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 Bradley 随机序列和随机场的强混合条件对于模拟现实世界中的现象非常有用，在现实世界中，在时间或位置上“相距很远”的观察结果彼此之间只有轻微的影响。 在这项研究中，研究了与强混合随机序列相关的几个问题，并简要讨论了与随机场的联系。 问题 1 涉及一种可能的“三分法”，用于从严格平稳、强混合的随机变量序列（在 Banach 空间中取值）得到的部分和的渐近行为。 问题2涉及（Rosenblatt）强混合条件下中心极限定理“边界线”部分的确切位置。 问题3涉及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存在的联系（或缺乏联系）； 这个问题与 I.A. 的一个古老猜想的相关性讨论了伊布拉吉莫夫。 在现实世界中，有许多现象似乎是“弱依赖”的，即时间或地点“接近”的观察结果可能相互之间有相当大的影响，而“弱依赖”的观察结果“时间或地点相距甚远，彼此影响甚微。 例如，1998年的年度失业率可能对1999年和2000年的失业率有相当大的影响，但对2050年的就业率可能几乎没有影响。同样，给定位置的风流可能与 10 英里外的（风流）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。 对于此类现象的建模，概率论中的很多注意力都集中在称为“强混合条件”的广泛类型的弱依赖性上。了解强混合条件的“结构”属性可以帮助您在评估它们对给定现实世界现象建模的适当性时，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。 这项研究涉及与各种强混合条件相关的“结构”特性和“平均定律”的几个问题。