## 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|Loynes’s scheme

Here we will consider the case where the state space $E$ is equipped with a partial ordering $\preceq$ (see section A.3), and admits a minimal point $\mathbf{0}$ such that $\mathbf{0} \preceq x$ for all $x \in E$. We will assume that on $E$ there exists a metric $d_{E}$ such that all $\preceq$-increasing sequences converge in $\bar{E}$, the adherence of $E$.
DEFINITION 2.5.- A function $\varphi: E \times F^{\mathbf{Z}} \rightarrow E$ is said $\preceq$-increasing when
$$\eta \preceq \eta^{\prime} \Longrightarrow \varphi(\eta, \omega) \preceq \varphi\left(\eta^{\prime}, \omega\right), \mathbf{P}{X}-a . s . .$$ It is said continuous with respect to its first variable when for $\mathbf{P}{X}$-almost all $\omega$, the function $(\eta \mapsto \varphi(\eta, \omega))$ is continuous for the metric $d_{E}$.

THEOREM $2.4$ (LOYNES’s THEOREM).- If $\varphi$ is $\preceq$-increasing and continuous, the equation [2.7] admits a solution $M_{\infty}$ with values in the adherence $\bar{E}$ of $E$.

Proof. Let us recall that we have assumed that we know the stimulus through the quadruple $\mathfrak{O}$, whose generic element is denoted $\omega$. We look for a random variable $Y$ valued in $E$ and satisfying [2.7]. We will get $Y$ as the limit of a sequence converging almost surely. To do this, we consider Loynes’s sequence $\left(M_{n}, n \in \mathbf{N}\right)$, defined by
$$M_{0}(\omega)=\mathbf{0} \text { and } M_{n+1}(\omega)=\varphi\left(M_{n} \circ \theta^{-1}(\omega), \theta^{-1} \omega\right), \forall n \geq 1 .$$
By the definition of $\mathbf{0}$, we have $M_{0}=\mathbf{0} \preceq M_{1}$, and assuming that for some $n>1$, $M_{n-1} \preceq M_{n}$ a.s., since $\varphi$ is increasing we have
$$M_{n}(\omega)=\varphi\left(M_{n-1}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right) \preceq \varphi\left(M_{n}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right)=M_{n+1}(\omega) \mathbf{P}_{X} \text {-a.s.. }$$

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

The idea of coupling plays a central role in the asymptotic study of SRS. It is in fact possible to state the conditions under which the trajectories of two SRS (or possibly those of the corresponding backward schemes) coincide at a certain point. These properties imply naturally, in particular, more traditional properties of convergence for random sequences such as convergence in distribution.

Hereafter we only state the results that will be useful to us in the applications to queueing, in their simplest form.

Secondly, we develop the theory of renovating events of Borovkov, which gives sufficient conditions for coupling, and even strong backward coupling. In addition, the results of Borovkov and Foss also allow in many cases to solve the equation [2.7], even when the conditions of continuity and monotonicity of Theorem $2.4$ are not satisfied. Particularly, in this framework we can also deal with the intricate question of the transient behavior depending on the initial conditions. In what follows, $\mathfrak{O}=$ $(\Omega, \mathcal{F}, \mathbf{P}, \theta)$ is a stationary ergodic quadruple.

## 统计代写|随机分析作业代写stochastic analysis代写|Loynes’s scheme

$$\eta \preceq \eta^{\prime} \Longrightarrow \varphi(\eta, \omega) \preceq \varphi\left(\eta^{\prime}, \omega\right), \mathbf{P} X-a . s . .$$

$$M_{0}(\omega)=\mathbf{0} \text { and } M_{n+1}(\omega)=\varphi\left(M_{n} \circ \theta^{-1}(\omega), \theta^{-1} \omega\right), \forall n \geq 1$$

$$M_{n}(\omega)=\varphi\left(M_{n-1}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right) \preceq \varphi\left(M_{n}\left(\theta^{-1} \omega\right), \theta^{-1} \omega\right)=M_{n+1}(\omega) \mathbf{P}_{X} \text {-a.s.. }$$

## 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|Fluid model

A fluid model consists of replacing a queue which is a discrete-time event system by a reservoir of infinite capacity which empties itself at unit speed and is fed by some continuous data flow. We can then obtain qualitative results on models whose study supports no other approaches. On the one hand, the method does not require precise knowledge about the rate of the input process, and on the other hand, it is particularly well adapted to the study of extreme cases: low and high loads, superposition of heterogeneous traffic.

We work in continuous time and we assume that all the processes are rightcontinuous with left limits. We denote:
1) $S(t)$ : the total service time for the requests arrived up to time $t$;
2) $W(t)$ : the virtual waiting time of a customer arriving at time $t$, that is the time that the customer must wait before starting to be served;
3) $X(t)=S(t)-t$.
As the system has no losses, we have
$$W(t)=X(t)-\left(t-\int_{0}^{t} \mathbf{1}_{{0}}(W(s)) \mathrm{d} s\right) .$$
We will focus on showing an equivalent formulation of this equation.

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

The concept of stationarity implies invariance in time, that is : a shift in time does not change the global picture. If the idea is easily understood, its formalization quickly clouds the basic concept.

Let us consider the set $F^{\mathbf{N}}$ of sequences of elements of a set $F$. The shift operator $\theta$ on $F^{\mathbf{N}}$ is then defined by
$$\theta: \begin{cases}F^{\mathbf{N}} & \longrightarrow F^{\mathbf{N}} \ \left(\omega_{n}, n \geq 0\right) & \longmapsto\left(\omega_{n+1}, n \geq 0\right)=\left(\omega_{n}, n \geq 1\right)\end{cases}$$
Defined in this way, this operator has the drawback of not being bijective: if we consider a sequence $\beta=\left(\beta_{n}, n \geq 0\right)$, all the sequences obtained by concatenation of any element of $F$ and $\beta$ are mapped onto $\beta$ by $\theta$. To overcome this problem, it is customary to work with sequences indexed by $\mathbf{Z}$ and not by $\mathbf{N}$. This change has no crucial mathematical consequence, as the indexation space remains countable. Philosophically, however, it implies that there is no more origin of time…
The shift operator is thus defined on $F^{\mathbf{Z}}$ by
$$\theta\left(\omega_{n}, n \in \mathbf{Z}\right)=\left(\omega_{n+1}, n \in \mathbf{Z}\right)$$
and thus becomes bijective!

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

1) $S(t)$ ：请求的总服务时间到达时间 $t$;
2) $W(t)$ ：客户到达时间的虚拟等待时间 $t$ ，即客户在开始服务之前必须等待的时间；
3) $X(t)=S(t)-t$.

$$W(t)=X(t)-\left(t-\int_{0}^{t} \mathbf{1}_{0}(W(s)) \mathrm{d} s\right) .$$

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

$$\theta:\left{F^{\mathbf{N}} \longrightarrow F^{\mathbf{N}}\left(\omega_{n}, n \geq 0\right) \longmapsto\left(\omega_{n+1}, n \geq 0\right)=\left(\omega_{n}, n \geq 1\right)\right.$$

$$\theta\left(\omega_{n}, n \in \mathbf{Z}\right)=\left(\omega_{n+1}, n \in \mathbf{Z}\right)$$

## 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|Traffic, load, Erlang, etc.

In electricity, we count the amps or volts; in meteorology, we measure the pressure; in telecommunications, we count the Erlangs.

