A famous result in renewal theory is the central limit theorem for renewal processes. Since, in applications, usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is desirable. We provide an explicit non-uniform bound for the renewal central limit theorem based on Stein’s method and track the explicit values of the constants. For this bound the inter-arrival time distribution is required to have only a second moment. As an intermediate result of independent interest we obtain explicit bounds in a non-central Berry–Esseen theorem under second moment conditions.

中文翻译：

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二阶矩条件下更新过程中心极限定理收敛速度的界

更新理论的一个著名结果是更新过程的中心极限定理。因为在应用中，通常只有有限时间间隔的观测值可用，所以需要 Kolmogorov 距离到正态分布的界限。我们基于 Stein 方法为更新中心极限定理提供了一个显式的非均匀界，并跟踪常数的显式值。对于这个界限，到达间隔时间分布只需要第二个时刻。作为独立兴趣的中间结果，我们在二阶矩条件下获得了非中心 Berry-Esseen 定理的明确界限。