On existence and uniqueness of univariate Stein kernels

TL;DR

This work fully characterizes when a univariate distribution admits a Stein kernel and provides an explicit formula for the kernel on the absolutely continuous part. The existence criterion requires the ac part to have a positive Lebesgue density on the support interval, and the kernel is given by $\tau_X(t)=\sigma^2 \frac{q(t)}{p(t)} h(t)$ on that interval, with $q$ the density of the $X^{nz}$-biased law and $h$ the Radon–Nikodym derivative of $\mu_{ac}$ with respect to $\mu$. A key equivalence $\mu^{nz} = \sigma^{-2} \tau_X \mu$ links the kernel to the non-zero biased distribution, yielding a μ-a.s. unique, nonnegative kernel that is λ-a.e. positive on the interior. As an application, the paper derives a rate-optimal $n^{-1/2}$ total variation CLT for i.i.d. samples from such a class, with the rate expressed in terms of $\mathrm{Var}(\tau_X(X))$.

Abstract

We completely characterize the class of univariate distributions allowing for a Stein kernel and illustrate our result by means of some concrete distributions. Moreover, we apply our findings to prove a quantitative version of the central limit theorem with optimal rate $n^{-1/2}$ in total variation distance for i.i.d. random variables whose distribution belongs to that class.

Theorems & Definitions (5)

• proof
• proof
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• proof : Proof of Theorem \ref{['maintheo']}
• proof : Proof of Theorem \ref{['app']}