The telephone came into existence in 1870. Most of the concepts and notations were derived during this period. Looking at a telephone connection over a time period of length $T$, we define its observed traffic flow as the percentage of time during which the connection is busy
$$\rho=\frac{\sum_{i} t_{i}}{T}$$
A priori, traffic is a dimensionless quantity since it is the ratio of the occupation time to the total time. However, it still has a unit, Erlang, in remembrance of Erlang who, along with Palm, was one of the pioneers of the performance assessment of telephone networks. Therefore, a load of 1 Erlang corresponds to an always busy connection.

Looking at several connections, the traffic carried by this trunk is the sum of the traffic of each connection
$$\rho_{\text {trunk }}=\sum_{\text {connections }} \rho_{\text {connection }}$$
This is no longer a percentage, but we can give a physical interpretation to this quantity according to the ergodic hypothesis. In fact, assume that the number of junctions is large, then we can calculate the average occupation rate in two different ways: either by calculating the percentage of the occupation time of a particular connection over a large period of time; or by computing the percentage of busy connections at a given time.

## 统计代写|随机分析作业代写stochastic analysis代写|Lindley and Beneˇs

We often consider the number of customers present in the system but the quantity that contains the most information is the system load, defined at each moment as the time required for the system to empty itself in the absence of new arrivals. The server works at unit speed: it serves a unit of work per unit time. Consequently, the load decreases with speed 1 between two arrivals. Figure $1.8$ which represents the load over time depending on the arrivals and required service times is easily constructed.

DEFINITION 1.2.- A busy period of a queue is a period that begins with the arrival of a customer in an empty system (server plus buffer) and ends with the end of a service after which the system is empty again.

A cycle is a time period that begins with the arrival of a customer in an empty system and ends on the next arrival of a customer in an empty system. This is the concatenation of a busy period and an idle period, that is the time elapsed between the departure of the last customer of the busy period and the arrival of the next customer.

NOTE.- In Figure 1.8, a busy period begins at $T_{1}$ and ends at $D_{4}$. The corresponding cycle begins at $T_{1}$ and ends at $T_{5}$.

Note that as long as a service policy is conservative, the size of a busy period is independent of it: for waiting rooms of infinite size, the busy periods have, for example, the same length for the FIFO policy as that for the non-preemptive or preemptive resume LIFO policy.

## 统计代写|随机分析作业代写stochastic analysis代写|Traffic, load, Erlang, etc.

$$\rho=\frac{\sum_{i} t_{i}}{T}$$

$$\rho_{\text {trunk }}=\sum_{\text {connections }} \rho_{\text {connection }}$$

## 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|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|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:
$$\mathbb{E} X=p, \quad \operatorname{Var}(X)=p q .$$

## 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 (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 $\mu \ll m$.

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

(i) $\Omega \in \mathcal{F}$
(ii) 如果 $A \in \mathcal{F}$ ，然后 $A^{c} \in \mathcal{F}$ ，在哪里 $A^{c}=\Omega \backslash A$ 是的补码 $A$ 在 $\Omega$;
(iii) 如果 $A_{1}, A_{2}, \ldots \in \mathcal{F}$ ，然后 $\bigcup_{n=1}^{\infty} A_{n} \in \mathcal{F}$.

$\Omega$. 我们表示 $\sigma(\mathcal{B})$ 最小的 $\sigma$ – 由集合生成的代数 $\mathcal{B}$ ，即最小的 $\sigma$-代数包含 $\mathcal{B}$. 这对 $(\Omega, \mathcal{F})$ 具有上述性质的空间称为可 测空间。

(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) 如果 $A_{1}, A_{2}, \ldots \in \mathcal{F}$ 是成对不相交的，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) 称为可数可加性或 $\sigma$-可加性。

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

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

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

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

## 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|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.

## 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.

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

(i) $\Omega \in \mathcal{F}$
(ii) 如果 $A \in \mathcal{F}$ ，然后 $A^{c} \in \mathcal{F}$ ，在哪里 $A^{c}=\Omega \backslash A$ 是的补码 $A$ 在 $\Omega$;
(iii) 如果 $A_{1}, A_{2}, \ldots \in \mathcal{F}$ ，然后 $\bigcup_{n=1}^{\infty} A_{n} \in \mathcal{F}$.

$\Omega$. 我们表示 $\sigma(\mathcal{B})$ 最小的 $\sigma$ – 由集合生成的代数 $\mathcal{B}$ ，即最小的 $\sigma$-代数包含 $\mathcal{B}$. 这对 $(\Omega, \mathcal{F})$ 具有上述性质的空间称为可 测空间。

(a) $\mathbb{P}(\emptyset)=0, \mathbb{P}(\Omega)=1$;
(b) 如果 $A_{1}, A_{2}, \ldots \in \mathcal{F}$ 是成对不相交的，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) 称为可数可加性或 $\sigma$-可加性。

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

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

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

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

## 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 $n \rightarrow \infty$

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

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. Noting that the distribution $\operatorname{Prob}\left{S_{2 n}=k\right}$ is unimodal and symmetric around the state $k=n$, we have
$\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \leq 2 \cdot \frac{1}{2^{2 n}} \sum_{k>n+2 n \epsilon} \frac{(2 n) !}{k !(2 n-k) !}$
$\leq 2(n-2 n \epsilon) \cdot \frac{1}{2^{2 n}} \frac{(2 n) !}{\lceil n+2 n \epsilon\rceil !\lfloor n-2 n \epsilon\rfloor !}$
$\sim \frac{2 \sqrt{1-2 \epsilon}}{\sqrt{\pi(1+2 \epsilon)}} \cdot \frac{\sqrt{n}}{(1-2 \epsilon)^{n(1-2 \epsilon)}(1+2 \epsilon)^{n(1+2 \epsilon)}} \rightarrow 0$
for sufficiently small $\epsilon$ and $n \gg 1$, where $\lceil\cdot\rceil$ and $\lfloor\cdot\rfloor$ are the ceil and floor functions, respectively, defined by $\lceil x\rceil=m+1$ and $\lfloor x\rfloor=m$ if $x \in[m, m+1)$ for $m \in \mathbb{Z}$. This is the weak law of large numbers for this particular example.
In the example of a fair coin, the number of outcomes in an experiment is finite. In contrast, the second class of examples involves a continuous set of possible outcomes. Consider the orientation of a unit vector $\boldsymbol{\tau}$. Denote by $\mathbb{S}^{2}$ the unit sphere in $\mathbb{R}^{3}$. Define $\rho(\boldsymbol{n}), \boldsymbol{n} \in \mathbb{S}^{2}$, as the orientation distribution density; i.e., for $A \subset \mathbb{S}^{2}$,
$$\operatorname{Prob}(\boldsymbol{\tau} \in A)=\int_{A} \rho(\boldsymbol{n}) d S,$$
where $d S$ is the surface area element on $\mathbb{S}^{2}$. If $\boldsymbol{\tau}$ does not have a preferred orientation, i.e., it has equal probability of pointing at any direction, then
$$\rho(\boldsymbol{n})=\frac{1}{4 \pi} .$$
In this case, we say that $\tau$ is isotropic. On the other hand, if $\boldsymbol{\tau}$ does have a preferred orientation, say $\boldsymbol{n}{0}$, then we expect $\rho(\boldsymbol{n})$ to be peaked at $\boldsymbol{n}{0}$.

