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**Laurent** Note that there is no conditionning by N here, hence this is not really a generalization of $\displaystyle E[X_1+\cdots+X_N|N]=NE[X_1]$.

A stopping time is what the name says: it is a time when you can decide to stop. In other words, suppose you discover the values X0,X1,... one after each other; then in order to stop at time n (i.e. to decide whether N=n), you can only look at the values X0,...Xn, not at the "future".

For instance, $\displaystyle N=\inf\{i\geq 0|X_1+\cdots+X_i>5\}$ is a stopping time: you can stop at time $\displaystyle N$ by waiting until a value exceeds 5.

On the other hand, $\displaystyle N=\sup\{i\leq 10|X_i<2\}$ is not a stopping time because you have to look at X0,...,X10 before you know where you should have stopped.

If you think about it, you'll see that the condition of "being able to stop at time N" is equivalent to "for all n, the event $\displaystyle \{N=n\}$ can be expressed in terms of X0,...,Xn".

The usual formal definition of a stopping time uses sigma-algebras (filtrations): for all $\displaystyle n$, $\displaystyle \{N=n\}\in\mathcal{F}_n$ where $\displaystyle \mathcal{F}_n=\sigma(X_0,\ldots,X_n)$ is the $\displaystyle \sigma$-algebra generated by $\displaystyle X_0,\ldots,X_n$.