Wald's martingale

In probability theory, Wald's martingale is the name sometimes given to a martingale used to study sums of i.i.d. random variables. It is named after the mathematician Abraham Wald, who used these ideas in a series of influential publications.

Wald's martingale can be seen as discrete-time equivalent of the Doléans-Dade exponential.

Formal statement
Let $$(X_n)_{n \geq 1}$$ be a sequence of i.i.d. random variables whose moment generating function $$M: \theta \mapsto \mathbb{E}(e^{\theta X_1})$$ is finite for some $$\theta > 0$$, and let $$S_n = X_1 + \cdots + X_n$$, with $$S_0 = 0$$. Then, the process $$(W_n)_{n \geq 0}$$ defined by
 * $$W_n = \frac{e^{\theta S_n}}{M(\theta)^n}$$

is a martingale known as Wald's martingale. In particular, $$\mathbb{E}(W_n) = 1$$ for all $$n \geq 0$$.