## 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.

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

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

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

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

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

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

\begin{aligned} &\operatorname{Prob}\left(\left|\frac{S_{2 n}}{2 n}-\frac{1}{2}\right|>\epsilon\right) \leq 2 \cdot \frac{1}{2^{2 n}} \sum_{k>n+2 n \epsilon} \frac{(2 n) !}{k !(2 n-k) !} \ &\leq 2(n-2 n \epsilon) \cdot \frac{1}{2^{2 n}} \frac{(2 n) !}{[n+2 n \epsilon] ![n-2 n \epsilon] !} \ &\sim \frac{2 \sqrt{1-2 \epsilon}}{\sqrt{\pi(1+2 \epsilon)}} \cdot \frac{\sqrt{n}}{(1-2 \epsilon)^{n(1-2 \epsilon)}(1+2 \epsilon \epsilon)^{n(1+2 \epsilon)}} \rightarrow 0 \ &\text { 对于足够小的 } 6 \text { 和 } n \gg 1 \text { ，在哪里 }\lceil\cdot\rceil \text { 和 }[\cdot \text { 分别是 ceil 和 floor 函数，由下式定义 }\lceil x\rceil=m+1 \text { 和 }[x\rfloor=m \text { 如果 } \end{aligned} $x \in[m, m+1)$ 为了 $m \in \mathbb{Z}$. 这是这个特定示例的弱大数定律。

$$\operatorname{Prob}(\boldsymbol{\tau} \in A)=\int_{A} \rho(\boldsymbol{n}) d S,$$

$$\rho(\boldsymbol{n})=\frac{1}{4 \pi} .$$

## 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|Theory and Applications of Infinite

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|Dimensional Oscillatory Integrals

Professor K. Itò’s work on the topic of infinite dimensional oscillatory integrals has been very germinal and stimulated much of the subsequent research in this area. It is therefore a special honour and pleasure to be able to dedicate the present pages to him. We shall give a short exposition of the theory of a particular class of functionals, the oscillatory integrals:
$$I^{\text {ᄒ}}(f)=\quad ” \int_{\Gamma} e^{i \frac{\psi}{*}(\gamma)} f(\gamma) d \gamma “$$
where $\Gamma$ denotes either a finite dimensional space (e.g. $\mathbb{R}^{s}$, or an s-dimensional differential manifold $M^{s}$ ), or an infinite dimensional space (e.g. a “path space”). $\Phi: \Gamma \rightarrow \mathbb{R}$ is called phase function, while $f: \Gamma \rightarrow \mathbb{C}$ is the function to be integrated and $\epsilon \in \mathbb{R} \backslash{0}$ is a parameter. The symbol $d \gamma$ denotes a “flat” measure. In particular, if $\operatorname{dim}(\Gamma)<\infty$ then $d \gamma$ is the Riemann-Lebesgue volume measure, while if $\operatorname{dim}(\Gamma)=\infty$ an analogue of Riemann-Lebesgue measure is not mathematically defined and $d \gamma$ is just a heuristic expression.

## 统计代写|随机分析作业代写stochastic analysis代写|Finite Dimensional Oscillatory Integrals

In the case where $\Gamma$ is a finite dimensional vector space, i.e. $\Gamma=\mathbb{R}^{s}, s \in \mathbb{N}$, the expression (1.1)
$$” \int_{\mathbb{R}^{}} e^{i \frac{\text { s্ }}{\varepsilon}(\gamma)} f(\gamma) d \gamma ”$$ can be defined as an improper Riemann integral. The study of finite dimensional oscillatory integrals of the type (1.2) is a classical topic, largely developed in connection with several applications in mathematics (such as the theory of Fourier integral operators $[48]$ ) and physics. Interesting examples of integrals of the form (1.2) in the case $s=1, \epsilon=1, f=\chi[0, w], w>0$, and $\Phi(x)=\frac{\pi}{2} x^{2}$, are the Fresnel integrals, that are applied in optics and in the theory of wave diffraction. If $\Phi(x)=x^{3}+a x, a \in \mathbb{R}$ we obtain the Airy integrals, introduced in 1838 in connection with the theory of the rainbow. Particular interest has been devoted to the study of the asymptotic behavior of integrals (1.2) when $\epsilon$ is regarded as a small parameter converging to 0 . Originally introduced by Stokes and Kelvin and successively developed by several mathematicians, in particular van der Corput, the “stationary phase method” provides a powerful tool to handle the asymptotics of (1.2) as $\epsilon \downarrow 0$. According to it, the main contribution to the asymptotic behavior of the integral should come from those points $\gamma \in \mathbb{R}^{}$ which belong to the critical manifold:
$$\Gamma_{c}^{\phi}:=\left{\gamma \in \mathbb{R}^{s}, \mid \Phi^{\prime}(\gamma)=0\right}$$
that is the points which make stationary the phase function $\Phi$. Beautiful mathematical work on oscillatory integrals and the method of stationary phase is connected with the mathematical classification of singularities of algebraic and geometric structures (Coxeter indices, catastrophe theory), see, e.g. [31].

## 统计代写|随机分析作业代写stochastic analysis代写|Infinite Dimensional Oscillatory Integrals

The extension of the results valid for $\Gamma=\mathbb{R}^{s}$ to the case where $\Gamma$ is an infinite dimensional space is not trivial. The main motivation is the study of the “Feynman path integrals”, a class of (heuristic) functional integrals introduced by R.P. Feynman in $1942^{1}$ in order to propose an alternative, Lagrangian, formulation of quantum mechanics. According to Feynman, the solution of the Schrödinger equation describing the time evolution of the state $\psi \in L^{2}\left(\mathbb{R}^{d}\right)$ of a quantum particle moying in a potential $V$
$$\left{\begin{array}{l} i \hbar \frac{\partial}{\partial t} \psi=-\frac{n^{2}}{2 m} \Delta \psi+V \psi \ \psi(0, x)=\psi_{0}(x) \end{array}\right.$$

(where $m>0$ is the mass of the particle, $\hbar$ is the reduced Planck constant, $t \geq 0, x \in \mathbb{R}^{d}$ ) can be represented by a “sum over all possible histories”, that is an integral over the space of paths $\gamma$ with fixed end point
$$\vartheta \gamma^{\prime}(t, x)=-\int_{{\gamma \mid \gamma(t)=x}} e^{\hbar S_{t}(\gamma)} \gamma_{\gamma}(\gamma(0)) d \gamma^{\eta}$$
$S_{t}(\gamma)=S^{0}(\gamma)-\int_{0}^{t} V(s, \gamma(s)) d s, S^{0}(\gamma)=\frac{m}{2} \int_{0}^{t}|\dot{\gamma}(s)|^{2} d s$, is the classical action of the system evaluated along the path $\gamma$ and $d \gamma$ a heuristic “flat” measure on the space of paths (see e.g. [40] for a physical discussion of Feynman’s approach and its applications). The Feynman path integrals (1.4) can be regarded as oscillatory integrals of the form (1.1), where
$$\Gamma=\left{\text { paths } \gamma:[0, t] \rightarrow \mathbb{R}^{s}, \gamma(t)=x \in \mathbb{R}^{s}\right}$$
the phase function $\Phi$ is the classical action functional $S_{t}, f(\gamma)=\psi_{0}(\gamma(0))$, the parameter $\epsilon$ is the reduced Planck constant $\hbar$ and $d \gamma$ denotes heuristically
$$d \gamma={ }^{\alpha} C \prod_{s \in[0, t]} d \gamma(s)^{“},$$
$C:=”\left(\int_{{\gamma \mid \gamma(t)=x}} e^{\frac{1}{\hbar} S_{0}(\gamma)} d \gamma\right)^{-1 “}$ being a normalization constant
The Feynman’s path integral representation (1.4) for the solution of the Schrödinger equation is particularly suggestive. Indeed it creates a connection between the classical (Lagrangian) description of the physical world and the quantum one and makes intuitive the study of the semiclassical limit of quantum mechanics, that is the study of the detailed behavior of the wave function $\psi$ in the case where the Planck constant $\hbar$ is regarded as a small parameter. According to an (heuristic) application of the stationary phase method, in the limit $\hbar \downarrow 0$ the main contribution to the integral (1.4) should come from those paths $\gamma$ which make stationary the action functional $S_{t}$. These, by Hamilton’s least action principle, are exactly the classical orbits of the system.

