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.

中文翻译：

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Stein 方法和 Narayana 数

Narayana 数在组合数学和概率论中出现很多地方，众所周知，它们是渐近正态的。使用 Stein 的可交换对方法，我们提供了对阶~$n^{-1}$ 的对称二项分布的总变差的近似误差，这也意味着阶~$n^{-1/2 的 Kolmogorov 界}$ 为正常近似值。我们的可交换对基于生死链并具有非凡的特性，这使我们能够执行一些其他棘手的时刻计算。尽管我们的主要兴趣是 Narayana 数，但我们表明我们的主要抽象结果也可以提高泊松二项式和超几何分布的收敛速度。