Stein's method and Narayana numbers

TL;DR

This paper applies Stein's method to analyze Narayana numbers, providing explicit error bounds for their approximation by a normal distribution, and extends these results to related distributions.

Contribution

It introduces a novel exchangeable pair construction based on a birth-death chain to obtain precise error bounds for Narayana numbers and related distributions.

Findings

Total variation error bound of order n^{-1} for Narayana numbers

Kolmogorov bound of order n^{-1/2} for normal approximation

Improved convergence rates for Poisson-binomial and hypergeometric distributions

Abstract

Narayana numbers appear in many places in combinatorics and probability, and it is known that they are asymptotically normal. Using Stein's method of exchangeable pairs, we provide an error of approximation in total variation to a symmetric binomial distribution of order~$n^{-1}$, which also implies a Kolmogorov bound of order~$n^{-1/2}$ for the normal approximation. Our exchangeable pair is based on a birth-death chain and has remarkable properties, which allow us to perform some otherwise tricky moment computations. Although our main interest is in Narayana numbers, we show that our main abstract result can also give improved convergence rates for the Poisson-binomial and the hypergeometric distributions.