## 统计代写|随机分析作业代写stochastic analysis代写|MA53200

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl相关的作业也就用不着说。

• Statistical Inference 统计推断
• 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|随机分析作业代写stochastic analysis代写|Continuous Distributions

Consider now the general case when $\Omega$ is not necessarily enumerable. Let us begin with the definition of a random variable. Denote by $\mathcal{R}$ the Borel $\sigma$-algebra on $\mathbb{R}$, the smallest $\sigma$-algebra containing all open sets.

Definition 1.10. A random variable $X$ is an $\mathcal{F}$-measurable real-valued function $X: \Omega \rightarrow \mathbb{R}$; i.e., for any $B \in \mathcal{R}, X^{-1}(B) \in \mathcal{F}$.

Definition 1.11. The distribution of the random variable $X$ is a probability measure $\mu$ on $\mathbb{R}$, defined for any set $B \in \mathcal{R}$ by
$$\mu(B)=\mathbb{P}(X \in B)=\mathbb{P} \circ X^{-1}(B) .$$
In particular, we define the distribution function $F(x)=\mathbb{P}(X \leq x)$ when $B=(-\infty, x]$

If there exists an integrable function $\rho(x)$ such that
$$\mu(B)=\int_B \rho(x) d x$$
for any $B \in \mathcal{R}$, then $\rho$ is called the probability density function ( $\mathrm{PDF}$ ) of $X$. Here $\rho(x)=d \mu / d m$ is the Radon-Nikodym derivative of $\mu(d x)$ with respect to the Lebesgue measure $m(d x)$ if $\mu(d x)$ is absolutely continuous with respect to $m(d x)$; i.e., for any set $B \in \mathcal{R}$, if $m(B)=0$, then $\mu(B)=0$ (see also Section C of the appendix) [Bil79]. In this case, we write $\mu \ll m$.
Definition 1.12. The expectation of a random variable $X$ is defined as
$$\mathbb{E} X=\int_{\Omega} X(\omega) \mathbb{P}(d \omega)=\int_{\mathbb{R}} x \mu(d x)$$
if the integrals are well-defined.
The variance of $X$ is defined as
$$\operatorname{Var}(X)=\mathbb{E}(X-\mathbb{E} X)^2 .$$
For two random variables $X$ and $Y$, we can define their covariance as (1.15) $\operatorname{Cov}(X, Y)=\mathbb{E}(X-\mathbb{E} X)(Y-\mathbb{E} Y)$.
$X$ and $Y$ are called uncorrelated if $\operatorname{Cov}(X, Y)=0$.

## 统计代写|随机分析作业代写stochastic analysis代写|Independence

We now come to one of the most distinctive notions in probability theory, the notion of independence. Let us start by defining the independence of events. Two events $A, B \in \mathcal{F}$ are independent if
$$\mathbb{P}(A \cap B)=\mathbb{P}(A) \mathbb{P}(B) .$$
Definition 1.21. Two random variables $X$ and $Y$ are said to be independent if for any two Borel sets $A$ and $B, X^{-1}(A)$ and $Y^{-1}(B)$ are independent; i.e.,
(1.30) $\quad \mathbb{P}\left(X^{-1}(A) \cap Y^{-1}(B)\right)=\mathbb{P}\left(X^{-1}(A)\right) \mathbb{P}\left(Y^{-1}(B)\right)$.

The joint distribution of the two random variables $X$ and $Y$ is defined to be the distribution of the random vector $(X, Y)$. Let $\mu_1$ and $\mu_2$ be the probability distribution of $X$ and $Y$, respectively, and let $\mu$ be their joint distribution. If $X$ and $Y$ are independent, then for any two Borel sets $A$ and $B$, we have
$$\mu(A \times B)=\mu_1(A) \mu_2(B) .$$
Consequently, we have
$$\mu=\mu_1 \mu_2 ;$$
i.e., the joint distribution of two independent random variables is the product distribution. If both $\mu_1$ and $\mu_2$ are absolutely continuous, with densities $p_1$ and $p_2$, respectively, then $\mu$ is also absolutely continuous, with density given by
$$p(x, y)=p_1(x) p_2(y) .$$
One can also understand independence from the viewpoint of expectations. Let $f_1$ and $f_2$ be two continuous functions. If $X$ and $Y$ are two independent random variables, then
$$\mathbb{E} f_1(X) f_2(Y)=\mathbb{E} f_1(X) \mathbb{E} f_2(Y) .$$
In fact, this can also be used as the definition of independence.

# 随机分析代考

## 统计代写|随机分析作业代写stochastic analysis代写|Continuous Distributions

$$\mu(B)=\mathbb{P}(X \in B)=\mathbb{P} \circ X^{-1}(B) .$$

$$\mu(B)=\int_B \rho(x) d x$$

$$\mathbb{E} X=\int_{\Omega} X(\omega) \mathbb{P}(d \omega)=\int_{\mathbb{R}} x \mu(d x)$$

$$\operatorname{Var}(X)=\mathbb{E}(X-\mathbb{E} X)^2$$

$\operatorname{Cov}(X, Y)=\mathbb{E}(X-\mathbb{E} X)(Y-\mathbb{E} Y)$.
$X$ 和 $Y$ 被称为不相关的，如果 $\operatorname{Cov}(X, Y)=0$.

## 统计代写|随机分析作业代写stochastic analysis代写|Independence

$$\mathbb{P}(A \cap B)=\mathbb{P}(A) \mathbb{P}(B) .$$

(1.30) $\mathbb{P}\left(X^{-1}(A) \cap Y^{-1}(B)\right)=\mathbb{P}\left(X^{-1}(A)\right) \mathbb{P}\left(Y^{-1}(B)\right)$.

$$\mu(A \times B)=\mu_1(A) \mu_2(B) .$$

$$\mu=\mu_1 \mu_2 ;$$

$$p(x, y)=p_1(x) p_2(y)$$

$$\mathbb{E} f_1(X) f_2(Y)=\mathbb{E} f_1(X) \mathbb{E} f_2(Y)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|随机分析作业代写stochastic analysis代写|MATH477

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl相关的作业也就用不着说。

• Statistical Inference 统计推断
• 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|随机分析作业代写stochastic analysis代写|Conditional Probability

Let $A, B \in \mathcal{F}$ and assume that $\mathbb{P}(B) \neq 0$. Then the conditional probability of $A$ given $B$ is defined as
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$
This is the proportion of events that both $A$ and $B$ occur given that $B$ occurs. For instance, the probability to obtain two tails in two tosses of a fair coin is $1 / 4$, but the conditional probability to obtain two tails is $1 / 2$ given that the first toss is a tail, and it is zero given that the first toss is a head.
Since $\mathbb{P}(A \cap B)=\mathbb{P}(A \mid B) \mathbb{P}(B)$ by definition, we also have
$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A \mid B \cap C) \mathbb{P}(B \mid C) \mathbb{P}(C),$$
and so on. It is straightforward to obtain
$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A) \mathbb{P}(B \mid A)}{\mathbb{P}(B)}$$
from the definition of conditional probability. This is called Bayes’s rule.

