Theorem 4.5 (Wald's Equations)

Consider a sequence  which are i.i.d, have a finite expectation and also assume that  is a stopping time for , such that , then

.

Proof of Theorem 4.5

Before we venture in trying to proof the above theorem note the interesting thing when both  and  are probabilistic.
Let us consider the indicator function, , where , such that we will have

.

Hence

We will stop if we successfully observe . Thus we do not terminate our experiment as and when we like but continue to observe and stop ONLY at the last reading which is . Therefore  is independent of .

We thus obtain



    (remember here the suffix  is not relevant as it can be any )

   

      
                  

Let  denote the inter arrival time of a renewal process. In this experiment let us consider we stop at the first renewal after , i.e., at the  renewal.

Note



 and

Thus the event  depends only on  and is definitely independent of  such that is the stopping time