### 统计代写|随机过程代写stochastic process代考|MATH3801

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• Statistical Computing 统计计算
• (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|The Kolmogorov Conditions

Kolmogorov stated the “Kolmogorov conditions”, which robustly ensure the good behavior of a stochastic process indexed by a subset of $\mathbb{R}^m$. These conditions are studied in any advanced probability course. If you have taken such a course, this section will refresh your memory about these conditions, and the next few sections will present the natural generalization of the chaining method in an abstract metric space, as it was understood in, say, 1970. Learning in detail about these historical developments now makes sense only if you have already heard of them, because the modern chaining method, which is presented in Chap. 2, is in a sense far simpler than the classical method. For this reason, the material up to Sect. $1.4$ included is directed toward a reader who is already fluent in probability theory. If, on the other hand, you have never heard of these things and if you find this material too difficult, you should start directly with Chap. 2 , which is written at a far greater level of detail and assumes minimal familiarity with even basic probability theory.

We say that a process $\left(X_t\right)_{t \in T}$, where $T=[0,1]^m$, satisfies the Kolmogorov conditions if
$$\forall s, t \in[0,1]^m, \mathrm{E}\left|X_s-X_t\right|^p \leq d(s, t)^\alpha .$$
where $d(s, t)$ denotes the Euclidean distance and $p>0, \alpha>m$. Here E denotes mathematical expectation. In our notation, the operator $\mathrm{E}$ applies to whatever expression is placed behind it, so that $\mathrm{E}|Y|^p$ stands for $\mathrm{E}\left(|Y|^p\right)$ and not for $(\mathrm{E}|Y|)^p$. This convention is in force throughout the book.

Let us apply the idea of chaining to processes satisfying the Kolmogorov conditions. The most obvious candidate for the approximating set $T_n$ is the set $G_n$ of points $x$ in $\left[0,1\left[^m\right.\right.$ such that the coordinates of $2^n x$ are positive integers. ${ }^1$ Thus, card $G_n=2^{n m}$. It is completely natural to choose $\pi_n(u) \in G_n$ as close to $u$ as possible, so that $d\left(u, \pi_n(u)\right) \leq \sqrt{m} 2^{-n}$ and $d\left(\pi_n(u), \pi_{n-1}(u)\right) \leq d\left(\pi_n(u), u\right)+$ $d\left(u, \pi_{n-1}(u)\right) \leq 3 \sqrt{m} 2^{-n}$.
For $n \geq 1$, let us then define
$$U_n=\left{(s, t) ; s \in G_n, t \in G_n, d(s, t) \leq 3 \sqrt{m} 2^{-n}\right} .$$
Given $s=\left(s_1, \ldots, s_m\right) \in G_n$, the number of points $t=\left(t_1, \ldots, t_m\right) \in G_n$ with $d(s, t) \leq 3 \sqrt{m} 2^{-n}$ is bounded independently of $s$ and $n$ because $\left|t_i-s_i\right| \leq d(s, t)$ for each $i \leq m$, so that we have the crucial property
$$\operatorname{card} U_n \leq K(m) 2^{n m},$$
where $K(m)$ denotes a number depending only on $m$, which need not be the same on each occurrence. Consider then the r.v.
$$Y_n=\max \left{\left|X_s-X_t\right| ;(s, t) \in U_n\right},$$
so that (and since $G_{n-1} \subset G_n$ ) for each $u$,
$$\left|X_{\pi_n(u)}-X_{\pi_{n-1}(u)}\right| \leq Y_n .$$