Proposition 1.6 (Bayes’s theorem). If $A_1, A_2, \ldots$ are disjoint sets such that $\bigcup_{j=1}^{\infty} A_j=\Omega$, then we have
$$\mathbb{P}\left(A_j \mid B\right)=\frac{\mathbb{P}\left(A_j\right) \mathbb{P}\left(B \mid A_j\right)}{\sum_{n=1}^{\infty} \mathbb{P}\left(A_n\right) \mathbb{P}\left(B \mid A_n\right)} \quad \text { for any } j \in \mathbb{N} \text {. }$$
This is useful in Bayesian statistics where $A_j$ corresponds to the hypothesis and $\mathbb{P}\left(A_j\right)$ is the prior probability of the hypothesis $A_j$. The conditional probability $\mathbb{P}\left(A_j \mid B\right)$ is the posterior probability of $A_j$ given that the event $B$ occurs.

## 统计代写|随机分析作业代写stochastic analysis代写|Discrete Distributions

If the elements in $\Omega$ are finite or enumerable, say, $\Omega=\left{\omega_1, \omega_2, \ldots\right}$, we have a situation of discrete probability space and discrete distribution. In this case, let $X\left(\omega_j\right)=x_j$ and
$$p_j=\mathbb{P}\left(X=x_j\right), \quad j=0,1, \ldots$$
Of course, we have to have
$$0 \leq p_j \leq 1, \quad \sum_j p_j=1 .$$
Given a function $f$ of $X$, its expectation is given by
$$\mathbb{E} f(X)=\sum_j f\left(x_j\right) p_j$$
if the sum is well-defined. In particular, the $p$ th moment of the distribution is defined as
$$m_p=\sum_j x_j^p p_j .$$
When $p=1$, it is called the mean of the random variable and is also denoted by mean $(X)$. Another important quantity is its variance, defined as
$$\operatorname{Var}(X)=m_2-m_1^2=\sum_j\left(x_j-m_1\right)^2 p_j .$$
Example 1.7 (Bernoulli distribution). The Bernoulli distribution has the form
$$\mathbb{P}(X=j)= \begin{cases}p, & j=1, \ q, & j=0,\end{cases}$$
$p+q=1$ and $p, q \geq 0$. When $p=q=1 / 2$, it corresponds to the toss of a fair coin. The mean and variance can be calculated directly:
$$\mathbb{E} X=p, \quad \operatorname{Var}(X)=p q$$

# 随机分析代考

## 统计代写|随机分析作业代写stochastic analysis代写|Conditional Probability

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)} .$$

$$\mathbb{P}(A \cap B \cap C)=\mathbb{P}(A \mid B \cap C) \mathbb{P}(B \mid C) \mathbb{P}(C),$$

$$\mathbb{P}(A \mid B)=\frac{\mathbb{P}(A) \mathbb{P}(B \mid A)}{\mathbb{P}(B)}$$

$$\mathbb{P}\left(A_j \mid B\right)=\frac{\mathbb{P}\left(A_j\right) \mathbb{P}\left(B \mid A_j\right)}{\sum_{n=1}^{\infty} \mathbb{P}\left(A_n\right) \mathbb{P}\left(B \mid A_n\right)} \quad \text { for any } j \in \mathbb{N} \text {. }$$

## 统计代写|随机分析作业代写stochastic analysis代写|Discrete Distributions

$$p_j=\mathbb{P}\left(X=x_j\right), \quad j=0,1, \ldots$$

$$0 \leq p_j \leq 1, \quad \sum_j p_j=1 .$$

$$\mathbb{E} f(X)=\sum_j f\left(x_j\right) p_j$$

$$m_p=\sum_j x_j^p p_j .$$

$$\operatorname{Var}(X)=m_2-m_1^2=\sum_j\left(x_j-m_1\right)^2 p_j .$$

$$\mathbb{P}(X=j)={p, \quad j=1, q, \quad j=0,$$
$p+q=1$ 和 $p, q \geq 0$. 什么时候 $p=q=1 / 2$ ，它对应于公平硬币的抛郑。可以直接计算均值和方差:
$$\mathbb{E} X=p, \quad \operatorname{Var}(X)=p q$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|随机分析作业代写stochastic analysis代写|STAT342

statistics-lab™ 为您的留学生涯保驾护航 在代写随机分析stochastic analysisl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分析stochastic analysisl代写方面经验极为丰富，各种代写随机分析stochastic analysisl相关的作业也就用不着说。

• Statistical Inference 统计推断
• 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|随机分析作业代写stochastic analysis代写|Elementary Examples

We will start with some elementary examples of probability. The most wellknown example is that of a fair coin: if flipped, the probability of getting a head or tail both equal to $1 / 2$. If we perform $n$ independent tosses, then the probability of obtaining $n$ heads is equal to $1 / 2^n$ : among the $2^n$ equally possible outcomes only one gives the result that we look for. More generally, let $S_n=X_1+X_2+\cdots+X_n$, where
$$X_j= \begin{cases}1, & \text { if the result of the } n \text {th trial is a head, } \ 0, & \text { if the result of the } n \text {th trial is a tail. }\end{cases}$$
Then the probability that we get $k$ heads out of $n$ tosses is equal to
$$\operatorname{Prob}\left(S_n=k\right)=\frac{1}{2^n}\left(\begin{array}{l} n \ k \end{array}\right) .$$
Applying Stirling’s formula
$$n ! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n, \quad n \rightarrow \infty,$$
we can calculate, for example, the asymptotic probability of obtaining heads exactly half of the time:
$$\operatorname{Prob}\left(S_{2 n}=n\right)=\frac{1}{2^{2 n}}\left(\begin{array}{c} 2 n \ n \end{array}\right)=\frac{1}{2^{2 n}} \frac{(2 n) !}{(n !)^2} \sim \frac{1}{\sqrt{\pi n}} \rightarrow 0,$$
as $n \rightarrow \infty$.