Despite its powerful physical applications, formula (1.4) lacks mathematical rigour, in particular the “flat” measure $d \gamma$ given by (1.5) has no mathematical meaning.

## 统计代写|随机分析作业代写stochastic analysis代写|Dimensional Oscillatory Integrals

K. Itò 教授关于无限维振荡积分的研究非常具有开创性，并激发了该领域的许多后续研究。因此，能够将本页献给他是一种特殊的荣幸和荣幸。我们将对一类特殊泛函的理论进行简短的阐述，即振荡积分：
ᄒ一世ᄒ(F)=”∫Γ和一世ψ∗(C)F(C)dC“

## 统计代写|随机分析作业代写stochastic analysis代写|Finite Dimensional Oscillatory Integrals

্”∫R和一世 s ্ e(C)F(C)dC”可以定义为不正确的黎曼积分。(1.2) 类型的有限维振荡积分的研究是一个经典课题，主要与数学中的几种应用（例如傅里叶积分算子理论[48]) 和物理学。本例中 (1.2) 形式的积分的有趣示例s=1,ε=1,F=χ[0,在],在>0， 和披(X)=圆周率2X2, 是菲涅耳积分，应用于光学和波衍射理论。如果披(X)=X3+一种X,一种∈R我们获得了 1838 年与彩虹理论相关的艾里积分。特别感兴趣的是积分（1.2）的渐近行为的研究，当ε被认为是一个收敛到 0 的小参数。最初由 Stokes 和 Kelvin 提出并由几位数学家，特别是 van der Corput 相继开发，“平稳相法”提供了一个强大的工具来处理 (1.2) 的渐近性：ε↓0. 据此，对积分渐近行为的主要贡献应该来自这些点C∈R属于临界流形：
\Gamma_{c}^{\phi}:=\left{\gamma \in \mathbb{R}^{s}, \mid \Phi^{\prime}(\gamma)=0\right}\Gamma_{c}^{\phi}:=\left{\gamma \in \mathbb{R}^{s}, \mid \Phi^{\prime}(\gamma)=0\right}

## 统计代写|随机分析作业代写stochastic analysis代写|Infinite Dimensional Oscillatory Integrals

$$\左{一世⁇∂∂吨ψ=−n22米Δψ+在ψ ψ(0,X)=ψ0(X)\对。$$

（在哪里米>0是粒子的质量，⁇是简化的普朗克常数，吨≥0,X∈Rd) 可以表示为“所有可能历史的总和”，即路径空间上的积分C带固定端点
ϑC′(吨,X)=−∫C∣C(吨)=X和⁇小号吨(C)CC(C(0))dC这

\Gamma=\left{\text { 路径} \gamma:[0, t] \rightarrow \mathbb{R}^{s}, \gamma(t)=x \in \mathbb{R}^{s}\对}\Gamma=\left{\text { 路径} \gamma:[0, t] \rightarrow \mathbb{R}^{s}, \gamma(t)=x \in \mathbb{R}^{s}\对}

dC=一种C∏s∈[0,吨]dC(s)“,
C:=”(∫C∣C(吨)=X和1⁇小号0(C)dC)−1“作为归一化常数

## 有限元方法代写

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## 统计代写|随机分析作业代写stochastic analysis代写|Homogenization of Diffusions on the Lattice

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• Statistical Inference 统计推断
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• 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|Periodic Drift Coefficients

In this paper we treat limit theorems for diffusions on the lattice $\mathbf{Z}^{d}$ of the form of those constituting the solution of the homogenization problem of diffusions. For finite dimensional diffusion processes, various models of homogenization (generalized in several directions) have been studied in detail (cf. eg. $[\mathrm{F} 2, \mathrm{FNT}$, FunU, O, PapV, Par] and references therein). On the other hand, for corresponding prohlems of infinite dimensional diffusions only fow results are known (cf. [FunU, ABRY1,2,3]). In this paper we consider a homogenization problem of infinite dimensional diffusion processes indexed by $\mathbf{Z}^{d}$ having periodic drift coefficients with the period $2 \pi$ (cf. (2.1)), by applying an $L^{2}$ type ergodic theorem for the corresponding quotient processes taking values in $[0,2 \pi)^{\mathbf{z}^{d}}$ (cf. Prop. 1). The ergodic theorem which is based on a (weak) Poincaré inequality.

In [ABRY3] the same problem has been discussed by applying the uniform ergodic theorem for the corresponding quotient process, that is available by assuming that the Markov semi-group of the quotient process of the original process satisfies a logarithmic Sobolev inequality. In the same paper it has also

been shown that a homogenization property of the processes starting from an almost every arbitrary point in the state space with respect to an invariant measure of the quotient process holds (cf. also [ABRY1, ABRY2]). In this occasion, the main purpose of the present paper is the comparison between the results derived under the assumption of logarithmic Sobolev inequality and the corresponding results proven by assuming $L^{2}$ ergodic theorem based on (weak) Poincaré inequality, which is strictly weaker than the one for logarithmic Sobolev inequality (cf. [AKR, G]). This paper is a series of works on the considerations of several types of homogenization models for infinite dimensional diffusion processes.

For an adequate understanding of crucial differences between homogenization problems in finite and infinite dimensional situations, we first brietly review a simple case of the homogenization problem for finite dimensional diffusions.

On some complete probability space, suppose that we are given a one dimensional standard Brownian motion process $\left{B_{t}\right}_{t \in \mathbf{R}{+}}$and consider the stochastic differential equation for each initial state $x \in \mathbf{R}$ and each scaling parameter $\epsilon>0$ given by \begin{aligned} X^{\epsilon}(t, x)=& x+\frac{1}{\epsilon} \int{0}^{t} b\left(\frac{X^{\epsilon}(s, x)}{\epsilon}\right) d s \ &+\sqrt{2} \int_{0}^{t} a\left(\frac{X^{\epsilon}(s, x)}{\epsilon}\right) d B_{s}, \quad t \in \mathbf{R}_{+}, \end{aligned}
where $a \in C^{\infty}(\mathbf{R} \rightarrow \mathbf{R})$ is a periodic function with period $2 \pi$ which satisfies
$$\lambda \leq a(x) \leq \lambda^{-1}, \quad \forall x \in \mathbf{R},$$
for some constant $\lambda>0$ and $b(x) \equiv \frac{d}{d x} a^{2}(x)$.

