Dependent summands

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.