A short proof of L\'{e}vy's continuity theorem without using tightness

TL;DR

This paper offers a concise, direct proof of Lévy's continuity theorem in any dimension, avoiding traditional theorems, and demonstrates how to establish absolute continuity of distributions with integrable characteristic functions.

Contribution

It introduces a novel proof method for Lévy's theorem that bypasses standard reliance on tightness and related theorems, using convolution with Gaussian distributions.

Findings

Provides a shorter proof of Lévy's continuity theorem.

Shows distributions with integrable characteristic functions are absolutely continuous.

Derives the density formula for such distributions.

Abstract

In this note we present a new short and direct proof of L\'{e}vy's continuity theorem in arbitrary dimension $d$, which does not rely on Prohorov's theorem, Helly's selection theorem or the uniqueness theorem for characteristic functions. Instead, it is based on convolution with a small (scalar) Gaussian distribution as well as on basic facts about weak convergence and measure theory. Moreover, we show how, by similar means, one may prove the fact that a distribution with integrable characteristic function is absolutely continuous with respect to $d$-dimensional Lebesgue measure and derive the formula for its density.