What's the expected stopping time for a sum of random variables to cross a threshold?

Question

I have a sequence of non-negative identically (and I believe independently) distributed random variables $X_i$. I know their expectation $\mu = \mathbb{E}[X_i]$. And I'm waiting for the sum to cross some threshold $t$. Define the stopping time $\tau$ as $$ \tau = \min_k : \sum_{i=1}^k X_i \geq t\,. $$ Then what's the expectation of $\tau$? Intuitively it should be $$ \mathbb{E}[\tau] = \frac{t}{\mu}\,. $$ But I'm not entirely convinced.

Answer 1

Yes, this is generally referred to as Wald's Equation or more generally Wald's Theorem. The assumptions are that $E[|X_1|]<\infty$ and $E[\tau]<\infty$. You can find a proof here.