Uniform asymptotic independence on essential parts of a sequence of random indices and sufficient conditions of limit theorems with random indices

在这项研究中，我们研究了{X-Y-D2τn-Y-D2}的（随机）限制理论的普遍化，Where {X_D2n_作为一个通用的版本的统一和不对称的独立条件，并在本研究中给出了这种条件的随机限制理论。这些条件和结果是统一ε-独立条件和在该条件上的随机限制理论的一般化。Moreover，我们展示了这些并不只是这样，但也是可能统一连续性的一般化（Anscombe）条件在很大程度上适用，而其他条件看起来不像所有情况的独立条件。我们定义了基本ε-独立条件的另一个版本，我们展示了这是等效的，因为它支持了一些类的所有随机指标的随机限制理论，如果度量空间{X-D2 n-D2}可以找到可分离的值。在此之后，我们将研究如何在这一研究中使用的停止规则集中使用的方法。

In this research, we studied a generalization of sufficient conditions for (random) limit theorems of {XィイD2τnィエD2} , Where {XィイD2nィエD2} was a lattice of random elements of a metric space and {τィイD2nィエD2} was a lattice of random multidimensional indices.We defined the Essential ε-independence condition of {XィイD2nィエD2} for {τィイD2nィエD2} as a generalized version of the uniform and asymptotic independence condition and gave the random limit theorem on this condition in this research. These condition and result are generalizations for the Uniform ε-independence condition and the random limit theorem on that condition. Moreover, we showed these were not only so, but also generalizations for the probabilistic uniform continuity (Anscombe) condition which was widely applicable and was another condition did not look like independence conditions at all. And we defined another version of the Essential ε-independence condition and we showed that this was equivalent that it held the random limit theorems for all lattice of random indices of some class, if the metric space {XィイD2nィエD2} took values on was separable.After this, we shall study for the methods of constitution of useful stopping rules in the set of random indices extended in this research.