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

Let $\mathbf{N}$ and $\mathbf{Z}$ be the set of natural numbers and integers respectively. For $d \in \mathbf{N}$ let $\mathbf{Z}^{d}$ be the $d$-dimensional lattice. We consider the problem for the diffusions taking values in $\mathbf{R}^{\mathbf{Z}^{d}}$. We use the following notions and notations:
By $\mathbf{k}$ we denote $\mathbf{k}=\left(k^{1}, \ldots, k^{d}\right) \in \mathbf{Z}^{d}$. For a subset $A \subseteq \mathbf{Z}^{d}$, we define $|A| \equiv \operatorname{card} A$. For $\mathbf{k} \in \mathbf{Z}^{d}$ and $A \subseteq \mathbf{Z}^{d}$ let
$$A+\mathbf{k} \equiv{\mathbf{l}+\mathbf{k} \mid \mathbf{l} \in A}$$
For any non-empty $A \subseteq \mathbf{Z}^{d}$, we assume that $\mathbf{R}^{A}$ is the topological space equipped with the direct product topology. For each non-empty $A \subseteq Z^{d}$, by $\mathbf{x}{A}$ we denote the image of the projection onto $\mathrm{R}^{A}$ : $$\mathbf{R}^{\mathbf{Z}^{d}} \ni \mathbf{x} \longmapsto \mathbf{x}{A} \in \mathbf{R}^{A}$$
For each $p \in N \cup{0} \cup{\infty}$ we define the set of $p$-times continuously differentiable functions with support $A: C_{A}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) \equiv\left{\varphi\left(\mathbf{x}{A}\right) \mid \varphi \in C P\left(\mathbf{R}^{A}\right)\right}$, where $C^{P}\left(\mathbf{R}^{A}\right)$ is the set of real valued $p$-times continuously differentiable functions on $\mathbf{R}^{A}$. For $p=0$, we simply denote $C{A}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$ by $C_{A}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) .$ Also we set
$$C_{0}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) \equiv\left{\varphi \in C_{A}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)|| A \mid<\infty\right}$$
$\mathcal{B}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$ is the Borel $\sigma$-field of $\mathbf{R}^{\mathbf{Z}^{d}}$ and $\mathcal{B}{A}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$ is the sub $\sigma$-field of $\mathcal{B}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$ that is generated by the family $C{A}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right)$. For each $\mathbf{k} \in \mathbf{Z}^{d}$, let $\vartheta^{\mathbf{k}}$ be the shift operator on $\mathbf{R}^{\mathbf{Z}^{d}}$ such that

$$\left(v^{\mathbf{k}} \mathbf{x}\right){{\mathbf{j}}} \equiv \mathbf{x}{{\mathbf{k}+\mathbf{j}}}, \mathbf{x} \in \mathbf{R}^{\mathbf{Z}^{d}}, \mathbf{j} \in \mathbf{Z}^{d},$$
where $\mathbf{x}_{{\mathbf{k}+\mathbf{j}}}$ is the $\mathbf{k}+\mathbf{j}$-th component of the vector $\mathbf{x}$.

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

In [ABRY3] we have considered the homogenization problem of the sequence of the diffusions $\left{\left{\mathbb{X}^{c}(t, \mathbf{x})\right}_{t \in \mathbf{R}}\right}_{\epsilon>0}$ in the case where the the following uniform ergodicity (3.1) holds for the quotient process $\left(\left{\eta_{t}\right}_{t \geq 0}, Q_{\mathbf{y}}: \mathbf{y} \in T^{\mathbf{z}^{d}}\right)$. Here we consider the same problem for $\left{\left{\mathbb{X}^{c}(t, \mathbf{x})\right}_{t \in \mathbb{R}{+}}\right}{e>0}$ in the case where the $L^{2}$-type ergodicity holds for $\left(\eta_{t}, Q_{\mathbf{y}}: \mathbf{y} \in T^{\mathbf{Z}^{d}}\right)$, and compare the results available under these two different assumptions of (3.1) and (3.2). Each comparison will be given as a Remark following each Theorem resp. Lemma.

In the sequel we denote the uniform ergodicity (3.1) as $(\mathrm{LS})$ and the $L^{2}$ type ergodicity $(3.2)$ as (WP) respectively. We have to remark that if the

potential $\mathcal{J}$, that satisfies J-1), J-2) and J-3), satisfies in addition DobrushinShlosman mixing condition, then (3.1) holds, more precisely in this case the logarithmic Sobolev inequality (LS) holds for the Dirichlet form $\mathcal{E}(u(\cdot), v(\cdot))$ defined in Remark 2, then the stronger inequality such that the term $(c+t)^{-\alpha}$ in (3.1) is replaced by $e^{-\alpha t}$ for some $\alpha>0$ holds (cf. [S]).

Correspondingly, if $\mathcal{E}(u(\cdot), v(\cdot))$ satisfies the weak Poincare (WP) inequality, then (3.2) holds. We remark that the logarithmic Sobolev inequality is strictly stronger than the the weak Poincare inequality (cf. [RWang]).
Precisely, we define the ergodicities (LS) and (WP) as follows:
(LS) For some Gibbs state $\mu$, there exists a $c=c(\mathcal{J})>0$ and an $\alpha=$ $\alpha(\mathcal{J})>1$ which depend only on $\mathcal{J}$, such that for each $A \in \mathbf{Z}^{d}$ with $|\Lambda|<\infty$ there exists $K(A) \in(0, \infty)$ and for $\forall t>0, \forall \varphi \in C_{A}^{\infty}\left(T^{\mathbf{Z}^{d}}\right)$ the following holds
$$\left|\int_{T^{\mathbf{z}}} \varphi\left(\mathbf{y}{A}\right) p\left(t,{ }^{,}, d \mathbf{y}\right)-\langle\varphi, \mu)\right|{L^{\infty}} \leq K(\Lambda)(c+t)^{-\alpha}\left(|\nabla \varphi|_{L^{\infty}}+|\varphi|_{L^{\infty}}\right)$$
(WP) There exist $c=c(\mathcal{J})>0, \alpha=\alpha(\mathcal{J})>1$ and $K>0$, that depends only on $\mathcal{J}$, and the following holds
$$\left|\mathcal{P}{t} \varphi-<\varphi, \mu>\right|{L^{2}(\mu)} \leq K(c+t)^{-\alpha}|\varphi|_{L^{2}(\mu)}, \forall t>0, \forall \varphi \in C\left(T^{\mathbf{Z}^{d}}\right)$$
We also remark that (3.1) or (3.2) gives the uniqueness of the Gibbs state, since by (3.1) or (3.2) we see that a Gibbs state $\mu$ that satisfies (3.1) or (3.2) is the only invariant measure for $p\left(t,{ }^{-}, d \mathbf{y}\right)$, but every Gibbs state is an invariant measure. From now on we denote the unique Gibbs measure by $\mu$ (cf. [ABRY3, $\mathrm{AKR}]$ ).

## 统计代写|随机分析作业代写stochastic analysis代写|Periodic Drift Coefficients

λ≤一种(X)≤λ−1,∀X∈R,

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

：ķ我们表示ķ=(ķ1,…,ķd)∈从d. 对于一个子集一种⊆从d，我们定义|一种|≡卡片⁡一种. 为了ķ∈从d和一种⊆从d让

C_{0}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) \equiv\left{\varphi \in C_{A}^{p}\left (\mathbf{R}^{\mathbf{Z}^{d}}\right)|| 一个 \mid<\infty\right}C_{0}^{p}\left(\mathbf{R}^{\mathbf{Z}^{d}}\right) \equiv\left{\varphi \in C_{A}^{p}\left (\mathbf{R}^{\mathbf{Z}^{d}}\right)|| 一个 \mid<\infty\right}

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

(LS) 对于某些吉布斯状态μ，存在一个C=C(Ĵ)>0和一种= 一种(Ĵ)>1这仅取决于Ĵ, 这样对于每个一种∈从d和|Λ|<∞那里存在ķ(一种)∈(0,∞)并且对于∀吨>0,∀披∈C一种∞(吨从d)以下成立
|∫吨和披(是一种)p(吨,,,d是)−⟨披,μ)|大号∞≤ķ(Λ)(C+吨)−一种(|∇披|大号∞+|披|大号∞)
(WP) 存在C=C(Ĵ)>0,一种=一种(Ĵ)>1和ķ>0，这仅取决于Ĵ, 并且以下成立
|磷吨披−<披,μ>|大号2(μ)≤ķ(C+吨)−一种|披|大号2(μ),∀吨>0,∀披∈C(吨从d)

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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|No–go Theorems

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|No–go Theorems

The first no-go theorem, showing that it is not true that, if a Lie algebra admits a Fock representation, then any associated current algebra also admits one was proved by Śniady [Śnia99]. In the terminology intruduced in the present paper Śniady’s result can be rephrased as follows:

Theorem 10. The Schrödinger algebra admits a Fock representation but its associated current algebra over $\mathbb{R}$ with Lebesgue measure doesn’t.

