On distance in total variation between image measures

TL;DR

This paper introduces a method to estimate the total variation distance between distributions of functions of random variables, achieving asymptotic optimality for Gaussian and trigonometric polynomial cases.

Contribution

It proposes a simple, general estimation method for total variation distance that is asymptotically optimal for certain classes of functions.

Findings

Effective estimation of total variation distance for Gaussian and trigonometric polynomial functions.

Asymptotic optimality of the proposed method as degree tends to infinity.

Applicable to a broad class of functions with potential for further extensions.

Abstract

We are interested in the estimation of the distance in total variation $$ \Delta := \|P_{f(X)} - P_{g(X)}\|_{\mathrm var} $$ between distributions of random variables $f(X)$ and $g(X)$ in terms of proximity of $f$ and $g.$ We propose a simple general method of estimating $\Delta$. For Gaussian and trigonometrical polynomials it gives an asymptotically optimal result (when the degree tends to $\infty$).