Jim Belk University of Glasgow

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.)
  2. 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 \(n\).