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On the Voigt profile and its dual

Massimo Cannas

TL;DR

The paper reframes the Voigt profile as a normal-scale mixture with Levy mixing, establishing a stochastic representation $U=ZL^{1/2}$ and showing $L\sim\mathrm{Levy}(\sigma^2,\gamma^2)$; it then introduces and analyzes the Dual Voigt, proving it also admits a normal-scale mixture form with mixing $V'=\tfrac{2}{\sigma^2}-L\,[L<\tfrac{2}{\sigma^2}]$, and discusses its (non)infinite divisibility. It provides theoretical results on existence, moments, and duality, and develops a practical parameter-estimation framework via truncated-normal transformations, complemented by an accept-reject sampling algorithm and public code. The work situates Voigt and Dual Voigt within the landscape of normal-scale mixtures, dual densities, and Bayesian prior predictive modeling, with implications for spectroscopy, stochastic processes, and statistical inference. Overall, the paper offers a coherent probabilistic foundation for the Voigt profile and its dual, plus concrete estimation techniques and computational tools for applications.

Abstract

The Voigt profile is the density obtained from the convolution of a Gaussian and a Cauchy and it is widely used in atomic and molecular spectroscopy. We show that the Voigt profile is a scale mixture of Gaussian distributions, with mixing Levy distribution. A consequence of this result is that there exists a dual of the Voigt distribution, which is itself a normal scale mixture. Both the Dual Voigt and its mixing are transformations, via truncation and reflection, of the Normal and Levy random variables. We discuss the dual Voigt characteristics, propose algorithms for parameter estimation and outline further developments.

On the Voigt profile and its dual

TL;DR

The paper reframes the Voigt profile as a normal-scale mixture with Levy mixing, establishing a stochastic representation and showing ; it then introduces and analyzes the Dual Voigt, proving it also admits a normal-scale mixture form with mixing , and discusses its (non)infinite divisibility. It provides theoretical results on existence, moments, and duality, and develops a practical parameter-estimation framework via truncated-normal transformations, complemented by an accept-reject sampling algorithm and public code. The work situates Voigt and Dual Voigt within the landscape of normal-scale mixtures, dual densities, and Bayesian prior predictive modeling, with implications for spectroscopy, stochastic processes, and statistical inference. Overall, the paper offers a coherent probabilistic foundation for the Voigt profile and its dual, plus concrete estimation techniques and computational tools for applications.

Abstract

The Voigt profile is the density obtained from the convolution of a Gaussian and a Cauchy and it is widely used in atomic and molecular spectroscopy. We show that the Voigt profile is a scale mixture of Gaussian distributions, with mixing Levy distribution. A consequence of this result is that there exists a dual of the Voigt distribution, which is itself a normal scale mixture. Both the Dual Voigt and its mixing are transformations, via truncation and reflection, of the Normal and Levy random variables. We discuss the dual Voigt characteristics, propose algorithms for parameter estimation and outline further developments.

Paper Structure

This paper contains 12 sections, 5 theorems, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Voigt as a normal scale mixture
  3. Composition of stable laws
  4. Stochastic processes
  5. Voigt as prior predictive distribution
  6. The Dual Voigt distribution
  7. The Dual Voigt as a normal scale mixture
  8. Infinite divisibility of the Voigt and Dual Voigt
  9. Parameter estimation for the Dual Voigt
  10. Discussion
  11. Existence of the Dual Voigt
  12. Code for Dual Voigt random samples and parameter estimation

Key Result

Lemma 2.1

Let $Z$, $Z_1$ and $Z_2$ be standard normal variables. Let also $L$ and $C$ be Levy and Cauchy distribution, respectively. The following representations hold:

Figures (3)

  • Figure B1: The dual Voigt density for $\sigma=\gamma$ pairs (left) and for $\sigma=1, \gamma=0.25, 0.5, 1, 2$ (right). Increasing values of $\gamma$ are associated with increasingly spiked densities. When $\sigma=\gamma$ the Dual Voigt peak exceeds the normal one when $\gamma<0.523$ (left).
  • Figure B2: Construction of the Dual Voigt mixing ($\sigma=1$). A Levy (gray) is truncated (gray, dashed) and reflected over its minimum (black).
  • Figure B3: Mixing distributions for the dual Voigt (left) and its dual (right).

Theorems & Definitions (5)

  • Lemma 2.1
  • Proposition 2.2: voigt as normal scale mixture
  • Proposition 3.1
  • Proposition 4.1
  • Proposition 6.1