Strong Mixing Conditions for Random Sequences

随机序列的强混合条件

• 批准号: 9703712
• 负责人: Richard Bradley
• 金额: $ 6万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703712&HistoricalAwards=false
• 关键词: Strong; Mixing; Conditions; Random; Sequences

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.

9703712随机序列和随机场的布拉德利强混合条件在模拟真实世界中的现象时很有用，在现实世界中，时间或位置上“相距很远”的观测彼此之间只有很小的影响。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讨论了在Banach空间中取值的严格平稳、强混合随机变量序列的部分和的渐近行为的一个可能的“三分法”。问题2涉及在(Rosenblatt)强混合条件下中心极限定理的“边界”的一部分的准确位置。问题3讨论在某种强混合条件下随机序列的谱密度。问题4涉及两个强混合条件之间可能的联系(或缺乏联系)；讨论了这个问题与I.A.Ibragimov的一个旧猜想的相关性。在现实世界中，有许多现象似乎是“弱依赖”的，即在时间或位置上“接近”的观测可能会对彼此产生相当大的影响，而在时间或位置上“相距遥远”的观测只会对彼此产生很小的影响。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，但想必对2050年的就业率几乎没有影响。同样，给定位置的风流波动可能与10英里外的(风流)波动高度相关，但与3000英里外的波动可能没有明显的相关性。对于这种现象的建模，概率论中已经投入了大量的注意力来研究一种广泛类型的弱相关性，这种类型的弱相关性被称为`强子g混合条件‘’。对强混合条件的“结构性”性质的理解有助于评估它们是否适合对给定的现实世界现象进行建模，而与这些条件有关的“平均定律”可以为给定现象的统计推断提供基础。这项研究涉及与各种强混合条件有关的“结构”性质和“平均定律”的几个问题。