Given a sequence
of random variable
, ... (the generalisation
to a sequence of sigma field
s is obvious
if you care about that sort of thing, and irrelevant
if you don't know or care
), a stopping time
is a random variable T
for which the event
} (for any fixed t
) depend only on the values of X1
, ..., Xt-1
A stopping time is just that: think that you participate in a sequence of trials, the result of trial #i being Xi. You may stop the sequence at any time. Then obviously your decision to stop at time t depends only on the first t values.
- Stop a sequence of coin tosses at the second head.
- Play a slot machine until either you lose all $100 you started off with, or you win $1000000.
- Stop a sequence of die rolls as soon as more than half the rolls came up 6.
- Employ a monkey to bang on a typewriter until it produces the complete text of Hamlet 2: The Prince is BACK.
- Repeatedly bet double or quits on a fair coin toss stopping just before you lose.
- Employ a monkey to bang on a typewriter until 30 minutes before it produces the complete text of Hamlet, then invite all your neighbours to see your educated monkey (bet on it producing Hamlet within 30 minutes).
As you can see, a stopping time is feasible, whereas other rules might not be. And, in fact, various theorems about betting are true for a fixed number of trials or for a stopping time (a fixed number is a stopping time!), but not for just any rule. Wald's theorem is probably the most important elementary result for stopping times.