# The Breuer-Major Theorem in total variation: improved rates under minimal regularity

## TL;DR

This paper improves the total variation distance estimates in the Breuer-Major theorem by combining Malliavin-Stein methods with Gebelein's inequality, under minimal regularity assumptions on the function involved.

## Contribution

It provides new bounds for the total variation distance in the Breuer-Major theorem using weaker regularity conditions and novel Malliavin operator estimates.

## Key findings

- Enhanced total variation bounds under minimal regularity
- Novel combination of Gebelein's inequality with Malliavin techniques
- Applicable to functions with weak differentiability and finite moments

## Abstract

In this paper we prove an estimate for the total variation distance, in the framework of the Breuer-Major theorem, using the Malliavin-Stein method, assuming the underlying function $g$ to be once weakly differentiable with $g$ and $g'$ having finite moments of order four with respect to the standard Gaussian density. This result is proved by a combination of Gebelein's inequality and some novel estimates involving Malliavin operators.