### 统计代写|随机过程代写stochastic process代考|STAT3021

statistics-lab™ 为您的留学生涯保驾护航 在代写随机过程stochastic process方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写随机过程stochastic process代写方面经验极为丰富，各种代写随机过程stochastic process相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机过程代写stochastic process代考|Does This Book Contain any Ideas?

At this stage, it is not really possible to precisely describe any of the new ideas which will be presented, but if the following statements are not crystal clear to you, you may have something to learn from this book:

Idea 1 It is possible to organize chaining optimally using increasing sequences of partitions.

Idea 2 There is an automatic device to construct such sequences of partitions, using “functionals”, quantities which measure the size of the subsets of the index set. This yields a complete understanding of boundedness of Gaussian processes.

Idea 3 Ellipsoids are much smaller than one would think, because they (and, more generally, sufficiently convex bodies) are thin around the edges. This explains the funny fractional powers of logarithms in certain matching theorems.

Idea 4 One may witness that a metric space is large by the fact that it contains large trees or equivalently that it supports an extremely scattered probability measure.
Idea 5 Consider a set $T$ on which you are given a distance $d$ and a random distance $d_\omega$ such that, given $s, t \in T$, it is rare that the distance $d_\omega(s, t)$ is much smaller than $d(s, t)$. Then if in the appropriate sense $(T, d)$ is large, it must be the case that $\left(T, d_\omega\right)$ is typically large. This principle enormously constrains the structure of many bounded processes built on random series.

Idea 6 There are different ways a random series might converge. It might converge because chaining witnesses that there is cancellation between terms, or it might converge because the sum of the absolute values of its terms already converges. Many processes built on random series can be split in two parts, each one converging according to one of the previous phenomena.

The book contains many more ideas, but you will have to read more to discover them.

## 统计代写|随机过程代写stochastic process代考|Gaussian Processes and the Generic Chaining

‘Ihis subsection gives an overview of Chap. 2. More generally, Sect. 1.7.n gives the overview for Chapter $n+1$.

The most important question considered in this book is the boundedness of Gaussian processes. The key object is the metric space $(T, d)$ where $T$ is the index set and $d$ the intrinsic distance (0.1). As investigated in Sect. 2.11, this metric space is far from being arbitrary: it is isometric to a subset of a Hilbert space. It is, however, a deadly trap to try to use this specific property of the metric space $(T, d)$. The proper approach is to just think of it as a general metric space.

After reviewing some elementary facts, in Sect. 2.4, we explain the basic idea of the “generic chaining”, one of the key ideas of this work. Chaining is a succession of steps that provide successive approximations of the index space $(T, d)$. In the Kolmogorov chaining, for each $n$, the difference between the $n$-th and the $(n+1)$-th approximation of the process, which we call here “the variation of the process during the $n$-th chaining step”, is “controlled uniformly over all possible chains”. Generic chaining allows that the variation of the process during the $n$-th chaining step “may depend on which chain we follow”. Once the argument is properly organized, it is not any more complicated than the classical argument. It is in fact exactly the same. Yet, while Dudley’s classical bound is not always sharp, the bound obtained through the generic chaining is optimal. Entropy numbers are reviewed in Sect. 2.5.

It is technically convenient to formulate the generic chaining bound using special sequences of partitions of the metric space $(T, d)$, that we shall call admissible sequences throughout the book. The key to make the generic chaining bound useful is then to be able to construct admissible sequences. These admissible sequences measure an aspect of the “size” of the metric space and are introduced in Sect. 2.7. In Sect. 2.8, we introduce another method to measure the “size” of the metric space, through the behavior of certain “functionals”, which are simply numbers attached to each subset of the entire space. The fundamental fact is that the two measures of the size of the metric space one obtains either through admissible sequences or through functionals are equivalent in full generality. This is proved in Sect. $2.8$ for the easy part (that the admissible sequence approach provides a larger measure of size than the functional approach) and in Sect. $2.9$ for the converse. This converse is, in effect, an algorithm to construct sequences of partitions in a metric space given a functional. Functionals are of considerable use throughout the book.

In Sect. 2.10, we prove that the generic bound can be reversed for Gaussian processes, therefore providing a characterization of their sample-boundedness. Generic chaining entirely explains the size of Gaussian processes, and the dream of Sect. $2.12$ is that a similar situation will occur for many processes.

In Sect. 2.11, we explain why a Gaussian process in a sense $i s$ nothing but a subset of Hilbert space. Remarkably, a number of basic questions remain unanswered, such as how to relate through geometry the size of a subset of Hilbert space seen as a Gaussian process with the corresponding size of its convex hull.

Dudley’s bound fails to explain the size of the Gaussian processes indexed by ellipsoids in Hilbert space. This is investigated in Sect. 2.13. Ellipsoids will play a basic role in Chap. 4.

# 随机过程代考

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## MATLAB代写

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