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 .
- 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 .
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.