On the other hand, since we have a fair coin, we do expect to obtain heads roughly half of the time; i.e.,
$$\frac{S_{2 n}}{2 n} \approx \frac{1}{2},$$
for large $n$. Such a statement is indeed true and is embodied in the law of large numbers that we will discuss in the next chapter. For the moment let us simply observe that while the probability that $S_{2 n}$ equals $n$ goes to zero as $n \rightarrow \infty$, the probability that $S_{2 n}$ is close to $n$ goes to 1 as $n \rightarrow \infty$. More precisely, for any $\epsilon>0$,
$$\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \rightarrow 0,$$
as $n \rightarrow \infty$. This can be seen as follows.

## 统计代写|随机分析作业代写stochastic analysis代写|Probability Space

It is useful to put these intuitive notions of probability on a firm mathematical basis, as was done by Kolmogorov. For this purpose, we need the notion of probability space, often written as a triplet $(\Omega, \mathcal{F}, \mathbb{P})$, defined as follows.
Definition $1.1$ (Sample space). The sample space $\Omega$ is the set of all possible outcomes. Each element $\omega \in \Omega$ is called a sample point.

Definition $1.2$ ( $\sigma$-algebra). A $\sigma$-algebra (or $\sigma$-field) $\mathcal{F}$ is a collection of subsets of $\Omega$ that satisfies the following conditions:
(i) $\Omega \in \mathcal{F}$;
(ii) if $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$, where $A^c=\Omega \backslash A$ is the complement of $A$ in $\Omega$;
(iii) if $A_1, A_2, \ldots \in \mathcal{F}$, then $\bigcup_{n=1}^{\infty} A_n \in \mathcal{F}$.
Each set $A$ in $\mathcal{F}$ is called an event. Let $\mathcal{B}$ be a collection of subsets of $\Omega$. We denote by $\sigma(\mathcal{B})$ the smallest $\sigma$-algebra generated by the sets in $\mathcal{B}$, i.e., the smallest $\sigma$-algebra that contains $\mathcal{B}$. The pair $(\Omega, \mathcal{F})$ with the above properties is called a measurable space.

Definition $1.3$ (Probability measure). The probability measure $\mathbb{P}: \mathcal{F} \rightarrow$ $[0,1]$ is a set function defined on $\mathcal{F}$ which satisfies
(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) if $A_1, A_2, \ldots \in \mathcal{F}$ are pairwise disjoint, i.e., $A_i \cap A_j=\emptyset$ if $i \neq j$, then
$$\mathbb{P}\left(\bigcup_{n=1}^{\infty} A_n\right)=\sum_{n=1}^{\infty} \mathbb{P}\left(A_n\right) .$$
(1.1) is called countable additivity or $\sigma$-additivity.

# 随机分析代考

## 统计代写|随机分析作业代写stochastic analysis代写|Elementary Examples

$X_j={1, \quad$ if the result of the $n$th trial is a head, $0, \quad$ if the result of the $n$th trial is a tail.

$$\operatorname{Prob}\left(S_n=k\right)=\frac{1}{2^n}(n k) .$$

$$n ! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n, \quad n \rightarrow \infty,$$

$$\operatorname{Prob}\left(S_{2 n}=n\right)=\frac{1}{2^{2 n}}(2 n n)=\frac{1}{2^{2 n}} \frac{(2 n) !}{(n !)^2} \sim \frac{1}{\sqrt{\pi n}} \rightarrow 0,$$

$$\frac{S_{2 n}}{2 n} \approx \frac{1}{2},$$

$$\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \rightarrow 0,$$

## 统计代写|随机分析作业代写stochastic analysis代写|Probability Space

(ii) 如果 $A \in \mathcal{F}$ ，然后 $A^c \in \mathcal{F}$ ，在哪里 $A^c=\Omega \backslash A$ 是的补充 $A$ 在 $\Omega$;
(iii) 如果 $A_1, A_2, \ldots \in \mathcal{F}$ ，然后 $\bigcup_{n=1}^{\infty} A_n \in \mathcal{F}$.

(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) 如果 $A_1, A_2, \ldots \in \mathcal{F}$ 成对不相交，即 $A_i \cap A_j=\emptyset$ 如果 $i \neq j$ ，然后
$$\mathbb{P}\left(\bigcup_{n=1}^{\infty} A_n\right)=\sum_{n=1}^{\infty} \mathbb{P}\left(A_n\right) .$$
(1.1) 称为可数加性或 $\sigma$-可加性。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|随机过程代写stochastic process代考|MTH7090

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
• (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|Matching Theorems

Chapter 4 makes the point that the generic chaining (or some equivalent form of it) is already required to really understand the irregularities occurring in the distribution of $N$ points $\left(X_i\right)_{i \leq N}$ independently and uniformly distributed in the unit square. These irregularities are measured by the “cost” of pairing (=matching) these points with $N$ fixed points that are very uniformly spread, for various notions of cost.
These optimal results involve mysterious powers of $\log N$. We are able to trace them back to the geometry of ellipsoids in Hilbert space, so we start the chapter with an investigation of these ellipsoids in Sect. 4.1. The philosophy of the main result, the ellipsoid theorem, is that an ellipsoid is in some sense somewhat smaller than it appears at first. This is due to convexity: an ellipsoid gets “thinner” when one gets away from its center. The ellipsoid theorem is a special case of a more general result (with the same proof) about the structure of sufficiently convex bodies, one that will have important applications in Chap. 19.

In Sect.4.3, we provide general background on matchings. In Sect.4.5, we investigate the case where the cost of a matching is measured by the average distance between paired points. We prove the result of Ajtai, Komlós and Tusnády that the expected cost of an optimal matching is at most $L \sqrt{\log N} / \sqrt{N}$ where $L$ is a number. The factor $1 / \sqrt{N}$ is simply a scaling factor, but the fractional power of $\log$ is optimal as shown in Sect. 4.6. In Sect. 4.7, we investigate the case where the cost of a matching is measured instead by the maximal distance between paired points. We prove the theorem of Leighton and Shor that the expected cost of a matching is at most $L(\log N)^{3 / 4} / \sqrt{N}$, and the power of $\log$ is shown to be optimal in Sect. 4.8. With the exception of Sect. 4.1, the results of Chap. 4 are not connected to any subsequent material before Chap. 17.