Since the Schrödinger algebra is contained in the full oscillator algebra, which clearly admits a Fock representation, Sniady’s theorem also rules out the possibility of a Fock representation for the current algebra of the full oscillator algebra over $\mathbb{R}$ with Lebesgue measure.

Recalling, from the examples at the end of Section (18), that the Schrödinger algebra is the smallest *-Lie algebra containing the oscillator algebra (with generators $\left{a^{+}, a, a^{+} a, 1\right}$ ) and the square-oscillator algebra, i.e. $s l(2, \mathbb{R}$ ) (with generators $\left{a^{+2}, a^{2}, a^{+} a, 1\right}$ ), we see that the difficulty comes from the combination of two closed Lie algebras. More precisely: consider the two sets of generators
$$\begin{gathered} \left{a^{+}, a, a^{+} a, 1\right} \ \left{a^{+2}, a^{2}, a^{+} a, 1\right} \end{gathered}$$
We know that the current algebra over $\mathbb{R}^{d}$ associated to each of them has a Fock representation. However the union of the two sets, i.e.
$$\left{a^{+}, a, a^{+2}, a^{2}, a^{+} a, 1\right}$$
is also a set of generators of a *-Lie algebra whose associated current algebra over $\mathbb{R}^{d}$ does not admit a Fock representation.

Notice that the first of the two algebras is generated by the first powers of the white noise and the number operator while the second one is generated by the second powers of the white noise and the number operator. An extrapolation of this argument suggested the hope that a similar thing could happen also for the higher powers, i.e. that, denoting $\mathcal{G}{3}$ the -Lie algebra generated by the cube of the white noise $b{t}^{3}$ and the number operator; and, for $n \geq 4, \mathcal{G}{n}$ the $$-Lie algebra generated by the number operator and the smallest power of the white noise not included in \bigcup{1 \leq k \leq n-1} \mathcal{G}{k}, the current algebra of \mathcal{G}{n} over \mathbb{R}^{d} admits a Fock representation. This hope was ruled out by the following generalization of Sniady’s theorem, due to Accardi, Boukas and Franz [AcBouFr05] and by its corollary reported below. ## 统计代写|随机分析作业代写stochastic analysis代写|Connection with an Old Open Problem in Classical Probability Since the vacuum distribution of the first order classical white noise is a Gaussian, any reasonable renormalization should lead to the conclusion that the n-th power of the first order classical white noise is still the n-th power of a Gaussian. But the \delta-correlation implies that the corresponding integrated process will be a stationary additive independent increment process on \mathbb{R}. These heuristic ideas, which can be put in a satisfactory mathematical form with some additional work, lead to the conjecture that a necessary condition for the existence of the n-th power of white noise, renormalized as in [AcBouFr05], is that the n-th power of a classical Gaussian random variable is infinitely divisible. The n-th powers of the standard Gaussian random variable \gamma and their distributions have been widely studied. It is known that, \forall k \geq 1 \gamma^{2 k} is infinitely divisible, but it is not known if, \forall k \geq 1 \gamma^{2 k+1} is infinitely divisible (and the experts conjecture that, at least for \gamma^{3}, the answer is negative). ## 统计代写|随机分析作业代写stochastic analysis代写|Renormalized Powers of White Noise and the Virasoro-Zamolodchikov Algebra In the present section we will use the notations of Section (20) and the results of the papers [AcBou06a, AcBou06b, AcBou06c] which contain the proofs of all the results discussed here. The formal extension of the white noise commutation relations to the associative *-algebra generated by b_{t}, b_{s}^{\dagger}, 1, called from now on the renormalized higher powers of (Boson) white noise (RHPWN) algebra, leads to the identities:$$
\begin{aligned}
{\left[b_{t}^{\dagger^{n}} b_{t}^{k}, b_{s}^{\dagger} b_{s}^{K}\right]=} & \epsilon_{k, 0} \epsilon_{N, 0} \sum_{L \geq 1}\left(\begin{array}{c}
k \
L
\end{array}\right) N^{(L)} b_{t}^{\dagger^{n}} b_{s}^{\dagger^{N-L}} b_{t}^{k-L} b_{s}^{K} \delta^{L}(t-s) \
&-\epsilon_{K, 0} \epsilon_{n, 0} \sum_{L \geq 1}\left(\begin{array}{c}
K \
L
\end{array}\right) n^{(L)} b_{s}^{\dagger} b_{t}^{\dagger^{n-L}} b_{s}^{K-L} b_{t}^{k} \delta^{L}(t-s)
\end{aligned}
$$In Section (20) we have given a meaning to these formal commutation relations, i.e. to the ill defined powers of the \delta-function, through the renormalization prescription (20.2). In the present note we will use a different renormalization rule, introduced in [AcBou06a] and whose motivations are discussed in [AcBou06b, AcBou06c], namely:$$
$$where the right hand side is defined as a convolution of distributions. Using this (23.1) can be rewritten in the form:$$
\begin{aligned}
{\left[b_{t}^{\dagger^{n}} b_{t}^{k}, b_{s}^{\dagger^{N}} b_{s}^{K}\right]=} & \epsilon_{k, 0} \epsilon_{N, 0}\left(k N b_{t}^{\dagger^{n}} b_{s}^{\dagger^{N-1}} b_{t}^{k-1} b_{s}^{K} \delta(t-s)\right.\
&\left.+\sum_{L \geq 2}\left(\begin{array}{l}
k \
L
\end{array}\right) N^{(L)} b_{t}^{\dagger^{n}} b_{s}^{\dagger^{N-L}} b_{t}^{k-L} b_{s}^{K} \delta(s) \delta(t-s)\right) \
&-\epsilon_{K, 0} \epsilon_{n, 0}\left(K n b_{s}^{\dagger^{N}} b_{t}^{\dagger^{\dagger-1}} b_{s}^{K-1} b_{t}^{k} \delta(t-s)\right.\
&\left.+\sum_{L \geq 2}\left(\begin{array}{c}
K \
L
\end{array}\right) n^{(L)} b_{s}^{\dagger^{N}} b_{t}^{\dagger^{n-L}} b_{s}^{K-L} b_{t}^{k} \delta(s) \delta(t-s)\right)
\end{aligned}
$$Introducing test functions and the associated smeared fields$$
B_{k}^{n}(f):=\int_{\mathbb{R}} f(t) b_{t}^{\dagger^{n}} b_{t}^{k} d t
$$The commutation relations (23.2) become:$$
\begin{aligned}
&{\left[B_{k}^{n}(\bar{g}), B_{K}^{N}(f)\right]=\left(\epsilon_{k, 0} \epsilon_{N, 0} k N-\epsilon_{K, 0} \epsilon_{n, 0} K n\right) B_{K+k-1}^{N+n-1}(\bar{g} f)} \
&\quad+\sum \sum_{L=2}^{(K \wedge n) \vee(k \wedge N)} \theta_{L}(n, k ; N, K) \bar{g}(0) f(0) b_{0}^{\dagger^{N+n-l}} b_{0}^{K+k-I} \
&\theta_{L}(N, K, n, k) \cdot-\varepsilon_{K, 0} \varepsilon_{n, 0}\left(\begin{array}{c}
K \
L
\end{array}\right) n^{(L)}-\tau_{k, 0} \kappa_{N, 0}\left(\begin{array}{c}
k \
L
\end{array}\right) N^{(L)}
\end{aligned}
$$The commutation relations (23.4) still contain the ill defined symbol b_{0}^{\dagger^{N+n-1}} b^{K+k-l}. However, if the test function space is chosen so that$$
f(0)=g(0)=0
$$then the singular term in (23.4) vanishes and the commutation relations (23.4) become:$$
\left[B_{k}^{n}(\bar{g}), B_{K}^{N}(f)\right]{R}:=(k N-K n) B{k+K-1}^{n+N-1}(\bar{g} f)

