Some Problems on the Convex Geometry of Probability Measures
This thesis consists of three main topics in which we explore the geometry and other features of certain convex sets arising in probabilistic contexts.
We first consider the set of laws of $K_n$, the number of distinct values among the first $n$ terms of a sequence, for infinite exchangeable sequences of random variables $(X_1,X_2,\ldots)$. We prove for $n=3$ that the extreme points of the convex set of all possible laws of $K_3$ are those derived from i.i.d. sampling from discrete uniform distributions and the limit case with $P(K_3=3)=1$. We also consider the problem in higher dimensions and variants of the problem for finite exchangeable sequences and exchangeable random partitions.
Second, we introduce the notion of a coherent pair of random variables, or two conditional probabilities of the same event, and study the convex set of laws on $[0,1]^2$ arising in this manner. We classify all extreme laws with a certain restriction on the support. We also discover a large class of extreme laws with finite and countably infinite support.
Third, we study the convex set of polynomial probability densities on $[0,1]$ of degree at most $n$. We review some known results, including characterization of the extreme points, a representation theorem, properties of the Bernstein polynomial basis, and the Lorentz degree which measures in some sense the representability of positive polynomials in the Bernstein basis. We map out the geometry for $n=2$, consider the uniform random sampling model, and compute the upper envelope for this set of polynomials.