Strong Mixing Conditions for Random Sequences

9703712 Bradley随机序列和随机场的强混合条件在模拟现实世界中的现象中是有用的，在这些现象中，在时间或位置上“相距很远”的观测彼此之间只有轻微的影响。本文研究了与强混合随机序列有关的几个问题，并简要讨论了与随机场的联系。问题1处理严格平稳、强混合随机变量序列在Banach空间中取其值的部分和的渐近行为的可能的“三分法”。问题2处理在（Rosenblatt）强混合条件下中心极限定理的部分“边界”的确切位置。问题3是在一定强混合条件下随机序列的谱密度问题。问题4涉及两个强混合条件之间可能存在的联系（或缺乏联系）；讨论了这个问题与伊布拉吉莫夫的一个老猜想的关系。在现实世界中，有许多现象似乎是“弱依赖”的，也就是说，在时间或位置上“接近”的观测结果可能会对彼此产生相当大的影响，而在时间或位置上“相距很远”的观测结果对彼此的影响很小。例如，1998年的年失业率可能对1999年和2000年的失业率有相当大的影响，但大概对2050年的就业率几乎没有影响。同样，某一地点的气流波动可能与10英里外的（气流）波动高度相关，但可能与3000英里外的波动没有明显的相关性。为了对这类现象进行建模，概率论中对一种广义的弱依赖性进行了大量的研究，这种弱依赖性被称为“强混合条件”。对强混合条件的“结构”特性的理解有助于评估它们对给定现实世界现象建模的适宜性，而与这些条件相关的“平均定律”可以为给定现象的统计推断提供基础。本研究涉及与各种强混合条件有关的“结构”性质和“平均规律”的几个问题。

9703712 Bradley Strong mixing conditions for random sequences and random fields have been useful in modeling phenomena in the real world in which observations that are ``far apart'' in time or location have only slight influence on each other. In this research, several questions are studied in connection with strongly mixing random sequences, and connections with random fields are discussed briefly. Question 1 deals with a possible ``trichotomy'' for the asymptotic behavior of the partial sums from a strictly stationary, strongly mixing sequence of random variables taking their values in a Banach space. Question 2 deals with the exact location of a part of the ``borderline'' of the central limit theorem under the (Rosenblatt) strong mixing condition. Question 3 deals with the spectral density of random sequences under a certain strong mixing condition. Question 4 deals with a possible connection (or lack of one) between two strong mixing conditions; the relevance of this question to an old conjecture of I.A. Ibragimov is discussed. In the real world, there are many phenomena which appear to be ``weakly dependent,'' in the sense that observations that are ``close'' in time or location may have considerable influence on each other, while observations that are ``far apart'' in time or location have only slight influence on each other. For example, the annual unemployment rate for the year 1998 may have considerable influence on the unemployment rates for the years 1999 and 2000, but will presumably have almost no influence on the employment rate in, say, the year 2050. Similarly, the fluctuations in the wind currents at a given location may be highly correlated with the (wind current) fluctuations ten miles away, but perhaps have no discernible correlation with the fluctuations 3000 miles away. For the modeling of such phenomena, much attention has been devoted in probability theory to a broad type of weak dependence known under the name ``stron g mixing conditions.'' An understanding of ``structural'' properties of strong mixing conditions can help in assessing their appropriateness for the modeling of a given real world phenomenon, and ``laws of averages'' connected with such conditions can provide the foundation for the statistical inference for the given phenomenon. This research deals with several questions concerning both ``structural'' properties and ``laws of averages'' in connection with various strong mixing conditions.