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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 ] . [ABSTRACT FROM AUTHOR]

A stick of length 1 is broken at random into n smaller sticks. How much inequality does this procedure produce? What happens if, instead of breaking a stick, we break a square? What happens asymptotically? Which is the most egalitarian distribution of the smaller sticks (or rectangles)? Usually, when studying inequality, one uses a Lorenz curve. The more egalitarian a distribution, the closer the Lorenz curve is to the first diagonal of [ 0 ,   1 ] 2 . This is why in the first section we study the space of Lorenz curves. What is the limit of a convergent sequence of Lorenz curves? We try to answer these questions, firstly, in the deterministic case and based on the results obtained there in the stochastic one. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let Δn be the n-dimensional simplex, ξ = (ξ1, ξ2,..., ξn) be an n-dimensional random vector, and U be a set of utility functions. A vector x* ∈ Δn is a U -absolutely optimal portfolio if E u ξ T x * ≥ E u ξ T x for every x ∈ Δn and u ∈ U . In this paper, we investigate the following problem: For what random vectors, ξ, do U -absolutely optimal portfolios exist? If U 2 is the set of concave utility functions, we find necessary and sufficient conditions on the distribution of the random vector, ξ, in order that it admits a U 2 -absolutely optimal portfolio. The main result is the following: If x0 is a portfolio having all its entries positive, then x0 is an absolutely optimal portfolio if and only if all the conditional expectations of ξ i , given the return of portfolio x0, are the same. We prove that if ξ is bounded below then CARA-absolutely optimal portfolios are also U 2 -absolutely optimal portfolios. The classical case when the random vector ξ is normal is analyzed. We make a complete investigation of the simplest case of a bi-dimensional random vector ξ = (ξ1, ξ2). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit U 2 -absolutely optimal portfolios. [ABSTRACT FROM AUTHOR]