Primitive Sequences for Probability Measures on Compact Intervals

TL;DR

This paper introduces primitive sequences as a novel sequence representation of probability measures on compact intervals, linking them to moments and enabling new insights into measure characterization and bounds.

Contribution

It develops the concept of primitive sequences for probability measures, connecting them to classical moment theory and establishing a homeomorphism with the space of measures.

Findings

Primitive sequences can be identified as factorially rescaled moments of reflected variables.

The map from probability measures to primitive sequences is a homeomorphism.

Bounds on measures are derived from fixed initial primitive sequence terms.

Abstract

We introduce a sequence representation of a random variable $X$ supported on a compact interval $[a,b]$, which we call a primitive sequence. We construct this sequence by repeatedly antidifferentiating the associated cumulative distribution function of $X$ and evaluating the antiderivatives at the endpoint $b$. We show that the primitive sequence of $X$ can be identified as a factorially rescaled moment sequence of the reflected random variable $b-X$. Through this identification, we show that the primitive sequence transparently captures qualitative features of the distribution of $X$. We then connect primitive sequences directly to classical moment theory and exploit this connection to characterize admissible primitive sequences and to show that under natural topologies, the map from probability measures to primitive sequences is a homeomorphism. We end by examining the set of probability measures whose first $m$ primitive sequence terms are fixed, and thereby obtaining sharp upper and lower bounds on two functionals of those measures.