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Injectivity of the Dubins-Freedman construction of random distributions
 

Summary: Injectivity of the Dubins-Freedman construction
of random distributions
Pieter Allart
University of North Texas
In the 1960s, L. Dubins and D. Freedman introduced the following
simple method to generate a probability distribution on the unit inter-
val at random. Starting with a base measure (a probability measure
on the unit square S = [0, 1]2
), pick a point at random according to
. This point divides S into four rectangles. Now map the measure
affinely onto the upper-right rectangle and iterate the process there;
and do the same for the lower-left rectangle. Iterating this procedure
ad infinitum one obtains the closed graph of a distribution function F.
This procedure defines a mapping P from the space of base
measures to the space P of probability measures on , where is
the space of all distribution functions on [0, 1] (suitably metrized). An
unsolved question is to find the largest possible "natural" domain for
this mapping on which the mapping is one-to-one. (It is clear that base
measures supported on the main diagonal of S must be excluded, as
must measures which give all their mass to two opposing points on the

  

Source: Assani, Idris - Department of Mathematics, University of North Carolina at Chapel Hill

 

Collections: Mathematics