# Spring 2018, Math 171

#### Week 7

1. 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) $$\sum_{k=0}^m {2m \choose 2k} t^{2k}$$
2. 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)