Rank Independence and Rearrangements of Random Variables

الأصل · EN

A rearrangement of n independent uniform [0,1] random variables is a sequence of n random variables Y₁,...,Yₙ whose vector of order statistics has the same distribution as that for the n uniforms. We consider rearrangements satisfying the strong rank independence condition, that the rank of Yₖ among Y₁,...,Yₖ is independent of the values of Y₁,...,Yₖ₋₁, for k=1,...,n. Nontrivial examples of such rearrangements are the travellers' processes defined by Gnedin and Krengel. We show that these are the only examples when n=2, and when certain restrictive assumptions hold for n≥ 3; we also construct a new class of examples of such rearrangements for which the restrictive assumptions do not hold.