[Paper Review] Multivariate approximation in total variation

A. D. Barbour, Malwina Luczak+1 more|arXiv (Cornell University)|Dec 23, 2015

Stochastic processes and statistical mechanics3 citations

TL;DR

This paper develops a Stein's method framework for multivariate discrete normal approximation in total variation distance for random vectors in $\mathbb{Z}^d$. By deriving a Stein equation with solution bounds and applying it to sums of independent integer-valued vectors, equilibrium distributions of Markov processes, and exchangeable pairs, it establishes a general approximation theorem with application to random colorings of regular graphs.

ABSTRACT

The paper lays the framework for the discrete normal approximation in total variation of random vectors in $Z^d$, using Stein's method. We derive an appropriate Stein equation, together with bounds on its solutions and their differences, and use them to formulate a general discrete normal approximation theorem. We illustrate the use of the method in three settings: sums of independent integer valued random vectors, equilibrium distributions of Markov population processes, and random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.

Motivation & Objective

• To establish a general framework for multivariate discrete normal approximation in total variation for $\mathbb{Z}^d$-valued random vectors.
• To address the lack of a systematic approach for total variation bounds in multivariate discrete settings using Stein's method.
• To derive a suitable Stein equation and solution bounds for discrete normal approximation in higher dimensions.
• To apply the framework to three distinct probabilistic models: sums of independent integer-valued vectors, equilibrium distributions of Markov population processes, and exchangeable pairs.
• To demonstrate the method’s utility through an application to random colorings of regular graphs.

Proposed method

• Derives a multivariate Stein equation tailored for discrete normal approximation in $\mathbb{Z}^d$.
• Establishes bounds on solutions of the Stein equation and their differences, essential for total variation error control.
• Applies the framework to sums of independent integer-valued random vectors using dependency structure and moment conditions.
• Adapts the method to equilibrium distributions of Markov population processes by leveraging generator properties.
• Utilizes the exchangeable pair approach to derive approximation bounds under symmetry and weak dependence.
• Validates the method’s robustness by applying it to random colorings of regular graphs, showing convergence to the multivariate normal distribution.

Experimental results

Research questions

• RQ1 How can Stein's method be extended to provide total variation bounds for multivariate discrete normal approximation in $\mathbb{Z}^d$?
• RQ2 What is the appropriate Stein equation and solution bounds for discrete multivariate normal approximation?
• RQ3 Can the framework be applied to sums of independent integer-valued random vectors with controlled dependency?
• RQ4 How does the method perform for equilibrium distributions of Markov population processes in the discrete setting?
• RQ5 What is the applicability of the method to exchangeable pairs and random graph colorings?

Key findings

• A general multivariate discrete normal approximation theorem in total variation is established using a tailored Stein equation and solution bounds.
• The method yields explicit total variation bounds for sums of independent integer-valued random vectors under moment and dependency conditions.
• The framework successfully extends to equilibrium distributions of Markov population processes, providing approximation guarantees via generator-based analysis.
• The exchangeable pair construction enables robust approximation bounds under weak dependence and symmetry.
• An application to random colorings of regular graphs shows convergence to the multivariate normal distribution, validating the method’s practical utility.