Optimal stopping of independent random variables and maximizing prophets. (English) Zbl 0563.60044

The prophet inequality for a sequence of independent nonnegative random variables shows that the ratio of the mean of the maximum of the sequence to the optimal expected return using stopping times is always bounded by 2; i.e., on average, the proportional advantage of a prophet with complete foresight over a gambler using nonanticipating stopping rules is at most 2.
Here, an inequality linking the mean of the sum of the k largest order statistics of the sequence and the optimal expected return is derived. This implies that if the k largest order statistics are close to the maximum in mean then the proportional advantage of the prophet is at most of order \((k+1)/k\). An extension of the additive prophet inequality for uniformly bounded independent random variables is also given.


60G40 Stopping times; optimal stopping problems; gambling theory
62L15 Optimal stopping in statistics
Full Text: DOI