## Bernoulli

### Prophet inequalities for optimal stopping rules with probabilistic recall

#### Abstract

Let Xi, i = 1, ..., n, be independent random variables, and consider an optimal stopping problem where an observation not chosen in the past is still available i steps later with some probability pi, 1 ≥ p1 ≥ ... ≥ pn -1 ≥ 0. Only one object may be chosen. After formulating the general solution to this optimal stopping problem, we consider prophet inequalities' for this situation. Let V\bf p (X1, ..., Xn) be the optimal value to the statistician. We show that for all non-trivial, non-negative Xi and all n ≥ 2, the ratio prophet inequality' \rm E[ \max (X1, ..., Xn)] < (2 - pn -1 ) V\bf p (X1, ..., Xn) holds, and 2 - pn -1 is the best constant'. This generalizes the classical prophet inequality with no recall, in which the best constant is 2. For any 0 ≤ Xi ≤ 1, the difference prophet inequality' \rm E[\max (X1, ..., Xn)] - V\bf p (X1, ..., Xn) ≤ (1- pn-1) [ 1 - (1 - pn - 1)1/2 ]2 / p2n-1 holds. Prophet regions are also discussed.

#### Article information

Source
Bernoulli, Volume 8, Number 1 (2002), 39-52.

Dates
First available in Project Euclid: 10 March 2004