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Examples of doubly stochastic measures supported on the graphs of two functions

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  • Holly Carley

Abstract

A doubly stochastic measure (DSM) is a measure μ on the unit square so that μ([0, 1] × A) = μ(A × [0, 1]) = m(A) where m is Lebesgue measure. The set of DSMs forms a convex set in the space of measures. It is known that DSMs supported on the union of two graphs of invertible functions are extreme points of that convex set (Seethoff and Shiflett, 1977/78). In general, there are few examples of extreme points in the literature. There are examples of so-called hairpins where the functions involved are inverses of each other, but there are also examples of the union of the graphs of a function and its inverse does not support a DSM (Sherwood and Taylor, 1988). In this paper, for a function f in a certain class, we find companion functions g so that the union of the graphs of f and g support a DSM even though the union of the graphs of f and f-inverse do not.

Suggested Citation

  • Holly Carley, 2017. "Examples of doubly stochastic measures supported on the graphs of two functions," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(17), pages 8628-8630, September.
    • Handle: RePEc:taf:lstaxx:v:46:y:2017:i:17:p:8628-8630
    DOI: 10.1080/03610926.2016.1186189
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