## 统计代写|随机分析作业代写stochastic analysis代写|Renormalized Powers of White Noise and the Virasoro-Zamolodchikov Algebra

[b吨†nb吨ķ,bs†bsķ]=εķ,0εñ,0∑大号≥1(ķ 大号)ñ(大号)b吨†nbs†ñ−大号b吨ķ−大号bsķd大号(吨−s) −εķ,0εn,0∑大号≥1(ķ 大号)n(大号)bs†b吨†n−大号bsķ−大号b吨ķd大号(吨−s)

dl(吨−s)=d(s)d(吨−s),l=2,3,4,…

[b吨†nb吨ķ,bs†ñbsķ]=εķ,0εñ,0(ķñb吨†nbs†ñ−1b吨ķ−1bsķd(吨−s) +∑大号≥2(ķ 大号)ñ(大号)b吨†nbs†ñ−大号b吨ķ−大号bsķd(s)d(吨−s)) −εķ,0εn,0(ķnbs†ñb吨††−1bsķ−1b吨ķd(吨−s) +∑大号≥2(ķ 大号)n(大号)bs†ñb吨†n−大号bsķ−大号b吨ķd(s)d(吨−s))

[乙ķn(G¯),乙ķñ(F)]=(εķ,0εñ,0ķñ−εķ,0εn,0ķn)乙ķ+ķ−1ñ+n−1(G¯F) +∑∑大号=2(ķ∧n)∨(ķ∧ñ)θ大号(n,ķ;ñ,ķ)G¯(0)F(0)b0†ñ+n−lb0ķ+ķ−一世 θ大号(ñ,ķ,n,ķ)⋅−eķ,0en,0(ķ 大号)n(大号)−τķ,0ķñ,0(ķ 大号)ñ(大号)

F(0)=G(0)=0

$$\left[B_{k}^{n}(\bar{g}), B_{K}^{N}(f) \right] {R}:=(k NK n) B {k+K-1}^{n+N-1}(\bar{g} f)$$

## 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|Current Representations of Lie Algebras

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|Current Representations of Lie Algebras

Intuitively, if ${\mathcal{L},[*,], *$,$} is a -Lie algebra, a current algebra of \mathcal{L}$ over $\mathbb{R}^{d}$ is a vector space $\mathcal{T}$ of $\mathcal{L}$-valued functions defined on $\mathbb{R}^{d}$ and closed under the pointwise operations: $$\varphi, \psi:=[\varphi(t), \psi(t)] ; \quad \varphi^{}(t):=\varphi(t)^{} ; \quad t \in \mathbb{R}, \varphi \in \mathcal{T}$$ For example, if $X_{1}, \ldots, X_{k}$ are generators of $\mathcal{L}$ one can fix a space $\mathcal{S}$, of complex valued test functions on $R$ and to each $\varphi \in \mathcal{S}$ and $j \in{1, \ldots, k}$ one can associate the $\mathcal{L}$-valued function on $\mathbb{R} X_{j}(\varphi)$ defined by: $$X_{j}(\varphi)(t):=\varphi(t) X_{j} ; \quad t \in \mathbb{R}$$ Definition 6. Let $\mathcal{G}$ be a complex-Lie algebra. A (canonical) set of generators of $\mathcal{G}$ is a linear basis of $\mathcal{G}$
$$l_{\alpha}^{+}, l_{\alpha}^{-}, l_{\beta}^{0}, \alpha \in I, \quad \beta \in I_{0}$$
where $I_{0}, I$ are sets, satzsfyng the following conditsons:
$$\begin{array}{ll} \left(l_{\beta}^{0}\right)^{+}=l_{\beta}^{0} ; & \forall \beta \in I_{0} \ \left(l_{\alpha}^{+}\right)^{*}=l_{\alpha}^{-} ; & \forall \alpha \in I \end{array}$$
and all the central elements among the generators are of $l^{0}$-type (i.e. selfadjoint).

We will denote $c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime}, \delta\right)$ the structure constants of $\mathcal{G}$ with respect to the generators $\left(l_{\alpha}^{\varepsilon}\right)$, i.e., with $\alpha, \beta \in I \cup I_{0}, \varepsilon, e^{\prime}, \delta \in{+,-, 0}$, and, assuming summation over repeated indices:
$$\begin{gathered} {\left[l_{\alpha,}^{z}, l_{\beta}^{\varepsilon^{\prime}}\right]=c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime}, \delta\right) l_{\gamma}^{\delta}=} \ :=\sum_{\gamma \in I_{0}} c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime}, 0\right) l_{\gamma}^{0}+\sum_{\gamma \in I} c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime},+\right) l_{\gamma}^{+}+\sum_{\gamma \in I} c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime},-\right) l_{\gamma}^{-} \end{gathered}$$
In the following we will consider only locally finite Lie algebras, i.e. those such that, for any pair $\alpha, \beta \in I \cup I_{0}$ only a finite number of the structure constants $c_{\alpha \beta}^{\gamma}\left(\varepsilon, \varepsilon^{\prime}, \delta\right)$ is different from zero.

## 统计代写|随机分析作业代写stochastic analysis代写|Connections with Classical Independent Increment Processes

In this section we look for some necessary conditions for the solution of the problem stated in the previous section. This will naturally lead to an interesting connection with the theory of classical independent increment processes which was first noticed in Araki’s thesis [Arak60]. We refer to the monographs of K.R. Parthasarathy and K. Schmidt [PaSch72] and of Guichardet [Gui72] for a systematic exposition. In the notations of Section (18) we consider:

• a pair $\left{\mathcal{G},\left(l_{\alpha}^{2}\right)\right}$ of a *-Lie algebra and a set of generators which admits a Fock representation.
• a measure space $(S, \mu)$
• a *-sub-algebra $\mathcal{C} \subseteq L_{\mathrm{C}}^{\infty}(S, \mathcal{B}, \mu)$
such that the current algebra
$$\left{l_{\alpha}^{c}(f): \varepsilon \in{+,-, 0}, \alpha \in I \text { or } \alpha \in I_{0}, f \in \mathcal{C}\right}$$
admits a Fock representation on some Hilbert space $\mathcal{H}$ with cyclic vector $\Phi$. We identify the elements of this current algebra with their images in this representation and we omit from the notation the symbol $\pi$ of the representation. Moreover we add the following assumptions:
(i) among the generators $\left(l_{\alpha}^{c}\right)$ there is exactly one (self-adjoint) central element, denoted $l_{0}^{0}$.
(ii) for any $f \in \mathcal{C}$ one has:
$$l_{0}^{0}(f)=\int_{S} f d \mu$$
where the scalar on the right hand side is identified to the corresponding multiple of the identity operator on $\mathcal{H}$. In particular the representation is weakly irreducible.

