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

In this research，we studied a generalization of sufficient conditions for (random) limit theorems of {X D2τn}，Where {X - D2n - D2} was a lattice of random elements of a metric space and {τ - D2n - D2} was a latticeof random multidimensional indices.We defined the Essential ε-independence condition of{X - D2n - D2} for {τ - D2n - D2} as a generalized version of the uniform and asymptotic independencecondition and gave the random limit theorem on this condition in this research.这些条件andresult are generalizations for the Uniform ε-independence条件和the random limit theorem onthat condition. Moreover，我们不应该只这样but also generalizations for the probabilistic uniform continuity （Anscombe）条件which waswidely applicable and was another condition not look like independence conditions at all. and we定义的另一个版本the Essential ε-independence条件，我们应该这样做equivalent that it held the random limit theorems for all lattice of random indices of some class，if the metric space {X - D2n - D2} took values on was separable.After this，我们应该研究的方法of constitution of useful stopping rules in the set of random indicesextended in this research

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.