 Proof of Wald's equation by stopping indicators

ID: proof-of-wald-s-equation-by-stopping-indicators

M

is a stopping time for independent identically distributed integrable variables

X_{i}

and

E M < \infty

, then

{M \geq i}

depends only on

X_{1}, \dots, X_{i - 1}

and is independent of

X_{i}

. Hence

E \sum_{i = 1}^{M} X_{i} = \sum_{i \geq 1} E [X_{i} 1_{{M \geq i}}] = E X_{1} \sum_{i \geq 1} P (M \geq i) = E X_{1} E M .

The interchange is justified because the same calculation with

∣ X_{i} ∣

gives finite expected absolute sum.

 New to topics? Read the docs here!