Stopping with Exact Confidence

Suppose a fixed but unknown number of events occur at times T1, T2, ..., these times being independent and identically distributed. The distribution may be known, or unknown. Choose m >= 0 and a. We derive several stopping rules with the property that for all n > m the probability that when stopping occurs, no more than m events have still to occur, is exactly a. The case m = 0 is especially interesting, since if such a rule is applied, one has confidence exactly 1 - a that all the events have happened.