Math 461 Bard College

# Homework 11

Due Date: Friday, May 13

1. Let $$\{a_n\}$$ be a sequence in $$[0,1]$$, and suppose that $$\sum_{n=1}^\infty a_n = \infty$$. Prove that $$\prod_{n=1}^\infty (1-a_n) = 0$$.
(Hint: Take the logarithm of the product.)
2. 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.)
1. 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.
1. Use Chebyshev's inequality to prove that $$P\bigl(|S_n| \leq 2\sqrt{n} \bigr) \,\geq\, 3/4$$ for all $$n\in\mathbb{N}$$.
2. Use the central limit theorem to guess the value of $\lim_{n\to\infty} \frac{E|S_n|}{\sqrt{n}}.$ You do not need to prove your answer.
3. If $$k\in\{-n,\ldots,n\}$$, find an explicit formula for $$P(S_{2n} = 2k)$$ involving a binomial coefficient.
4. Use question 1 and your formula from part (c) to prove that, almost surely, $$S_{2n} = 0$$ for infinitely many values of $$n$$.