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

Random signs are obviously important r.v.s and occur frequently in connection with “symmetrization procedures”, a very useful tool. In a Bernoulli process, the individual random variables $X_t$ are linear combinations of independent random signs. Each Bernoulli process is associated with a Gaussian process in a canonical manner, when one replaces the random signs by independent standard Gaussian r.v.s. The Bernoulli process has better tails than the corresponding Gaussian process (it is “sub-Gaussian”) and is bounded whenever the corresponding Gaussian process is bounded. There is, however, a completely different reason for which a Bernoulli process might be bounded, namely, that the sum of the absolute values of the coefficients of the random signs remain bounded independently of the index $t$. A natural question is then to decide whether these two extreme situations are the only fundamental reasons why a Bernoulli process can be bounded, in the sense that a suitable “mixture” of them occurs in every bounded Bernoulli process. This was the “Bernoulli conjecture” (to be stated formally on page 179), which has been so brilliantly solved by W. Bednorz and R. Latała.

It is a long road to the solution of the Bernoulli conjecture, and we start to build the main tools hearing on Rernoulli processes. A linear combination of independent random signs looks like a Gaussian r.v. when the coefficients of the random signs are small. We can expect that a Bernoulli process will look like a Gaussian process when these coefficients are suitably small. This is a fundamental idea: the key to understanding Rernoulli processes is to reduce to situations where these coefficients are small.

The Bernoulli conjecture, on which the author worked so many years, greatly influenced the way he looked at various processes. In the case of empirical processes, this is explained in Sect. $6.8$.

# 随机过程代考

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 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
• (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

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

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
• (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

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

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

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|随机过程代写stochastic process代考|MTH3016

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
• (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|Hexagonal Lattice, Nearest Neighbors

Here I dive into the details of the processes discussed in Section 1.5.3. I also discuss Figure 2. The source code to produce Figure 2 is discussed in Sections $6.4$ (nearest neighbor graph) and $6.7$ (visualizations). Some elements of graph theory are discussed here, as well as visualization techniques.

Surprisingly, it is possible to produce a point process with a regular hexagonal lattice space using simple operations on a small number $(m=4)$ of square lattices: superimposition, stretching, and shifting. A stretched lattice is a square lattice turned into a rectangular lattice, by applying a multiplication factor to the $\mathrm{X}$ and/or Y coordinates. A shifted lattice is a lattice where the grid points have been shifted via a translation.

Each point of the process almost surely (with probability one) has exactly one nearest neighbor. However, when the scaling factor $s$ is zero, this is no longer true. On the left plot in Figure 2, each point (also called vertex when $s=0$ ) has exactly 3 nearest neighbors. This causes some challenges when plotting the case $s=0$. The case $s>0$ is easier to plot, using arrows pointing from any point to its unique nearest neighbor. I produced the arrows in question with the arrow function in R, see source code in Section $6.7$, and online documentation here. $\mathrm{A}$ bidirectional arrow between points $\mathrm{A}$ and $\mathrm{B}$ means that $\mathrm{B}$ is a nearest neighbor of $\mathrm{A}$, and $\mathrm{A}$ is a nearest neighbor of B. All arrows on the left plot in Figure 2 are bidirectional. Boundary effects are easily noticeable, as some arrows point to nearest neighbors outside the window. Four colors are used for the points, corresponding to the 4 shifted stretched Poisson-binomial processes used to generate the hexagon-based process. The color indicates which of these 4 process, a point is attached to.

The source code in Section $6.4$ handles points with multiple nearest neighbors. It produces a list of all points with their nearest neighbors, using a hash table. A point with 3 nearest neighbors has 3 entries in that list: one for each nearest neighbor. A group of points that are all connected by arrows, is called a connected component [Wiki]. A path from a point of a connected component to another point of the same connected component, following arrows while ignoring their direction, is called a path in graph theory.

In my definition of connected component, the direction of the arrow does not matter: the underlying graph is considered undirected [Wiki]. An interesting problem is to study the size distribution, that is, the number of points per connected component, especially for standard Poisson processes. See Exercise 20. In graph theory, a point is called a vertex or node, and an arrow is called an edge. More about nearest neighbors is discussed in Exercises 18 and 19.

Finally, if you look at Figure 2, the left plot seems to have more points than the right plot. But they actually have roughly the same number of points. The plot on the right seems to be more sparse, because there are large areas with no points. But to compensate, there are areas where several points are in close proximity.

## 统计代写|随机过程代写stochastic process代考|Modeling Cluster Systems in Two Dimensions

There are various ways to create points scattered around a center. When multiple centers are involved, we get a cluster structure. The point process consisting of the centers is called the parent process, while the point distribution around each center, is called the child process. So we are dealing with a two-layer, or hierarchical structure, referred to as a cluster point process. Besides clustering, many other types of point process operations [Wiki] are possible when combining two processes, such as thinning or superimposition. Typical examples of cluster point processes include Neyma-Scott (see here) and Matérn (see here).

Useful references include Baddeley’s textbook “Spatial Point Processes and their Applications” [4] available online here, Sigman’s course material (Columbia University) on one-dimensional renewal processes for beginners, entitled “Notes on the Poisson Process” [71], available online here, Last and Kenrose’s book “Lectures on the Poisson Process” [52], and Cressie’s comprehensive 900-page book “Statistics for Spatial Data” [16]. Cluster point processes are part of a larger field known as spatial statistics, encompassing other techniques such as geostatistics, kriging and tessellations. For lattice-based processes known as perturbed lattice point processes, more closely related to the theme of this textbook (lattice processes), and also more recent with applications to cellular networks, see the following references:

