random variables with distribution \(U(0,1)\).
Set
\[ M_n=\max\{\xi_1,\ldots,\xi_n\}. \]
Prove that
\[ n(1-M_n)\xrightarrow{d}\operatorname{Exp}(1). \]
Let \(\{\xi_n\}\) be a sequence of independent identically distributed random variables such that
\[ \mathsf{P}(\xi_1=1)=p, \qquad \mathsf{P}(\xi_1=-1)=1-p. \]
Let \(\eta_n\) denote the number of times that the sign changes in the sequence
\[ \xi_1,\ldots,\xi_n. \]
Prove that the sequence
\[ \frac{\eta_n}{n} \]
converges almost surely and find its limit.
Compute the following limit:
\[ \lim_{n\to\infty} \int_{[0,1]^n} \frac{x_1^5+x_2^5+\cdots+x_n^5} {x_1^4+x_2^4+\cdots+x_n^4} \,dx. \]
Let \(\{\xi_n\}\) be a sequence of independent identically distributed non-negative random variables with
\[ \mathsf{E}\xi_1=a>0. \]
Set
\[ N_t=\max\left\{n\geq 0: \xi_1+\cdots+\xi_n\leq t \right\}, \qquad t>0. \]
Prove that
\[ \frac{N_t}{t}\xrightarrow{\mathrm{a.s.}}\frac1a \]
as \(t\to\infty\).
Let \(\{\xi_n,\ n\in\mathbb{N}\}\) be a sequence of i.i.d. random variables with
\[ \mathsf{E}\xi_1=a\neq0, \qquad \mathsf{D}\xi_1>0. \]
Set
\[ S_n=\xi_1+\cdots+\xi_n. \]
Find the convergence in distribution of the sequence
\[ \sqrt{n} \left( \frac{n}{S_n}-\frac1a \right). \]
Let \(\{\xi_n,\ n\in\mathbb{N}\}\) be a sequence of i.i.d. random variables with distribution
\[ \operatorname{Exp}(\lambda), \qquad \lambda>0. \]
Set
\[ Y_n=\frac{S_n}{n}. \]
Find values \(a(\lambda)\) and \(\sigma^2(\lambda)\) such that, as \(n\to\infty\),
\[ \sqrt{n}\left( Y_n\sin Y_n-a(\lambda) \right) \xrightarrow{d} \mathcal{N}\left(0,\sigma^2(\lambda)\right). \]
Let \(\{\xi_n\}\) be a sequence of independent identically distributed random variables with characteristic function \(\phi(t)\).
Prove that if, for some \(C>0\) and \(0<\alpha\leq2\),
\[ \phi(t) = 1-C|t|^\alpha(1+o(1)), \qquad t\to0, \]
then, as \(n\to\infty\), there exists a limit distribution of the random variables
\[ \eta_n= \frac{\xi_1+\cdots+\xi_n}{n^{1/\alpha}}, \]
and find the characteristic function of this distribution.
Let \(\{\xi_n,\ n\in\mathbb{N}\}\) be a sequence of i.i.d. random variables with finite variance.
Prove that, for any \(x\in\mathbb{R}\), the following limit is equal to \(0\), \(1\), or \(\frac12\):
\[ \lim_{n\to\infty} \mathsf{P} \left( \xi_1+\cdots+\xi_n\leq x \right). \]
Let
\[ \xi_n\xrightarrow{d}\xi. \]
Consider a function \(h(x)\) differentiable at a point \(a\in\mathbb{R}\).
Find the convergence in distribution of the sequence
\[ \frac{h(a+b_n\xi_n)-h(a)}{b_n}, \]
where \(b_n\to0\) is an arbitrary sequence of positive numbers.
Let
\[ \xi_n\sim\operatorname{Pois}(\lambda_n), \qquad n\in\mathbb{N}. \]
Prove that if \(\lambda_n\to\infty\), then
\[ \frac{\xi_n-\lambda_n}{\sqrt{\lambda_n}} \xrightarrow{d} \mathcal{N}(0,1). \]
Let \(X_1,\ldots,X_n\) be a sequence of random variables with Laplace distribution with parameter \(\sigma\), so that their density is
\[ p(x)=\frac{1}{2\sigma}e^{-|x|/\sigma}. \]
Set
\[ Y=\frac1n\sum_{i=1}^n |X_i|, \qquad Z=\frac1n\sum_{i=1}^n X_i^2. \]
Find the limiting distribution of
\[ \sqrt{n}(T-\sigma), \]
where
\[ T=\frac{Z^2}{4Y^3}. \]
Пусть последовательность случайных векторов \(\xi_1,\ldots,\xi_n,\ldots\) сходится по распределению к константе \(C\). Докажите, что тогда \[ \xi_n \xrightarrow{\mathsf{P}} C. \]
Пусть \(\{X_n,\ n\in\mathbb N\}\) — независимые одинаково распределенные случайные величины с распределением \(\mathcal N(0,\sigma^2)\). Рассмотрим \[ Y_n=\frac1n\sum_{i=1}^n |X_i|, \qquad Z_n=\frac1n\sum_{i=1}^n X_i^2 \] и \[ T_n=\sqrt{\frac{2}{\pi}}\frac{Z_n}{Y_n}. \]
Найдите предел сходимости по распределению у последовательности \[ \sqrt n\,(T_n-\sigma). \]
Пусть \(\{\xi_n,\ n\in\mathbb N\}\) и \(\{\eta_n,\ n\in\mathbb N\}\) — две последовательности случайных величин, причем для каждого \(n\geq 1\) величины \(\xi_n\) и \(\eta_n\) независимы. Пусть \[ \xi_n\xrightarrow{\mathsf{P}}\xi, \qquad \eta_n\xrightarrow{\mathsf{P}}\eta. \]
Докажите, что \(\xi\) и \(\eta\) — тоже независимы.