Strictly subgaussian probability distributions

TL;DR

This paper investigates a class of probability distributions with subgaussian Laplace transforms, extending previous work, and explores their properties, including conditions on zeros of characteristic functions and implications for the central limit theorem with Rényi entropy.

Contribution

It extends the class of subgaussian distributions using Hadamard's theorem and provides new conditions based on zeros of characteristic functions, also analyzing distributions with periodic components.

Findings

Extended the class of subgaussian distributions using Hadamard's theorem.

Identified conditions on zeros of characteristic functions for subgaussianity.

Established CLT with Rényi entropy divergence for distributions with periodic components.

Abstract

We explore the class of probability distributions on the real line whose Laplace transform admits a strong upper bound of subgaussian type. Using Hadamard's factorization theorem, we extend the class $\mathfrak L$ of Newman and propose new sufficient conditions for this property in terms of location of zeros of the associated characteristic functions in the complex plane. The second part of this note deals with Laplace transforms of strictly subgaussian distributions with periodic components. This subclass contains interesting examples, for which the central limit theorem with respect to the R\'enyi entropy divergence of infinite order holds.