Subdiagonal and almost uniform distributions
The space \(W\) of almost uniform distributions contains those distributions whose functions \(F\) are almost surely (a.s.) the identity, i.e. for which \(F\circ F=F\). A distribution function \(F\) on \([0,1]\) is called subdiagonal if \(F([0,t])\leq t\) for all \(t\in [0,1]\). \textit{U. Hirth} and the author [J. Theor. Probab. 13, No. 3, 609-634 (2000; Zbl 0971.60016)] have shown a close connection between \(W\) and the extreme exchangeable random total orders on the natural number set \(N\). The present paper wants to point out another interesting property of the space \(W\), viz. that the compact and convex set \(K\) of all subdiagonal distributions on \([0,1]\) has as extreme points the almost uniform distributions of the space \(W\). A simple example shows that \(K\) is not a simplex.
scientific article; zbMATH DE number 1873761