New error bounds for Laplace approximation via Stein's method

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 \cite{pike} 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 it first two derivatives, of the Rayleigh Stein equation.

23 pages. To appear in ESAIM: Probability and Statistics, 2021+

References in corpus (4)

• On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands
• Stein's method and the Laplace distribution
• Weak Limit of the Geometric Sum of Independent But Not Identically Distributed Random Variables
• High order steady-state diffusion approximation of the Erlang-C system