## 统计代写|随机过程代写stochastic process代考|Chaining in a Metric Space: Dudley’s Bound

Suppose now that we want to study the uniform convergence on $[0,1]$ of a random Fourier series $X_t=\sum_{k \geq 1} a_k g_k \cos (2 \pi i k t)$ where $a_k$ are numbers and $\left(g_k\right)$ are independent standard Gaussian r.v.s. The Euclidean structure of $[0,1]$ is not intrinsic to the problem. Far more relevant is the distance $d$ given by
$$d(s, t)^2=\mathrm{E}\left(X_s-X_t\right)^2=\sum_k a_k^2(\cos (2 i \pi k s)-\cos (2 i \pi k t))^2 .$$
This simple idea took a very long time to emerge. Once one thinks about the distance $d$, then in turn the fact that the index set $T$ is $[0,1]$ is no longer very relevant because this particular structure does not connect very well with the distance $d$. One is then led to consider Gaussian processes indexed by an abstract set $T .{ }^4 \mathrm{We}$ say that $\left(X_I\right){I \in T}$ is a Gaussian process when the family $\left(X_I\right){I \in T}$ is jointly Gaussian and centered. ${ }^5$ Then, just as in (1.16), the process induces a canonical distance $d$ on $T$ given by $d(s, t)=\left(\mathrm{E}\left(X_s-X_t\right)^2\right)^{1 / 2}$. We will express that Gaussian r.v.s have small tails by the inequality
$$\forall s, t \in T, \mathrm{E} \varphi\left(\frac{\left|X_s-X_t\right|}{d(s, t)}\right) \leq 1,$$
where $\varphi(x)=\exp \left(x^2 / 4\right)-1$. This inequality holds because if $g$ is a standard Gaussian r.v., then $E \exp \left(g^2 / 4\right) \leq 2 .^6$

To perform chaining for such a process, in the absence of further structure on our metric space $(T, d)$, how do we choose the approximating sets $T_n$ ? Thinking back to the Kolmogorov conditions, it is very natural to introduce the following definition:

Definition 1.4.1 For $\epsilon>0$, the covering number $N(T, d, \epsilon)$ of a metric space $(T, d)$ is the smallest integer $N$ such that $T$ can be covered by $N$ balls of radius $\epsilon .^7$
Equivalently, $N(T, d, \epsilon)$ is the smallest number $N$ such that there exists a set $V \subset T$ with card $V \leq N$ and such that each point of $T$ is within distance $\epsilon$ of $V$.

Let us denote by $\Delta(T)=\sup _{s, t \in T} d(s, t)$ the diameter of $T$ and observe that $N(T, d, \Delta(T))=1$. We construct our approximating sets $T_n$ as follows: Consider the largest integer $n_0$ with $\Delta(T) \leq 2^{-n_0}$. For $n \geq n_0$, consider a set $T_n \subset T$ with card $T_n=N\left(T, d, 2^{-n}\right)$ such that each point of $T$ is within distance $2^{-n}$ of a point of $T_n .{ }^8$ In particular $T_0$ consists of a single point.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|The Kolmogorov Conditions

Kolmogorov 陈述了“Kolmogorov 条件”，它有力地确保了由一个子集索引的随机过程的良好行为 $\mathbb{R}^m$. 这 些条件在任何高级概率课程中都有研究。如果你上过这样的课程，本节将刷新你对这些条件的记忆，接 下来的几节将展示链接方法在抽象度量空间中的自然推广，正如 1970 年人们所理解的那样。学习现在只 有当你已经听说过这些历史发展的细节时才有意义，因为第 1 章中介绍的现代链接方法。2，在某种意义 上远比经典方法简单。出于这个原因，材料高达教派。1.4包括在内的是针对已经精通概率论的读者。另 一方面，如果您从末听说过这些东西，并且觉得这些材料太难，则应该直接从第 1 章开始。2，它的详细 程度要高得多，并且假定您对基本概率论的了解最少。

$U_{-} n=\backslash l f t\left{(s, t) ; s \backslash\right.$ in G_n, $\left.t \backslash i n G_{-} n, d(s, t) \backslash e q 3 \backslash s q r t{m} 2^{\wedge}{-n} \backslash r i g h t\right}$.

$$\operatorname{card} U_n \leq K(m) 2^{n m}$$

$Y_{-} n=\backslash \max \backslash$ \eft $\left{\right.$ \eft $\mid X_{-} s-X_{-} \backslash \backslash$ ight $\left.\mid:(s, t) \backslash i n U_{-} n \backslash r i g h t\right}$,

$$\left|X_{\pi_n(u)}-X_{\pi_{n-1}(u)}\right| \leq Y_n .$$

## 统计代写|随机过程代写stochastic process代考|Chaining in a Metric Space: Dudley’s Bound

$$d(s, t)^2=\mathrm{E}\left(X_s-X_t\right)^2=\sum_k a_k^2(\cos (2 i \pi k s)-\cos (2 i \pi k t))^2 .$$

$$\forall s, t \in T, \mathrm{E} \varphi\left(\frac{\left|X_s-X_t\right|}{d(s, t)}\right) \leq 1$$

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## MATLAB代写

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