Under these conditions it is not difficult to see that the general principle that algebra implies statistics can be applied and that the vacuum mixed moments of the operators $l_{\alpha}^{\varepsilon}(f)$ are uniquely determined by the structure constants of the Lie algebra. Another important property is that, by fixing a measurable subset $I \subseteq S$ such that
$$\mu(I)=1$$
and denoting $\chi_{I}$ the corresponding characteristic function, the $*$-Lie algebra generated by the operators $l_{\alpha}^{\varepsilon}\left(\chi_{I}\right)$ is isomorphic to $\mathcal{G}$ and therefore it has the same vacuum statistics.

Finally the commutation relations (18.1) imply that the maps $f \mapsto l_{\alpha}^{\varepsilon}(f)$ define an independent increment process of boson type, i.e. the restriction of the vacuum state on the polynomial algebra generated by two families

$\left(l_{\alpha}^{\varepsilon}(f)\right){\varepsilon, \alpha}$ and $\left(l{\alpha}^{\varepsilon}(g)\right)_{\varepsilon, \alpha}$ with $f$ and $g$ having disjoint supports, coincides with the tensor product of the restrictions on the single algebras.

In particular, if $X(I)$ is any self-adjoint linear combination of operators of the form $l_{\alpha}^{\varepsilon}\left(\chi_{I}\right)$, then the map $I \subseteq S \mapsto X(I)$ defines an additive independent increment process on $(S, \mathcal{B}, \mu)$. Thus the law of every random variable of the form $X(I)$ will be an infinitely divisible law on $\mathbb{R}$ whenever the set $I$ can be written as a countable union of subsets of nonzero $\mu$-measure.

If $S=\mathbb{R}^{d}$ and $\mu$ is the Lebesgue measure, then any such process $X(I)$ $\left(I \subseteq \mathbb{R}^{d}\right)$ will also be translation invariant.

Combining together all the above remarks one obtains a necessary condition for the existence of the Fock representation of the current algebra of a *-Lie algebra $\mathcal{G}$ and a set of generators namely: the pair $\left{\mathcal{G},\left(l_{\alpha}^{\varepsilon}\right)\right}$ must admit a Fock representation and the vacuum distribution of any self-adjoint linear combination $X$ of generators must be infinitely divisible

Since there is no reason to expect that any pair $\left{\mathcal{G},\left(l_{\alpha}^{z}\right)\right}$ will have this property, this gives a probabilistic intuition of the reason why it might happen that a *-Lie algebra and a set of generators $\left{\mathcal{G},\left(l_{\alpha}^{z}\right)\right}$ might admit a Fock representation without this being true for the associated current algebra.
In the following section we review some progresses made in the past few years in one important special case: the full oscillator algebra.

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

We have seen how the developments reviewed in the previous sections naturally lead to the following problem: can we extend to the renormalized higher powers of quantum white noise what has been achieved for the second powers? To answer this question we start with the Heisenberg algebra
$$\left[a, a^{+}\right]=1$$
Its universally enveloping algebra is generated by the products of monomials of the form
$$a^{n}, a^{+m}$$
and their commutation relations are deduced from (20.1) and the derivation property of the commutator. The problem we want to study is the following: does there exist a current representation of this algebra over $\mathbb{R}^{d}$ for some $d>0$ ?

In order to define the current algebra of the full oscillator algebra, we have first to overcome the renormalization problem, illustrated in Section (14) in the case of the second powers of white noise. In fact, dealing with higher powers of white noise we meet higher powers of the $\delta$-function. A natural way out is to write
$$\delta^{n}=\delta^{2}\left(\delta^{n-2}\right) ; \quad n \geq 2 ; \quad \delta^{0}:=1$$

and to apply iteratively the renormalization prescription used in Section (14). This leads to the following:

Definition. The boson Fock white noise, renormalized with the prescription:
$$\delta(t)^{l}=c^{l-1} \delta(t), c>0, l=2,3, \ldots$$
simply called $R B F W N$ in the following, over a Hilbert space $\mathcal{H}$ with vacuum (unit) vector $\Phi$ is the locally finite *-Lie algebra canonically associated to the associative unital *-algebra of operator-valued distributions on $\mathcal{H}$ with generators
$$b_{t}^{+n} b_{t}^{k}, \quad k, n \in \mathbb{N}, \quad t \in \mathbb{R}^{d}$$
and relations deduced from:
$$\begin{gathered} {\left[b_{t}, b_{s}^{+}\right]=\delta(t-s)} \ {\left[b_{t}^{+}, b_{s}^{+}\right]=\left[b_{t}, b_{s}\right]=0} \ \left(b_{s}\right)^{*}=b_{s}^{+} \ b_{t} \Phi=0 \end{gathered}$$
Here locally finite méans thàt the commutator of any pair of generators is a finite linear combination of generators.

## 统计代写|随机分析作业代写stochastic analysis代写|Current Representations of Lie Algebras

[l一种,和,lbe′]=C一种bC(e,e′,d)lCd= :=∑C∈一世0C一种bC(e,e′,0)lC0+∑C∈一世C一种bC(e,e′,+)lC++∑C∈一世C一种bC(e,e′,−)lC−

## 统计代写|随机分析作业代写stochastic analysis代写|Connections with Classical Independent Increment Processes

• 一双\left{\mathcal{G},\left(l_{\alpha}^{2}\right)\right}\left{\mathcal{G},\left(l_{\alpha}^{2}\right)\right}*-Lie 代数和一组接受 Fock 表示的生成器。
• 测度空间(小号,μ)
• *-子代数C⊆大号C∞(小号,乙,μ)
使得当前代数
\left{l_{\alpha}^{c}(f): \varepsillon \in{+,-, 0}, \alpha \in I \text { 或 } \alpha \in I_{0}, f \in \mathcal{C}\右}\left{l_{\alpha}^{c}(f): \varepsillon \in{+,-, 0}, \alpha \in I \text { 或 } \alpha \in I_{0}, f \in \mathcal{C}\右}
承认某个希尔伯特空间上的 Fock 表示H带循环向量披. 我们在这个表示中用它们的图像来识别这个当前代数的元素，我们从符号中省略了符号圆周率的表示。此外，我们添加了以下假设：
（i）在生成器中(l一种C)有一个（自伴的）中心元素，记为l00.
(ii) 对于任何F∈C一个有：
l00(F)=∫小号Fdμ
其中右侧的标量被标识为对应的身份运算符的倍数H. 特别是表示是弱不可约的。

μ(一世)=1

$\left(l_{\alpha}^{\varepsilon}(f)\right) {\varepsilon, \alpha}一种nd\left(l {\alpha}^{\varepsilon}(g)\right)_{\varepsilon, \alpha}在一世吨HF一种ndg$ 具有不相交的支持，与单个代数限制的张量积一致。

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

[一种,一种+]=1

dn=d2(dn−2);n≥2;d0:=1

d(吨)l=Cl−1d(吨),C>0,l=2,3,…

b吨+nb吨ķ,ķ,n∈ñ,吨∈Rd

[b吨,bs+]=d(吨−s) [b吨+,bs+]=[b吨,bs]=0 (bs)∗=bs+ b吨披=0

## 有限元方法代写

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## 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。