Why is stopping time defined as a random variable?

Solution 1:

It sometimes is a useful exercise to separate the random from the non-random pieces of the puzzle.

Let's build up a stopping time, starting without randomness, along the lines of your intuition. Suppose that the observations $X_j$ take values in the space $S$, and let $S^\mathbb{N}$ be the space of $S$-valued sequences.

For any strategy or stopping policy, and any $0\leq n<\infty$ we may define a two-valued map $\phi_n:S^\mathbb{N}\to\{\mbox{GO},\mbox{STOP}\}$ which tells me what to do at time $n$ if I were to observe $s=(s_0,s_1,\dots)$. We require that $\phi_n(s)$ only depends on the first part of the sequence $(s_0,s_1,\dots,s_n)$. That is, the decision to stop at time $n$ must only depend on the observations up to time $n$. No peeking into the future!

Now define $\phi(s)=\inf(n\geq 0: \phi_n(s)=\mbox{STOP})$, where the infimum over the empty set is $\infty$. This gives a map $\phi:S^\mathbb{N}\to \mathbb{N}\cup \{\infty\}$ which expresses our policy, by telling us when to stop.

Finally we can put probability back into the picture by defining $\tau:\Omega\to \mathbb{N}\cup \{\infty\}$ by $$\tau(\omega)=\phi(X_0(\omega), X_1(\omega), X_2(\omega), \dots ).$$ This random variable is the stopping strategy applied to the random sequence $(X_0(\omega), X_1(\omega), X_2(\omega), \dots)$.

Every stopping time $\tau$ can be expressed like this for some such $\phi$.
For $0\leq n< \infty$, by the Doob-Dynkin lemma, there is a measurable map $\varphi_n:(S^\mathbb{N},{\cal G}_n) \to \{0,1\}$ so that $1_{[\tau=n]}=\varphi_n(X_0,X_1,X_2,\dots)$. Here ${\cal G}_n$ is the $\sigma$-field generated by the coordinate maps $s_j$ for $0\leq j\leq n$. Now let $\phi(s)=\inf(n\geq 0: \varphi_n(s)=1)$.

Solution 2:

Whether $\tau\le n$ is determined by $X_1,\ldots,X_n$. It depends on those, and those are random, so $\tau$ is random. One can ask, for example, for the expected value of $\tau$ or the probability that $\tau=6$. One could not do that if $\tau$ were not a random variable, which has a probability distribution.

Notice, though, that the value of $\tau$ is not determined by $X_1,\ldots,X_n$ if they determine (as they sometimes do) that $\tau>n$.

An unpleasant complication is that a different conventional definition of "stopping time" is sometimes also used: A random variable $\tau$ is a stopping time for a sequence $X_1,X_2,X_3,\ldots$ if for each $n$ the even $\tau=n$ is independent of $X_{n+1},X_{n+2},X_{n+3},\ldots$.