• “On Comparison of Clustering Properties of Point Processes” [12]. Online PDF here.
• “Clustering and percolation of point processes” [11]. Online version here.
• “Clustering comparison of point processes, applications to random geometric models” [13]. Online version here.
• “Stochastic Geometry-Based Tools for Spatial Modeling and Planning of Future Cellular Networks” [51]. Online version here.
• “Hyperuniform and rigid stable matchings” [54]. Online PDF here. Short presentation available here.
• “Rigidity and tolerance for perturbed lattices” [68]. Online version here.
• “Cluster analysis of spatial point patterns: posterior distribution of parents inferred from offspring” [66].
• “Recovering the lattice from its random perturbations” [79]. Online version here.
• “Geometry and Topology of the Boolean Model on a Stationary Point Processes” [81]. Online version here.
• “On distances between point patterns and their applications” [56]. Online version here.
More general references include two comprehensive volumes on point process theory by Daley and Vere-Jones [20, 21], a chapter by Johnson [45] (available online here or here), books by Møller and Waagepetersen, focusing on statistical inference for spatial processes [60, 61], and “Point Pattern Analysis: Nearest Neighbor Statistics” by Anselin [3] focusing on point inhibition/aggregation metrics, available here. See also [58] by Møller, available online here, and “Limit Theorems for Network Dependent Random Variables” [48], available online here.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Modeling Cluster Systems in Two Dimensions

• 《论点过程的聚类特性比较》[12]。此处为在线 PDF。
• “点过程的聚类和渗透”[11]。在线版本在这里。
• “点过程的聚类比较，在随机几何模型中的应用”[13]。在线版本在这里。
• “用于未来蜂窝网络空间建模和规划的基于随机几何的工具”[51]。在线版本在这里。
• “超均匀和刚性稳定匹配”[54]。此处为在线 PDF。此处提供简短演示。
• “扰动格子的刚度和容忍度”[68]。在线版本在这里。
• “空间点模式的聚类分析：从后代推断出父母的后验分布”[66]。
• “从随机扰动中恢复晶格”[79]。在线版本在这里。
• “驻点过程布尔模型的几何和拓扑”[81]。在线版本在这里。
• “关于点模式之间的距离及其应用”[56]。在线版本在这里。
更一般的参考资料包括 Daley 和 Vere-Jones [20, 21] 的两本关于点过程理论的综合性著作，Johnson [45] 的一章（可在此处或此处在线获取），Møller 和 Waagepetersen 的书籍，侧重于空间的统计推断过程 [60、61] 和 Anselin [3] 的“点模式分析：最近邻统计”重点关注点抑制/聚合指标，可在此处获取。另请参见 Møller 的 [58]，可在此处在线获取，以及“网络相关随机变量的极限定理”[48]，可在此处在线获取。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|随机过程代写stochastic process代考|MTH7090

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
• (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|Rotation, Stretching, Translation and Standardization

In two dimensions, rotating a Poisson-binomial process is equivalent to rotating its underlying lattice attached to the index space. Rotating the points has the same effect as rotating the lattice locations, because $F$ (the distribution attached to the points) belongs to a family of location-scale distributions [Wiki]. For instance, a $\pi / 4$ rotation will turn the square lattice into a centered-square lattice [Wiki], but it won’t change the main properties of the point process. Both processes, the original one and the rotated one, may be indistinguishable for all practical purposes unless the scaling factor $s$ is small, creating model identifiability [Wiki] issues. For instance, the theoretical correlation between the point coordinates $\left(X_h, Y_k\right)$ or the underlying lattice point coordinates $(h / \lambda, k / \lambda)$, measured on all points, remains equal to zero after rotation, because the number of points is infinite (this may not be the case if you observe points through a small window, because of boundary effects). Thus, a Poisson-binomial process has a point distribution invariant under rotations, on a macro-scale. This property is called anisotropy [Wiki]. On a micro-scale, a few changes occur though: for instance the twodimensional version of Theorem $4.1$ no longer applies, and the distance between the projection of two neighbor points on the $\mathrm{X}$ or $\mathrm{Y}$ axis, shrinks after the rotation.

Applying a translation to the points of the process, or to the underlying lattice points, results in a shifted point process. It becomes interesting when multiple shifted processes, with different translation vectors, are combined together as in Section 1.5.3. Theorem $4.1$ may not apply to the shifted process, though it can easily be adapted to handle this situation. One of the problems is to retrieve the underlying lattice space of the shifted process. This is useful for model fitting purposes, as it is easier to compare two processes once they have been standardized (after removing translations and rescaling). Estimation techniques to identify the shift are discussed in Section 3.4.

By a standardized Poisson-binomial point process, I mean one in its canonical form, with intensity $\lambda=1$, scaling factor $s=1$, and free of shifts or rotations. Once two processes are standardized, it is easier to compare them, assess if they are Poisson-binomial, or perform various machine learning procedures on observed data, such as testing, computing confidence intervals, cross-validation, or model fitting. In some way, this is similar to transforming and detrending time series to make them more amenable to statistical inference. There is also some analogy between the period or quasi-period of a time series, and the inverse of the intensity $\lambda$ of a Poisson-binomial process: in fact, $1 / \lambda$ is the fixed increment between the underlying lattice points in the lattice space, and can be viewed as the period of the process.

Finally, a two dimensional process is said to be stretched if a different intensity is used for each coordinate for all the points of the process. It turns the underlying square lattice space into a rectangular lattice, and the homogeneous process into a non-homogeneous one, because the intensity varies locally. Observed data points can be standardized using the Mahalanobis transformation [Wiki], to remove stretching (so that variances are identical for both coordinates) and to decorrelate the two coordinates, when correlation is present.

## 统计代写|随机过程代写stochastic process代考|Superimposition and Mixing

Here we are working with two-dimensional processes. When the points of $m$ independent point processes with same distribution $F$ and same index space $\mathbb{Z}^2$ are bundled together, we say that the processes are superimposed. These processes are no longer Poisson-binomial, see Exercise 14. Indeed, if the scaling factor $s$ is small and $m>1$ is not too small, they exhibit clustering around each lattice location in the lattice space. Also, the intensities or scaling factors of each individual point process may be different, and the resulting combined process may not be homogeneous. Superimposed point processes also called interlaced processes.
A mixture of $m$ point processes, denoted as $M$, is defined as follows:

• We have $m$ independent point processes $M_1, \ldots, M_m$ with same distribution $F$ and same index space $\mathbb{Z}^2$,
• The intensity and scaling factor attached to $M_i$ are denoted respectively as $\lambda_i$ and $s_i(i=1, \ldots, m)$,
• The points of $M_i(i=1, \ldots, m)$ are denoted as $\left(X_{i h}, Y_{i k}\right)$; the index space consists of the $(h, k)$ ‘s,
• The point $\left(X_h, Y_k\right)$ of the mixture process $M$ is equal to $\left(X_{i h}, Y_{i k}\right)$ with probability $\pi_i>0, i=1, \ldots, m$.
While mixing or superimposing Poisson-binomial processes seem like the same operation, which is true for stationary Poisson processes, in the case of Poisson-binomial processes, these are distinct operations resulting in significant differences when the scaling factors are very small (see Exercise 18). The difference is most striking when $s=0$. In particular, superimposed processes are less random than mixtures. This is due to the discrete nature of the underlying lattice space. However, with larger scaling factors, the behavior of mixed and superimposed processes tend to be similar.

