On the Probability of Finding Extremes in a Random Set

We consider a sequence Z j j ≥ 1 of i.i.d. d -dimensional random vectors and for every n ≥ 1 consider the sample S n = { Z 1 , Z 2 , … , Z n } . We say that Z j is a “leader” in the sample S n if Z j ≥ Z k , ∀ k ∈ { 1 , 2 , … , n } and Z j is an “anti-leader” if Z j ≤ Z k , ∀ k ∈ { 1 , 2 , … , n } . After all, the leader and the anti-leader are the naive extremes. Let a n be the probability that S n has a leader, b n be the probability that S n has an anti-leader and c n be the probability that S n has both a leader and an anti-leader. One of the aims of the paper is to compute, or, at least to estimate, or if even that is not possible, to estimate the limits of this quantities. Another goal is to find conditions on the distribution F of Z j j ≥ 1 so that the inferior limits of a n , b n , c n are positive. We give examples of distributions for which we can compute these probabilities and also examples when we are not able to do that. Then we establish conditions, unfortunately only sufficient when the limits are positive. Doing that we discovered a lot of open questions and we make two annoying conjectures—annoying because they seemed to be obvious but at a second thought we were not able to prove them. It seems that these problems have never been approached in the literature.