The Leader Property in Quasi Unidimensional Cases

The following problem was studied: let Z j j ≥ 1 be a sequence of i.i.d. d -dimensional random vectors. Let F be their probability distribution and for every n ≥ 1 consider the sample S n = { Z 1 , Z 2 , … , Z n } . Then Z j was called a “leader” in the sample S n if Z j ≥ Z k , ∀ k ∈ { 1 , 2 , … , n } and Z j was an “anti-leader” if Z j ≤ Z k , ∀ k ∈ { 1 , 2 , … , n } . The comparison of two vectors was the usual one: if Z j = Z j 1 , Z j 2 , … , Z j d , j ≥ 1 , then Z j ≥ Z k means Z j i ≥ Z k i , while Z j ≤ Z k means Z j i ≤ Z k i , ∀ 1 ≤ i ≤ d , ∀ j , k ≥ 1 . 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. Sometimes these probabilities can be computed or estimated, for instance in the case when F is discrete or absolutely continuous. The limits a = lim inf a n , b = lim inf b n , c = lim inf c n were considered. If a > 0 it was said that F has the leader property, if b > 0 they said that F has the anti-leader property and if c > 0 then F has the order property. In this paper we study an in-between case: here the vector Z has the form Z = f X where f = f 1 , … , f d : 0 , 1 → R d and X is a random variable. The aim is to find conditions for f in order that a > 0 , b > 0 or c > 0 . The most examples will focus on a more particular case Z = X , f 2 X , … , f d X with X uniformly distributed on the interval [ 0 , 1 ] .