Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent integral-valued random variables, and let $S_{n}=\sum_{j=1}^{n}X_{j}$ . One of the interesting probabilities is the probability at a particular point, i.e., the density of $S_{n}$ . The theorem that gives the estimation of this probability is called the local limit theorem. This theorem can be useful in finance, biology, etc. Petrov (Sums of Independent Random Variables, 1975) gave the rate $O (\frac{1}{n} )$ of the local limit theorem with finite third moment condition. Most of the bounds of convergence are usually defined with the symbol O. Giuliano Antonini and Weber (Bernoulli 23(4B):3268–3310, 2017) were the first who gave the explicit constant C of error bound $\frac{C}{\sqrt{n}}$ . In this paper, we improve the convergence rate and constants of error bounds in local limit theorem for $S_{n}$ . Our constants are less complicated than before, and thus easy to use.

中文翻译：

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积分值随机变量局部极限定理收敛速度的改进

假设$ X_ {1}，X_ {2}，\ ldots，X_ {n} $是独立的整数值随机变量，并让$ S_ {n} = \ sum_ {j = 1} ^ {n} X_ {j } $。有趣的概率之一是特定点的概率，即$ S_ {n} $的密度。估计该概率的定理称为局部极限定理。这个定理在金融，生物学等领域可能是有用的。Petrov（独立随机变量的总和，1975）给出了具有有限第三矩条件的局部极限定理的比率$ O（\ frac {1} {n}）$。收敛的大多数边界通常用符号O定义。Giuliano Antonini和Weber（Bernoulli 23（4B）：3268–3310，2017）是第一个给出显式常数C的错误约束$ \ frac {C} { \ sqrt {n}} $。在本文中，我们针对$ S_ {n} $提高了局部极限定理的收敛速度和误差范围常数。我们的常数没有以前那么复杂，因此易于使用。