Normal approximation for partial sums: general convex costs

TL;DR

This paper establishes explicit non-asymptotic bounds and asymptotic limits for convex transport costs between partial sums of i.i.d. random variables and the normal distribution, using a novel approach based on ideal distances and Lindeberg's method.

Contribution

It introduces a new method to derive explicit bounds and limits for convex transport costs in the context of partial sums of i.i.d. variables.

Findings

Provides explicit non-asymptotic bounds for convex transport costs.

Derives asymptotic limits with explicit constants.

Uses an adaptation of Lindeberg's method for the analysis.

Abstract

We provide non-asymptotic bounds and asymptotic limits for convex transport costs between the distribution of partial sums of independent and identically distributed square integrable and centered random variables and the normal distribution with mean zero and the same variance. The proof relies on controlling the transport cost by an appropriate ideal distance, combined with an adaptation of Lindeberg's method. The numerical constants and the asymptotic constants are explicit.