Long-Memory Processes: Probabilistic Properties and Statistical Methods
Long-Memory Processes: Probabilistic Properties and Statistical Methods
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DOI：
10.1007/978-3-642-35512-7
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发表时间：
2013-05
期刊：
影响因子：
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通讯作者：
J. Beran;Yuanhua Feng;Sucharita Ghosh;Rafal Kulik
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文献类型：
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作者：
J. Beran;Yuanhua Feng;Sucharita Ghosh;Rafal Kulik
已知长记忆或更一般地分形过程在许多科学学科和应用领域中发挥重要作用，例如物理学、地球物理学、水文学、经济学、金融学、气候学、环境科学、生物学、医学、电信、网络工程等。在长记忆领域中，进程之所以普遍存在，有几个原因。首先，在部分和的极限定理中，双曲标度自然发生（直到缓慢变化的函数的修改），因为在非常一般的条件下，极限过程必然是自相似的。事实上，人们可以说，在随机过程的世界中，自相似过程扮演着与有限维分布世界中的稳定分布（包括正态分布）相同的基本角色。双曲标度现象也是统计物理学中的一个重要组成部分（一个相关的概念是，例如，所谓的重整化群）。这是，至少部分地，与自相似过程在极限定理中的作用。长记忆现象发生的另一个原因是聚集。这一点，再加上异质性，是在经济背景下对长期依赖的一种常见解释。在电信和计算机网络中，等待时间的分布特性可以导致类似的结果。最后，还有与分形的联系（尽管并不总是直接的，取决于更具体的分布假设）。尽管长记忆和相关主题的概念可以追溯到20世纪初甚至19世纪末，但公平地说，曼德尔布罗特和他的同事的开创性工作已经引起了更广泛的数学受众（特别是概率学家和统计学家）的注意。类似的开创性作用可以归功于经济学领域的格兰杰，物理学领域的多布鲁申（以及更早的科尔莫戈罗夫），甚至更早的水文学领域的赫斯特。这些早期的贡献激励了一些杰出的probabilists发展理论的随机过程领域的随机自相似性，标度律和非标准极限定理。统计方法的发展也随之而来。例如，可以在Beran（1994a）中找到20世纪90年代初最新技术水平的概述。其他关于该主题的书籍和专著，其中大多数特别侧重于某些应用领域或特定方法或过程，vii
Long-memory, or more generally fractal, processes are known to play an important role in many scientific disciplines and applied fields such as physics, geophysics, hydrology, economics, finance, climatology, environmental sciences, biology, medicine, telecommunications, network engineering, to name a few. There are several reasons for the ubiquitous occurrence of processes in the realm of long memory. First of all, hyperbolic scaling occurs naturally (up to modifications by slowly varying functions) in limit theorems for partial sums, since, under very general conditions, the limiting processes are necessarily self-similar. One may in fact say that in the world of stochastic processes, self-similar processes play the same fundamental role as stable distributions (including the normal) in the world of finitedimensional distributions. Hyperbolic scaling phenomena are also an essential ingredient in statistical physics (a related notion is, for example, the so-called renormalization group). This is, at least partially, connected with the role of self-similar processes in limit theorems. Another reason for the occurrence of long-memory phenomena is aggregation. This, together with heterogeneity, is a frequent explanation of long-range dependence in an economic context. In telecommunications and computer networks, distributional properties of waiting times can lead to similar results. Finally, there is also a connection to fractals (though not always direct, depending on more specific distributional assumptions). Although the notion of long memory and related topics can be traced far back into the early 20th or even the late 19th century, it is probably fair to say that the subject has been brought to the attention of a wider mathematical audience (and, in particular, probabilists and statisticians) by the pioneering work of Mandelbrot and his coworkers. A similar pathbreaking role can be attributed to Granger in economics, to Dobrushin (and before, to Kolmogorov) in physics and, even earlier, to Hurst in hydrology. These early contributions motivated a number of eminent probabilists to develop a theory of stochastic processes in the realm of stochastic self-similarity, scaling laws and nonstandard limit theorems. The development of statistical methods followed. An overview of the state of the art in the early 1990s can be found, for instance, in Beran (1994a). Other books and monographs on the topic, most of them with a special focus on certain areas of application or specific methods or processes, vii