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

摘要本文主要研究满足强混合条件的随机序列和随机场。如果（根据适当的标准）“过去”和“未来”之间的随机“依赖”量随着“过去”和“未来”之间的“时间”量变得足够大而变得均匀任意小，那么一个随机序列就被称为满足“强混合条件”。对于随机场，“强混合”的概念是类似的，涉及到当它们之间的“距离”变大时，观测集之间的一致渐近独立性。对于这类随机序列和随机场，研究了它们的结构性质和极限定理。特别地，这个建议涉及到一些最近的工作的延续，主要研究者在强混合假设下证明了某一类随机场的中心极限定理，这比其他研究人员早期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，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，全国平均年失业率是许多似乎符合这种模式的现象之一；1995年的失业率可能会对1996年或1997年的失业率产生相当大的影响，但对2050年的失业率影响不大。这一建议涉及对弱依赖性概率模型的各种“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”。特别地，这个建议的一部分涉及到首席研究员最近的一些工作的延续，这些工作大大扩大了已知满足某一重要平均定律的弱依赖模型的类别。这一建议还涉及对类似类型的概率模型的研究，在这些模型中，观测是按位置而不是时间分开的。

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.