Homework 11
Due Date: Friday, May 13

 Let \(\{a_n\}\) be a sequence in \([0,1]\), and suppose that \(\sum_{n=1}^\infty a_n = \infty\). Prove that \(\prod_{n=1}^\infty (1a_n) = 0\).
(Hint: Take the logarithm of the product.)
 Let \(\{E_n\}\) be a sequence of independent events, and suppose that
\[
\sum_{n=1}^\infty P(E_n) \;=\; \infty.
\]
Prove that, almost surely, infinitely many of the events \(E_n\) occur.
(Hint: Start by proving that at least one of the events occurs.)
 Let \(\{X_n\}\) be a sequence of independent, identically distributed random variables with
\[
P(X_n = 1) \;=\; P(X_n = 1) \;=\; \frac{1}{2}\text{,}
\]
and let \(S_n = X_1 + \cdots + X_n\). The sequence \(\{S_n\}\) is known as a simple random walk.
 Use Chebyshev's inequality to prove that \(P\bigl(S_n \leq 2\sqrt{n} \bigr) \,\geq\, 3/4\) for all \(n\in\mathbb{N}\).
 Use the central limit theorem to guess the value of
\[
\lim_{n\to\infty} \frac{ES_n}{\sqrt{n}}.
\]
You do not need to prove your answer.
 If \(k\in\{n,\ldots,n\}\), find an explicit formula for \(P(S_{2n} = 2k)\) involving a binomial coefficient.
 Use question 1 and your formula from part (c) to prove that, almost surely, \(S_{2n} = 0\) for infinitely many values of \(n\).