# Non-monotone convergence in the quadratic Wasserstein distance

## TL;DR

This paper provides a simple counter-example demonstrating that the quadratic Wasserstein distance does not always decrease monotonically when considering n-fold normalized convolutions of two measures, challenging an existing assumption.

## Contribution

It presents the first explicit counter-example to the presumed monotonicity of quadratic Wasserstein distance under convolution.

## Key findings

- Quadratic Wasserstein distance can increase under convolution.
- Counter-example disproves the monotonicity assumption.
- Challenges previous beliefs in mass transport theory.

## Abstract

We give an easy counter-example to Problem 7.20 from C. Villani's book on mass transport: in general, the quadratic Wasserstein distance between $n$-fold normalized convolutions of two given measures fails to decrease monotonically.