Mathematical Sciences: Strong Mixing Conditions for Random Sequences and Random Fields

9500307布拉德利摘要这一建议涉及研究满足强混合条件的随机序列和随机场。一个随机序列被称为满足“强混合条件”，如果(根据适当的标准)“过去”和“未来”之间的随机“依赖”量变得一致地任意小，而过去和未来之间的“时间”量变得足够大。对于随机场，“强混合”的概念是相似的，包括随着观测集合之间的“距离”变大，它们之间的一致渐近独立性。对于这种随机序列和随机场，我们研究了它们的“结构性质”和极限定理。特别地，这一建议涉及到主要研究者最近的一些工作的继续，在强混合假设下证明了某类随机场的中心极限定理，比其他研究者以前的类似定理要弱得多。这一建议涉及涉及“弱依赖”的概率模型的研究，在该模型中，“过去”和“现在”可能对“近未来”有相当大的影响，但对“远未来”的影响要小得多。例如，全国年平均失业率是似乎符合这种模型的许多现象之一；1995年的失业率可能对1996年或1997年的失业率有相当大的影响，但对2050年的失业率影响不大。这一建议涉及对弱相依性概率模型的各种“结构性质”的调查，以及这些模型可能满足的各种长期“平均定律”。特别是，这一建议的一部分涉及首席研究员最近的一些工作的继续，这项工作大大扩大了已知满足某一重要平均定律的弱相关性模型的类别。这一建议还涉及对类似类型的概率模型的研究，在这种模型中，观测按位置而不是按时间分开。

9500307 Bradley Abstract This proposal concerns the study of random sequences and random fields satisfying strong mixing conditions. A random sequence is said to satisfy a "strong mixing condition" if (according to an appropriate criterion) the amount of stochastic "dependence" between the "past" and the "future" becomes uniformly arbitrarily small as the amount of "time" between past and future becomes sufficiently large. For random fields, the notions of "strong mixing" are similar, involving the uniform asymptotic independence between sets of observations as the "distance" between them becomes large. For such random sequences and random fields, both "structural properties" and limit theorems are investigated. In particular, this proposal involves a continuation of some recent work by the principal investigator in which the central limit theorem is proved for a certain class of random fields under strong mixing assumptions considerably weaker than in earlier similar theorems by other researchers. This proposal concerns the study of probabilistic models that involve "weak dependence," in which the "past" and "present" might have considerable influence on the "near future" but much less influence on the "far future." For example, the average annual national unemployment rate is one of many phenomena which appear to fit such a model; the unemployment rate for the year 1995 could presumably have considerable effect on the rate for the year 1996 or 1997 but not much effect on the rate for the year 2050. This proposal involves an investigation into various kinds of "structural properties" of probabilistic models of weak dependence, and also various long-term "laws of averages" that such models might satisfy. In particular, part of this proposal involves a continuation of some recent work by the principal investigator which considerably enlarges the class of weak dependence models that are known to satisfy a certain important law of averages. This proposal also concerns the study of similar types of probabilistic models in which observations are separated by location instead of time.