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

9500307布拉德利摘要 这个建议涉及随机序列和随机场满足强混合条件的研究。 一个随机序列被称为满足“强混合条件”，如果（根据一个适当的标准）“过去”和“未来”之间的随机“依赖”的量变得均匀任意小，因为过去和未来之间的“时间”的量变得足够大。 对于随机场，“强混合”的概念是类似的，涉及到观测集之间的一致渐近独立性，因为它们之间的“距离”变得很大。 对于这样的随机序列和随机场，“结构性质”和极限定理进行了研究。 特别是，这项建议涉及到最近的一些工作的主要研究者的延续，其中的中心极限定理证明了强混合假设下的某一类随机场大大弱于早期类似的定理由其他研究人员。 这个建议涉及到概率模型的研究，涉及“弱相关”，其中“过去”和“现在”可能对“近期”有相当大的影响，但对“远期”的影响要小得多。“例如，全国年平均失业率是似乎符合这一模式的许多现象之一; 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。 这一提议涉及到对弱相关概率模型的各种“结构性质”的研究，以及这些模型可能满足的各种长期“平均律”。 特别是，这个建议的一部分涉及到继续最近的一些工作的主要研究人员，大大扩大了类弱依赖模型，已知满足一定的重要法律的平均值。该建议还涉及研究类似类型的概率模型，其中观测值按位置而不是时间分开。

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.