论文标题
Edgeworth的扩展，用于独立有限整数有价值的随机变量
Edgeworth expansions for independent bounded integer valued random variables
论文作者
论文摘要
我们获得了均匀界限的局部总和的局部概率的渐近扩展，但不一定是分布的整数值随机变量。膨胀涉及多项式和三角多项式的产物。我们的结果不需要任何其他假设。作为我们扩展的应用，我们发现经典Edgeworth扩展的必要条件。事实证明，对于订单$ r $扩展的有效性，有两个可能的障碍。首先，基础部分总和之间的分布之间的距离Modulo在\ bbn $和均匀分布中可能无法为$ O（σ_n^{1-r}）$，其中$σ_n$是部分总和的标准偏差。其次，这种分布可能具有所需的亲密关系，但是这种亲密关系是不稳定的，因为它可以通过有限的许多术语来破坏它。在第一种情况下，订单$ r $的扩展失败。在第二种情况下，它可能会或可能不会取决于汇总导致均匀分布的分解的汇总功能的衍生物的行为。我们还表明，经典prokhorov条件的定量版本（对于强局中心限制定理）足以用于Edgeworth的扩展，而且从某种意义上说，这种情况是最佳的。
We obtain asymptotic expansions for local probabilities of partial sums for uniformly bounded independent but not necessarily identically distributed integer-valued random variables. The expansions involve products of polynomials and trigonometric polynomials. Our results do not require any additional assumptions. As an application of our expansions we find necessary and sufficient conditions for the classical Edgeworth expansion. It turns out that there are two possible obstructions for the validity of the Edgeworth expansion of order $r$. First, the distance between the distribution of the underlying partial sums modulo some $h\in \bbN$ and the uniform distribution could fail to be $o(σ_N^{1-r})$, where $σ_N$ is the standard deviation of the partial sum. Second, this distribution could have the required closeness but this closeness is unstable, in the sense that it could be destroyed by removing finitely many terms. In the first case, the expansion of order $r$ fails. In the second case it may or may not hold depending on the behavior of the derivatives of the characteristic functions of the summands whose removal causes the break-up of the uniform distribution. We also show that a quantitative version of the classical Prokhorov condition (for the strong local central limit theorem) is sufficient for Edgeworth expansions, and moreover this condition is, in some sense, optimal.
