Some ancillary results

Show that the probability that infinitely many of the events $\left\{ A_{n}: n \geq 1\right\}$ occur satisfies $\mathbb {P}\left(A_{n} \text{ i.o.}\right) \geq \lim \sup_{n \rightarrow \infty } \mathbb {P}\left(A_{n}\right)$.

Let $\left\{ S_{n}: n \geq 0\right\}$ be a simple random walk which moves to the right with probability $p$ at each step, and suppose that $S_{0} = 0$. Write $X_{n} = S_{n}-S_{n-1}$.

Let $X_{1}, X_{2}, \ldots$ be independent identically distributed random variables. The event $A$, defined in terms of the $X_{n}$, is called exchangeable if $A$ is invariant under finite permutations of the coordinates, which is to say that its indicator function $I_{A}$ satisfies $I_{A}\left(X_{1}, X_{2}, \ldots , X_{n}, \ldots \right) = I_{A}\left(X_{i_{1}}, X_{i_{2}}, \ldots , X_{i_{n}}, X_{n+1}, \ldots \right)$ for all $n \geq 1$ and all permutations $\left(i_{1}, i_{2}, \ldots , i_{n}\right)$ of $(1,2, \ldots , n)$. Show that all exchangeable events $A$ are such that either $\mathbb {P}\left(A\right) = 0$ or $\mathbb {P}\left(A\right) = 1$.

Returning to the simple random walk $S$ of Exercise (7.3.3), show that $\left\{ S_{n} = 0 \text{ i.o.}\right\}$ is an exchangeable event with respect to the steps of the walk, and deduce from the Hewitt-Savage zero-one law that it has probability either 0 or 1.

Let $f:[0,1] \rightarrow \mathbb {R}$ be a continuous function, and let $S_{n}$ be a random variable having the binomial distribution with parameters $n$ and $x$. Using the formula $\mathbb {E}\left[Z\right] = \mathbb {E}\left[Z I_{A}\right]+\mathbb {E}\left[Z I_{A^{\complement }}\right]$ with $Z = f(x)-f\left(n^{-1} S_{n}\right)$ and $A = \left\{ \left|n^{-1} S_{n}-x\right| > \delta \right\}$, show that

You have proved Weierstrass's approximation theorem, which states that every continuous function on $[0,1]$ may be approximated by a polynomial uniformly over the interval.

A sequence $X_{1}, X_{2}, \ldots$ of random variables is said to be completely convergent to $X$ if

Show that, for sequences of independent variables, complete convergence is equivalent to a.s. convergence. Find a sequence of (dependent) random variables which converges a.s. but not completely.

Let $X_{1}, X_{2}, \ldots$ be independent identically distributed random variables with common mean $\mu$ and finite variance. Show that

Let $\left\{ X_{n}: n \geq 1\right\}$ be independent and exponentially distributed with parameter 1. Show that

Let $\left\{ X_{n}: n \geq 1\right\}$ be independent $N(0,1)$ random variables. Show that:

Construct an example to show that the convergence in distribution of $X_{n}$ to $X$ does not imply the convergence of the unique medians of the sequence $X_{n}$.

(i) Let $\left\{ X_{r}: r \geq 1\right\}$ be independent, non-negative and identically distributed with infinite mean. Show that $\lim_{\sup_{r \rightarrow \infty }} X_{r} / r = \infty$ almost surely.

(ii) Let $\left\{ X_{r}\right\}$ be a stationary Markov chain on the positive integers with transition probabilities

Let $\left\{ X_{r}: 1 \leq r \leq n\right\}$ be independent and identically distributed with mean $\mu$ and finite variance $\sigma^{2}$. Let $\overline{X} = n^{-1} \sum_{r = 1}^{n} X_{r}$. Show that

converges in distribution to the $N(0,1)$ distribution as $n \rightarrow \infty$.

For a random variable $X$ with mean $0$, variance $\sigma^{2}$, and $\mathbb {E}\left[X^{4}\right] < \infty$, show that