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布拉德利 随机序列和随机场的强混合条件已经被证明是 在模拟真实的世界中的现象时很有用，在这个世界中，观察结果“相距甚远” 在时间或地点上彼此之间只有轻微的影响。 在这项研究中，几个 研究了强混合随机序列的问题， 与随机场的连接进行了简要讨论。 问题1：一个可能的 “渐近性”的部分和的渐近行为，从严格平稳， 取值于Banach空间的随机变量的强混合序列。 问题2涉及中央界限的“边界线”的一部分的确切位置 Rosenblatt强混合条件下的一个定理。 问题3涉及 强混合条件下随机序列的谱密度。 问题4 处理两个强混合条件之间可能的联系（或缺乏联系）; 这个问题与I.A.的一个古老猜想的相关性。吉莫夫讨论。 在真实的世界中，有许多现象似乎是“弱”的 依赖，“在这个意义上，观察是“接近”的时间或位置可能有 相互之间有相当大的影响，而观察结果在时间上或 位置对彼此的影响很小。 例如，每年的失业率 1998年的失业率可能对2000年的失业率有很大影响， 1999年和2000年，但可能对就业几乎没有影响。 比如说，2050年。同样，在给定的风速下， 位置可能与十英里外的（风流）波动高度相关，但 也许与3000英里外的波动没有明显的相关性。 为 由于对这种现象的建模，在概率论中， 广义的弱相关性，称为强混合条件。安 了解强混合条件的“结构”特性有助于评估 它们对于给定的真实的世界现象的建模的适当性，以及 与这些条件相关的“平均数”可以为统计提供基础。 对给定现象的推论。 这项研究涉及几个问题 关于“结构”性质和“平均律”， 强混合条件