The Law of Large Numbers

Let $\nu_n$ be the number of successes in $n$ Bernoulli trials with success probability $p$. Prove that for every function $f(x)$ continuous on $[0,1]$ the following convergence holds:

Let $f: [0,1] \to \mathbb {R}$ be a continuous function. Prove that the Bernstein polynomials

converge to $f(x)$ as $n \to \infty$ uniformly in $x \in [0,1]$:

Let $f: \mathbb {R}^{+} \to \mathbb {R}$ be a continuous bounded function, and let $K$ be a compact set in $[0,\infty )$. Prove that the polynomials

converge to $f(x)$ as $n \to \infty$ uniformly in $x \in K$.

Let $f: \mathbb {R}^{+} \to \mathbb {R}$ be a continuous bounded function, and let $K$ be a compact set in $[0,\infty )$. Prove that the polynomials

converge to $f(x)$ as $n \to \infty$ uniformly in $x \in K$.

Under the hypotheses of problem

Let $f(x,y)$ be a function continuous on the triangle $x \geqslant 0$, $y \geqslant 0$, $x+y \leqslant 1$. Prove that the Bernstein polynomials in $\mathbb {R}^{2}$

converge to $f(x,y)$ as $n \to \infty$ uniformly in $x, y$.

Let $\xi_1, \xi_2, \ldots$ be a sequence of independent random variables with the standard Cauchy distribution. Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\xi_1, \xi_2, \ldots$ be a sequence of independent identically distributed random variables having the standard normal distribution. Set

Find the limiting distribution of $\tau_n$ as $n \to \infty$.

Let $\xi_1, \xi_2, \ldots$ be a sequence of independent identically distributed random variables with $\mathbb {E}\left[\xi_1\right] = a$, $\operatorname {Var}\left[\xi_1\right] = \sigma^2$. Prove that the sequence of random variables $\eta_n$, defined by

converges in probability. Find the limit.

Let the random variable $\eta_n$ have:

Let $\operatorname {Var}\left[\xi_n\right] < \infty$ for all $n$ and $\operatorname {Var}\left[S_n\right] = o(n^2)$ as $n \to \infty$. Prove that the LLN holds for the sequence $\{ \xi_n\}$.

Let $\xi_n$ take the values $\sqrt{n}$, $-\sqrt{n}$, and $0$ with probabilities $1/2n$, $1/2n$, and $1-1/n$ respectively. Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\xi_n$ take the values $2^n$ and $-2^n$ with probability $1/2$ each. Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\xi_n$ take the values $n$, $-n$, and $0$ with probabilities:

Let $\xi_n$ take the values $2^n$, $-2^n$, and $0$ with probabilities $2^{-(2n+1)}$, $2^{-(2n+1)}$, and $1-2^{-2n}$ respectively. Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\xi_n$ take the values $3^n$, $-3^n$, and $0$ with probabilities $3^{-(2n+2)}$, $3^{-(2n+2)}$, and $1 - 2 \cdot 3^{-(2n+2)}$ respectively. Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\xi_n$ take the values $1$ and $-1$ with probabilities $(1-2^{-n})/2$ each, and the values $2^n$ and $-2^n$ with probabilities $2^{-n-1}$ each. Prove that the LLN holds for the sequence $\{ \xi_n\}$.

Let $\xi_n$ take the values $n^{\lambda }$ and $-n^{\lambda }$ with probability $1/2$ each. Determine for which values of $\lambda$ the LLN holds for the sequence $\{ \xi_n\}$.

Let $\xi_n$ take $2n+1$ values $0, \pm L_n, \pm 2L_n, \pm 3L_n, \ldots , \pm nL_n$ with probability $1/(2n+1)$ each. Find conditions on the constants $L_n$ ensuring that the LLN holds for the sequence $\{ \xi_n\}$.

Let $\xi_n$ take the values $a_n$, $-a_n$, and $0$ with probabilities $p_n$, $p_n$, and $1-2p_n$ respectively. Find conditions on the constants $a_n$ and $p_n$ ensuring that the LLN holds for the sequence $\{ \xi_n\}$.

If $n$ is a perfect square, the random variable $\xi_n$ takes the values $-\sqrt{n}$ and $\sqrt{n}$ with probability $1/2$ each; otherwise $\xi_n$ takes the values $-1$ and $1$ with probability $1/2$ each. Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\xi_n$ have the Poisson distribution with parameter $\alpha_n$, where

Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\{ \xi_n\}$ be a sequence of independent identically distributed random variables, and let $\{ a_n\}$ be a bounded sequence of nonnegative numbers. Is it true that if the LLN holds for the sequence $\{ \xi_n\}$, then it also holds for the sequence $\{ \eta_n\}$, where $\eta_n = a_n \xi_n$?

