Last quarter, I did a DRP on martingales, and at the start of this quarter, gave a short talk on what I learned. Here are the notes from that talk.

**1. Conditional Expectation**

**Definition 1.** Suppose is a probability space, is a sub -algebra, and is a random variable.

- The conditional expectation of given , , is any -measurable random variable such that
- If is another random variable, then is defined as where is the -algebra generated by .

*Fact.* The conditional expectation exists, is unique, and is integrable.

Intuitively, we can think of as the best guess of the value of given the information available in .

**Example 1.**

- If is -measurable, then .
- If and are independent, then .
- Let be random variables with mean , and let be the partial sums. Then, if ,
The idea is that you know your position at , and you take steps whose sizes are, on average, , so your best guess for your position is .

**2. Martingales **

A martingale is a model of a fair game.

**Definition 2.** Consider a filtration (increasing sequence of -algebras) and a sequence of random variables , each measurable with respect to and integrable. Then, if for all , we say is a martingale.

**Example 2.** Let be random variables that take only the values and with probability each. Then is a martingale, because

**Theorem 3.** * If is a martingale with , then converges a.s. to a limit with . *

I’ll only sketch this proof, because even though the idea is nice, the details are a little annoying. The idea is to set up a way of betting on a martingale, show that you can’t make money off such a betting system, and then use this to draw conclusions about the martingales behavior.

**Definition 4.** Let be a filtration. A sequence of random variables is said to be predictable if is measurable for all .

That is, the value of can always be predicted with certainty at time . Then, if we “bet” an amount on the martingale , our total winnings will be

If is a supermartingale and is bounded, then is a supermartingale as well.

Now, fix an interval and choose in the following way: if , then bet 1, and continue to do so until . Then bet zero until you fall back below , and repeat. Every time you go from below to above , you will make a profit of at least . Let be the number of these upcrossings that occur. Then, you can use the previous fact, together with the bound on , to show that the number of these upcrossings is finite with probability 1. Since this is true for arbitrary , the must converge.

Finally, we give a brief application of the martingale convergence theorem.

**Proposition 5.** * Let be a sequence of random variables such that Then converges with probability 1. *

*Proof:* Let It’s easy to check that is a martingale, and that is bounded (in fact, it’s always 0), so the converge almost surely. Thus the random harmonic series converges almost surely.