Several of the concepts discussed in Section $1.5$ are illustrated in Figure 2, representing a realization of $m$ superimposed shifted stretched Poisson-binomial processes, called $m$-interlacing. For each individual process $M_i, i=1, \ldots, m$, the distribution attached to the point $\left(X_{i h}, X_{i k}\right)$ (with $h, k \in \mathbb{Z}$ ) is
$$P\left(X_{i h}<x, Y_{i k}<y\right)=F\left(\frac{x-\mu_i-h / \lambda}{s}\right) F\left(\frac{y-\mu_i^{\prime}-k / \lambda^{\prime}}{s}\right), \quad i=1, \ldots, m$$
This generalizes Formula (2). The parameters used for the model pictured in Figure 2 are:

• Number of superimposed processes: $m=4$; each one displayed with a different color,
• Color: red for $M_1$, blue for $M_2$, orange for $M_3$, black for $M_4$,
• scaling factor: $s=0$ (left plot) and $s=5$ (right plot),
• Intensity: $\lambda=1 / 3$ ( $\mathrm{X}$-axis) and $\lambda^{\prime}=\sqrt{3} / 3$ ( $\mathrm{Y}$-axis),
• Shift vector, $\mathrm{X}$-coordinate: $\mu_1=0, \mu_2=1 / 2, \mu_3=2, \mu_4=3 / 2$,
• Shift vector, Y-coordinate: $\mu_1^{\prime}=0, \mu_2^{\prime}=\sqrt{3} / 2, \mu_3^{\prime}=0, \mu_4^{\prime}=\sqrt{3} / 2$,
• $F$ distribution: standard centered logistic with zero mean and variance $\pi^2 / 3$.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Superimposition and Mixing

• 我们有 $m$ 独立点过程 $M_1, \ldots, M_m$ 具有相同的分布 $F$ 和相同的索引空间 $\mathbb{Z}^2$ ，
• 附加的强度和比例因子 $M_i$ 分别记为 $\lambda_i$ 和 $s_i(i=1, \ldots, m)$,
• 的要点 $M_i(i=1, \ldots, m)$ 表示为 $\left(X_{i h}, Y_{i k}\right)$ ；索引空间由 $(h, k)$ 的，
• 重点 $\left(X_h, Y_k\right)$ 混合过程 $M$ 等于 $\left(X_{i h}, Y_{i k}\right)$ 有概率 $\pi_i>0, i=1, \ldots, m$. 虽然混合或諿加泊松二项式过程看起来像是相同的操作，对于平稳泊松过程也是如此，但在泊松二 项式过程的情况下，这些是不同的操作，当缩放因子非常小时会导致显着差异 (参见练习 18). 当 $s=0$. 特别是，咺加过程的随机性低于混合过程。这是由于底层晶格空间的离散性质。然而，对 于较大的比例因子，混合过程和䝁加过程的行为往往相似。
本节中讨论的几个概念 $1.5$ 如图 2 所示，代表了一种实现 $m$ 呾加的偏移拉伸泊松二项式过程，称为 $m$ – 交 错。对于每个单独的过程 $M_i, i=1, \ldots, m$, 分布附加到点 $\left(X_{i h}, X_{i k}\right.$ ) (和 $h, k \in \mathbb{Z}$ ) 是
$$P\left(X_{i h}<x, Y_{i k}<y\right)=F\left(\frac{x-\mu_i-h / \lambda}{s}\right) F\left(\frac{y-\mu_i^{\prime}-k / \lambda^{\prime}}{s}\right), \quad i=1, \ldots, m$$
这推广了公式 (2)。用于图 2 中所示模型的参数是:
• 㠬加进程数： $m=4$; 每一个都以不同的颜色显示，
• 颜色: 红色为 $M_1$ ，蓝色为 $M_2$ ，橙色为 $M_3$ ，黑色为 $M_4$ ，
• 比例因子: $s=0$ (左图) 和 $s=5$ (右图),
• 强度: $\lambda=1 / 3$ ( $\mathrm{X}$-轴) 和 $\lambda^{\prime}=\sqrt{3} / 3$ (Y-轴)，
• 移位向量， $\mathrm{X}$-协调: $\mu_1=0, \mu_2=1 / 2, \mu_3=2, \mu_4=3 / 2$,
• 移位向量， Y坐标: $\mu_1^{\prime}=0, \mu_2^{\prime}=\sqrt{3} / 2, \mu_3^{\prime}=0, \mu_4^{\prime}=\sqrt{3} / 2$,
• $F$ 分布: 均值和方差为零的标准中心逻辑 $\pi^2 / 3$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 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
• (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|Point Count and Interarrival Times

An immediate result is that $F_s(x-k / \lambda)$ is centered at $k / \lambda$. Also, if $s=0$, then $X_k=k / \lambda$. If $s$ is very small, $X_k$ is very close to $k / \lambda$ most of the time. But when $s$ is large, the points $X_k$ ‘s are no longer ordered, and the larger $s$, the more randomly they are permutated (or shuffled, or mixed) on the real line.
Let $B=[a, b]$ be an interval on the real line, with $a2$. This is due to the combinatorial nature of the Poisson-binomial distribution. But you can easily obtain approximated values using simulations.

Another fundamental, real-valued random variable, denoted as $T$ or $T(\lambda, s)$, is the interarrival times between two successive points of the process, once the points are ordered on the real line. In two dimensions, it is replaced by the distance between a point of the process, and its nearest neighbor. Thus it satisfies (see Section $4.2$ ) the following identity:
$$P(T>y)=P[N(B)=0],$$
with $\left.B=] X_0, X_0+y\right]$, assuming it is measured at $X_0$ (the point of the process corresponding to $k=0$ ). See Formula (38) for the distribution of $T$. In practice, this intractable exact formula is not used; instead it is approximated via simulations. Also, the point $X_0$ is not known, since the $X_k$ ‘s are in random order, and retrieving $k$ knowing $X_k$ is usually not possible. The indices (the $k$ ‘s) are hidden. However, see Section $4.7$. The fundamental question is whether using $X_0$ or any $X_k$ (say $X_5$ ), matters for the definition of $T$. This is discussed in Section $1.4$ and illustrated in Table 4.

