How many Laplace transforms of probability measures are there?

TL;DR

This paper establishes a bound on the complexity of Laplace transforms of probability measures by linking it to small deviation probabilities of a specific Gaussian process, offering insights into their structure.

Contribution

It introduces a novel bracketing metric entropy bound for Laplace transforms of probability measures using Gaussian process small deviation analysis.

Findings

Bound on the metric entropy of Laplace transforms

Connection between Laplace transforms and Gaussian process deviations

Potential applications in probability measure analysis

Abstract

A bracketing metric entropy bound for the class of Laplace transforms of probability measures on [0,\infty) is obtained through its connection with the small deviation probability of a smooth Gaussian process. Our results for the particular smooth Gaussian process seem to be of independent interest.