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

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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.