## 统计代写|随机过程代写stochastic process代考|Limiting Distributions, Speed of Convergence

I prove in Theorem $4.5$ that Poisson-binomial processes converge to ordinary Poisson processes. In this section, I illustrate the rate of convergence, both for the interarrival times and the point count in one dimension.

In Figure 1 , we used $\lambda=1$ and $B=[-0.75,0.75] ; \mu(B)=1.5$ is the length of $B$. The limiting values (combined with those of Table 3), as $s \rightarrow \infty$, are in agreement with $N(B)$ ‘s moments converging to those of a Poisson distribution of expectation $\lambda \mu(B)$, and $T$ ‘s moments to those of an exponential distribution of expectation $1 / \lambda$. In particular, it shows that $P[N(B)=0] \rightarrow \exp [-\lambda \mu(B)]$ and $E\left[T^2\right] \rightarrow 2 / \lambda$ as $s \rightarrow \infty$. These limiting distributions are features unique to stationary Poisson processes of intensity $\lambda$.

Figure 1 illustrates the speed of convergence of the Poisson-binomial process to the stationarity Poisson process of intensity $\lambda$, as $s \rightarrow \infty$. Further confirmation is provided by Table 3 , and formally established by Theorem 4.5. Of course, when testing data, more than a few statistics are needed to determine whether you are dealing with a Poisson process or not. For a full test, compare the empirical moment generating function (the estimated $\mathrm{E}\left[T^r\right]^{\prime}$ s say for all $r \in[0,3]$ ) or the empirical distribution of the interarrival times, with its theoretical limit (possibly obtained via simulations) corresponding to a Poisson process of intensity $\lambda$. The parameter $\lambda$ can be estimated based on the data. See details in Section 3.

In Figure 1, the values of $\mathrm{E}\left[T^2\right]$ are more volatile than those of $P[N(B)=0]$ because they were estimated via simulations; to the contrary, $P[N(B)=0]$ was computed using the exact Formula (6), though truncated to 20,000 terms. The choice of a Cauchy or logistic distribution for $F$ makes almost no difference. But a uniform $F$ provides noticeably slower, more bumpy convergence. The Poisson approximation is already quite good with $s=10$, and only improves as $s$ increases. Note that in our example, $N(B)>0$ if $s=0$. This is because $X_k=k$ if $s=0$; in particular, $X_0=0 \in B=[-0.75,0.75]$. Indeed $N(B)>0$ for all small enough $s$, and this effect is more pronounced (visible to the naked eye on the left plot, blue curve in Figure 1 ) if $F$ is uniform. Likewise, $E\left[T^2\right]=1$ if $s=0$, as $T(\lambda, s)=\lambda$ if $s=0$, and here $\lambda=1$.

The results discussed here in one dimension easily generalize to higher dimensions. In that case $B$ is a domain such as a circle or square, and $T$ is the distance between a point of the process, and its nearest neighbor. The limit. Poisson process is stationary with intensity $\lambda^d$, where $d$ is the dimension.

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Point Count and Interarrival Times

$$P(T>y)=P[N(B)=0],$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 统计代写|回归分析作业代写Regression Analysis代考|STAT2220

statistics-lab™ 为您的留学生涯保驾护航 在代写回归分析Regression Analysis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。我们的专家在代写回归分析Regression Analysis代写方面经验极为丰富，各种代写回归分析Regression Analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• 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|回归分析作业代写Regression Analysis代考|BRM with a Known Dispersion Matrix

It should be stressed that the multivariate model illustrated in Fig. $2.5$ is a special case of the model given in (1.9), which will serve as a basic model for the presentation of the subject matter of this book. Before starting the technical presentation, a formal definition of the $B R M$ is provided.

Definition 2.1 (BRM) $\quad$ Let $\boldsymbol{X}: p \times n, \boldsymbol{A}: p \times q, q \leq p, \boldsymbol{B}: q \times k, \boldsymbol{C}: k \times n$, $r(\boldsymbol{C})+p \leq n$ and $\boldsymbol{\Sigma}: p \times p$ be p.d. Then
$$X=A B C+E$$
defines the $B R M$, where $\boldsymbol{E} \sim N_{p, n}(\mathbf{0}, \boldsymbol{\Sigma}, \boldsymbol{I}), \boldsymbol{A}$ and $\boldsymbol{C}$ are known matrices, and $\boldsymbol{B}$ and $\boldsymbol{\Sigma}$ are unknown parameter matrices.

The condition $r(\boldsymbol{C})+p \leq n$ is an estimability condition when $\boldsymbol{\Sigma}$ is unknown. However, for ease of presentation in this section, it is assumed that the dispersion matrix $\boldsymbol{\Sigma}$ is known. The idea is to give a general overview and leave many details for the subsequent sections.
For the likelihood, $L(\boldsymbol{B})$, we have
$$L(\boldsymbol{B}) \propto|\boldsymbol{\Sigma}|^{-n / 2} e^{-1 / 2 \mathrm{tr}\left[\boldsymbol{\Sigma}^{-1}\left(\boldsymbol{X}_o-\boldsymbol{A B C}\right)\left(\boldsymbol{X}_o-\boldsymbol{A B C}\right)^{\prime}\right] .}$$
From (2.16) it is seen that there exists a design matrix $\boldsymbol{A}$ which describes the expectation of the rows of $\boldsymbol{X}$ (a within-individuals design matrix), as well as a design matrix $\boldsymbol{C}$ which describes the mean of the columns of $\boldsymbol{X}$ (a between-individuals design matrix). It is known that if one pre- and post-multiplies a matrix, a bilinear transformation is performed. Thus, in a comparison of (1.7) and (2.16), instead of a linear model in (1.7), there is a bilinear one in (2.16). The previous techniques used when $R^n$ was decomposed into $\mathcal{C}\left(\boldsymbol{C}^{\prime}\right) \boxplus \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)^{\perp}$ are adopted; i.e. due to bilinearity the tensor product $R^p \otimes R^n$ is decomposed as
$$\left(\mathcal{C}(\boldsymbol{A}) \otimes \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)\right) \boxplus\left(\mathcal{C}(\boldsymbol{A}) \otimes \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)^{\perp}\right) \boxplus\left(\mathcal{C}(\boldsymbol{A})^{\perp} \otimes \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)\right) \boxplus\left(\mathcal{C}(\boldsymbol{A})^{\perp} \otimes \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)^{\perp}\right)$$