Let $\xi_n$ be uniformly distributed on $[-n, n]$. Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\xi_n$ have the uniform distribution on $[-n^{\alpha }, n^{\alpha }]$, $\alpha \in \mathbb {R}$. Determine for which values of $\alpha$ the LLN holds for the sequence $\{ \xi_n\}$.

Let $\xi_n$ have the binomial distribution with parameters $n$ and $1/n^{\alpha }$, where $\alpha > 0$. Determine for which values of $\alpha$ the LLN holds for the sequence $\{ \xi_n\}$.

Let $\{ \xi_n\}$ and $\{ \eta_n\}$ be two sequences, in each of which the random variables are independent. Suppose the series $\sum_{n=1}^{\infty } \mathbb {P}\left(\xi_n \neq \eta_n\right)$ converges. Prove that if $\mathbb {E}\left[\xi_n\right] = \mathbb {E}\left[\eta_n\right]$ for every $n$ and the LLN holds for the sequence $\{ \xi_n\}$, then it also holds for the sequence $\{ \eta_n\}$.

Let $\xi$ be a random variable. Define the random sequence $\xi_n$ by $\xi_n = \xi$. Does the sequence $\{ \xi_n\}$ satisfy the LLN?

Bernoulli trials are performed with success probability $p$. Set $\xi_n = 1$ if the $n$-th and $(n+1)$-th trials both end in success, and $\xi_n = 0$ otherwise. Does the LLN hold for the sequence $\{ \xi_n\}$?

Let $\{ \xi_n\}$ be a sequence of random variables such that $\operatorname {Var}\left[\xi_n\right] \leqslant c < \infty$, and $\xi_n$ is independent of $\xi_m$ whenever $|n-m| > 1$. Prove that the LLN holds for the sequence $\{ \xi_n\}$.

Let $\{ \xi_n\}$ be a sequence of random variables such that $\operatorname {Var}\left[\xi_n\right] \leqslant c < \infty$, and the covariances are negative. Prove that the LLN holds for the sequence $\{ \xi_n\}$.

Let $\{ \xi_n\}$ be a sequence of random variables such that $\operatorname {Var}\left[\xi_n\right] \leqslant c < \infty$, and the covariances $r_{ij} = \operatorname {Cov}\left[\xi_i, \xi_j\right]$ satisfy $r_{ij} \to 0$ uniformly as $|i-j| \to \infty$. Prove that the LLN holds for the sequence $\{ \xi_n\}$.

Prove that if $|S_n| < cn$ and $\operatorname {Var}\left[S_n\right] > \alpha n^2$, then the LLN does not hold for the sequence $\{ \xi_n\}$.

Let $\{ \xi_n\}$ be a sequence of independent identically distributed random variables with finite nonzero variance. Prove that the LLN does not hold for the sequence $\{ S_n\}$.

Let $\{ \xi_n\}$ be a sequence of independent identically distributed random variables with finite variance. Prove that the LLN holds for the sequence $\{ a_n S_n\}$ if $a_n \sqrt{n} \to 0$ as $n \to \infty$.

Die $A$ has four red faces and two white faces, while die $B$ has two red faces and four white faces. A coin is tossed once. If it comes up heads, die $A$ is rolled repeatedly thereafter; if tails, only die $B$ is rolled. Set $\xi_n = 1$ if the $n$-th roll of the die shows a red face, and $\xi_n = 0$ otherwise. Prove that the LLN does not hold for the sequence $\{ \xi_n\}$.

An urn contains $b$ black and $r$ red balls. A ball is drawn at random. It is returned to the urn, and, in addition, $c$ balls of the same color as the one drawn are added. A new random draw is then made from the urn (now containing $b+r+c$ balls), and the described procedure is repeated. Let the random variable $\xi_n$ equal $1$ or $0$ according to whether the $n$-th ball drawn was black or red. Prove that the LLN does not hold for the sequence $\{ \xi_n\}$.

Let $\{ \xi_n\}$ be a sequence of random variables for which the law of large numbers holds. Must the LLN hold for the sequence $\{ |\xi_n|\}$?

Let $\{ \xi_n\}$ be a sequence of random variables with zero expectations. Does the convergence $\xi_n \xrightarrow {p} 0$ imply that the LLN holds for the sequence $\{ \xi_n\}$?

Let the random variable $\xi_1$ be uniformly distributed on the set $S = \{ 1, 2, \ldots , s\}$, and let $f: S \to S$ be a bijective function. Define the random sequence $\xi_n$ by $\xi_n = f(\xi_{n-1})$. Determine under what conditions on $f$ the sequence $\{ \xi_n\}$ satisfies the LLN.