## Martingales

Here, I give a quick review of the concept of a Martingale. A Martingale is a sequence of random variables satisfying a specific expectation conservation law. If one can identify a Martingale relating to some other sequence of random variables, its use can sometimes make quick work of certain expectation value evaluations.

This note is adapted from Chapter 2 of Stochastic Calculus and Financial Applications, by Steele.

## Stochastic geometric series

Let $a_1, a_2, \ldots$ be an infinite set of non-negative samples taken from a distribution $P_0(a)$, and write
$$\tag{1} \label{problem} S = 1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \ldots.$$
Notice that if the $a_i$ were all the same, $S$ would be a regular geometric series, with value $S = \frac{1}{1-a}$. How will the introduction of $a_i$ randomness change this sum? Will $S$ necessarily converge? How is $S$ distributed? In this post, we discuss some simple techniques to answer these questions.

Note: This post covers work done in collaboration with my aged p, S. Landy.