## 统计代写|回归分析作业代写Regression Analysis代考|EBRM with a Known Dispersion Matrix

In Sect. $1.5$ two extensions of the $B R M$ were presented, i.e. the $E B R M_B^m$ and $E B R M_W^m$, together with examples of the application of these models. In this section the reader is introduced to the mathematics concerning the $E B R M_B^m$, with $m=3$, which will also be used later when studying the model without a known dispersion matrix. Now (2.16) is formally generalized and the $E B R M_B^m$ is specified in detail.
Definition $2.2\left(E B R M_B^m\right) \quad$ Let $\boldsymbol{X}: p \times n, \boldsymbol{A}i: p \times q_i, q_i \leq p, \boldsymbol{B}_i: q_i \times k_i, \boldsymbol{C}_i$ : $k_i \times n, i=1,2, \ldots, m, r\left(\boldsymbol{C}_1\right)+p \leq n, \mathcal{C}\left(\boldsymbol{C}_i^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}{i-1}^{\prime}\right), i=2,3, \ldots, m$, and $\Sigma: p \times p$ be p.d. Then
$$\boldsymbol{X}=\sum_{i=1}^m \boldsymbol{A}i \boldsymbol{B}_i \boldsymbol{C}_i+\boldsymbol{E}$$ defines the $E B R M_B^m$, where $\boldsymbol{E} \sim N{p, n}(\mathbf{0}, \boldsymbol{\Sigma}, \boldsymbol{I}),\left{\boldsymbol{A}_i\right}$ and $\left{\boldsymbol{C}_i\right}$ are known matrices, and $\left{\boldsymbol{B}_i\right}$ and $\boldsymbol{\Sigma}$ are unknown parameter matrices.

In the present book it is usually assumed that $m=2,3$, and in this section $\boldsymbol{\Sigma}$ is supposed to be known. In that case, $r\left(\boldsymbol{C}1\right)+p \leq n, \mathcal{C}\left(\boldsymbol{C}_i^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}{i-1}^{\prime}\right), i=$ $2,3, \ldots, m$ are not needed when estimating $\boldsymbol{B}i$. However, since the results from this chapter will be utilized in the next chapter, it is assumed that $\mathcal{C}\left(\boldsymbol{C}_i^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}{i-1}^{\prime}\right)$, $i=2,3, \ldots, m$, holds. Thus, the following model will be handled:
$$\boldsymbol{X}=\boldsymbol{A}1 \boldsymbol{B}_1 \boldsymbol{C}_1+\boldsymbol{A}_2 \boldsymbol{B}_2 \boldsymbol{C}_2+\boldsymbol{A}_3 \boldsymbol{B}_3 \boldsymbol{C}_3+\boldsymbol{E}, \quad \boldsymbol{E} \sim N{p, n}(\mathbf{0}, \boldsymbol{\Sigma}, \boldsymbol{I}),$$
where $\mathcal{C}\left(\boldsymbol{C}_3^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}_2^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}_1^{\prime}\right), \boldsymbol{A}_i: p \times q_i$, the parameter $\boldsymbol{B}_i: p \times q_i$, is unknown, $\boldsymbol{C}_i: k_i \times n$ and the dispersion matrix $\boldsymbol{\Sigma}$ is supposed to be known. It has already been noted in Sect. $1.5$ that without the subspace condition on $\mathcal{C}\left(\boldsymbol{C}_i\right)$, we would have the general “sum of profiles model” (a multivariate seemingly unrelated regression (SUR) model). Later (2.20) is studied when $\mathcal{C}\left(\boldsymbol{A}_3\right) \subseteq \mathcal{C}\left(\boldsymbol{A}_2\right) \subseteq \mathcal{C}\left(\boldsymbol{A}_1\right)$ replaces $\mathcal{C}\left(\boldsymbol{C}_3^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}_2^{\prime}\right) \subseteq \mathcal{C}\left(\boldsymbol{C}_1^{\prime}\right)$, i.e. we have an $E B R M_W^3$. Since the model under the assumption $\mathcal{C}\left(\boldsymbol{A}_3\right) \subseteq \mathcal{C}\left(\boldsymbol{A}_2\right) \subseteq \mathcal{C}\left(\boldsymbol{A}_1\right)$ through a reparametrization can be converted to (2.20) and vice versa, i.e. $E B R M_B^3 \rightleftarrows E B R M_W^3$, the models are in some sense equivalent. However, because of non-linearity in estimators of mean parameters, this does not imply that all the results for the models can easily be transferred from one model to the other.

# 回归分析代写

## 统计代写|回归分析作业代写Regression Analysis代考|BRM with a Known Dispersion Matrix

$$X=A B C+E$$

$$L(\boldsymbol{B}) \propto|\boldsymbol{\Sigma}|^{-n / 2} e^{-1 / 2 \operatorname{tr}\left[\boldsymbol{\Sigma}^{-1}\left(\boldsymbol{X}_o-\boldsymbol{A B C}\right)\left(\boldsymbol{X}_o-\boldsymbol{A B C}\right)^{\prime}\right] .}$$

$$\left(\mathcal{C}(\boldsymbol{A}) \otimes \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)\right) \boxplus\left(\mathcal{C}(\boldsymbol{A}) \otimes \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)^{\perp}\right) \boxplus\left(\mathcal{C}(\boldsymbol{A})^{\perp} \otimes \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)\right) \boxplus\left(\mathcal{C}(\boldsymbol{A})^{\perp} \otimes \mathcal{C}\left(\boldsymbol{C}^{\prime}\right)^{\perp}\right)$$

## 统计代写|回归分析作业代写Regression Analysis代考|EBRM with a Known Dispersion Matrix

$$\boldsymbol{X}=\boldsymbol{A} 1 \boldsymbol{B}_1 \boldsymbol{C}_1+\boldsymbol{A}_2 \boldsymbol{B}_2 \boldsymbol{C}_2+\boldsymbol{A}_3 \boldsymbol{B}_3 \boldsymbol{C}_3+\boldsymbol{E}, \quad \boldsymbol{E} \sim N p, n(\mathbf{0}, \boldsymbol{\Sigma}, \boldsymbol{I})$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。