I am studying martingales and I have a few conceptual questions regarding why we need stopping times. My book (Probability and Computing by Mitzenmacher and Upfal) defines a martingale as follows:
A sequence of random variables $Z_0,Z_1,\ldots$ is a martingale with respect to the sequence $X_0,X_1,\ldots$ if $\forall n\geq 0$, the following condition holds:
- $Z_n$ is a function of $X_0,X_1,\ldots, X_n$
- $\mathbb{E}(|Z_n|) < \infty$
- $\mathbb{E}(Z_{n+1}\mid X_0,\ldots, X_n) = Z_n$
Here is what I don't get: It seems to me, you could just pick any random variable and symbolically assert the following:
$\forall n \geq 0, \mathbb{E}(Z_n)=\mathbb{E}(Z_0) $ using the tower of expectations property recursively, so how do I symbolically verify the need to worry about the stopping time and develop the martingale stopping theorem?
PS: Is the following guess as to why we need the stopping theorem correct? We need it because the original theorem might be defined for a countably infinite number of random variables and stopping it at a random time might break the conditions under which it holds?