Statement

Let $Y_1,Y_2,\dots$ be i.i.d. random variables with

$$ \mathbb E[|Y_1|]<\infty. $$

Let $T$ be a stopping time with $\mathbb E[T]<\infty$, and define

$$ S_n=\sum_{i=1}^n Y_i. $$

Then

$$ \mathbb E[S_T]=\mathbb E[T]\cdot \mathbb E[Y_1]. $$

Martingale interpretation

If

$$ \mu=\mathbb E[Y_1], $$

then

$$ S_n-n\mu $$

is a martingale. Applying OST to this martingale gives Wald’s identity.

Why it can fail

The formula may fail if:

  • the increments are not i.i.d.
  • $T$ is not a stopping time
  • $\mathbb E[T]=\infty$