New error bounds for Laplace approximationviaStein’s method

Robert E Gaunt

doi:10.1051/ps/2021012

Abstract

We use Stein’s method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [ALEA Lat. Am. J. Probab. Math. Stat.11(2014) 571–587] for Stein’s method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.

Highlights

The central limit theorem states that for a sequence of independent and identically distribution (i.i.d.) random variables, X1, X2, . . ., with zero mean and variance σ2 ∈(0, ∞), the standardised sum Wn = √1 σn n i=1Xi convergences in distribution to the standard normal distribution, as n → ∞
Stein’s method was adapted to the Laplace distribution by [38], and as an application they derived an explicit bound on the bounded Wasserstein distance between the distribution of Sp and its limiting Laplace distribution
Their approach, which involves the introduction of the so-called centered equilibrium transformation for Laplace approximation, mirrored that of [35], who used Stein’s method for exponential approximation to give explicit bounds on the rate of convergence in a generalisation of a well-known result of Renyi [40] concerning the convergence of geometric sums of positive random variables to the exponential distribution

Summary

Introduction

The central limit theorem states that for a sequence of independent and identically distribution (i.i.d.) random variables, X1, X2, . . ., with zero mean and variance σ2. 2), and as an application they derived an explicit bound on the bounded Wasserstein distance between the distribution of Sp and its limiting Laplace distribution Their approach, which involves the introduction of the so-called centered equilibrium transformation for Laplace approximation, mirrored that of [35], who used Stein’s method for exponential approximation to give explicit bounds on the rate of convergence in a generalisation of a well-known result of Renyi [40] concerning the convergence of geometric sums of positive random variables to the exponential distribution.

Stein’s method for the Laplace distribution

The Rayleigh Stein equation