Spring 2018, Math 171

Martingales
1. Let \((X_n)_{n \ge 0}\) be i.i.d. uniform on \([-1, 0) \cup (0, 1]\)
  1. Show that \((M_n)_{n \ge 0}\) with \(M_n = X_0 + \dots + X_n\) is a Martingale.
  2. Show that \((M_n)_{n \ge 0}\) with \(M_n = \frac{1}{X_0} + \dots + \frac{1}{X_n}\) is not a Martingale.
    - (Answer) \(\mathbb{E}|M_n| = \infty\)
2. Let \((X_n)_{n \ge 1}\) be i.i.d. uniform on \(\{-1, 1\}\). AKA Rademacher distributed. Let \(S_n = X_1 + \dots + X_n\), \(S_0 = 0\).
  1. Compute the moment generating function of \(S_n\). That is, \(\mathbb{E}[e^{tS_n}]\)
    - (Answer) \(\left(\frac{e^t + e^{-t}}{2}\right)^n\)
  2. Find the odd moments of \(S_n\). That is, \(\mathbb{E}[S_n^{2m+1}]\). Explain briefly why the result makes sense.
    - (Answer) 0. Makes sense because \(S_n\) is symmetric about 0.
  3. Find a formula for the even moments of \(S_n\). That is, \(\mathbb{E}[S_n^{2m}]\).
    - (Answer) \(\sum_{k=0}^n \binom{n}{k}\left(\frac{1}{2}\right)^n (2k-n)^{2m}\)
  4. For what values of \(c_n\) is \(M_n = S_n^2 - c_n\) a martingale?
    - (Answer) \(c_n = n\)
  5. Find a formula for \(\mathbb{E}[S_{n+1}^{2m} \mid S_n=t]\) which depends only on \(m\) and \(t\)
    - (Answer) \(\sum_{k=0}^m {2m \choose 2k} t^{2k}\)
Optional Stopping
1. Problem 5.16 from the textbook (2nd edition)
2. Problem 5.17 from the textbook (2nd edition)
3. Problem 5.8 from the textbook (2nd edition)