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• 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

## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|Classification of States

Let $X$ be a Markov chain with discrete state space $\mathscr{E}$ and transition matrix $P$. We can characterize the relations between states in the following way: If states $i$ and $j$ are such that $P^t(i, j)>0$ for some $t \geqslant 0$, we say that $i$ leads to $j$ and write $i \rightarrow j$. We say that $i$ and $j$ communicate if $i \rightarrow j$ and $j \rightarrow i$, and write $i \leftrightarrow j$. Using the relation ” $\leftrightarrow$ “, we can divide $\mathscr{E}$ into equivalence classes such that all the states in an equivalence class communicate with each other but not with any state outside that class. If there is only one equivalent class $(=\mathscr{E})$, the Markov chain is said to be irreducible. If a set of states $\mathscr{A}$ is such that $\sum_{j \in \mathscr{A}} P(i, j)=1$ for all $i \in \mathscr{A}$, then $\mathscr{A}$ is called a closed set. A state $i$ is called an absorbing state if ${i}$ is closed. For example, in the transition graph depicted in Figure 1.5, the equivalence classes are ${1,2},{3}$, and ${4,5}$. Class ${1,2}$ is the only closed set: the Markov chain cannot escape from it. If state 1 were missing, state 2 would be absorbing. In Example $1.10$ the Markov chain is irreducible since all states communicate.

Another classification of states is obtained by observing the system from a local point of view. In particular, let $T$ denote the time the chain first visits state $j$, or first returns to $j$ if it started there, and let $N_j$ denote the total number of visits to $j$ from time 0 on. We write $\mathbb{P}_j(A)$ for $\mathbb{P}\left(A \mid X_0=j\right)$ for any event $A$. We denote the corresponding expectation operator by $\mathbb{E}_j$. State $j$ is called a recurrent state if $\mathbb{P}_j(T<\infty)=1$; otherwise, $j$ is called transient. A recurrent state is called positive recurrent if $\mathbb{E}_j[T]<\infty$; otherwise, it is called null recurrent. Finally, a state is said to be periodic, with period $\delta$, if $\delta \geqslant 2$ is the largest integer for which $\mathbb{P}_j(T=n \delta$ for some $n \geqslant 1)=1$; otherwise, it is called aperiodic. For example, in Figure $1.5$ states 1 and 2 are recurrent, and the other states are transient. All these states are aperiodic. The states of the random walk of Example $1.10$ are periodic with period 2.

It can be shown that recurrence and transience are class properties. In particular, if $i \leftrightarrow j$, then $i$ recurrent (transient) $\Leftrightarrow j$ recurrent (transient). Thus, in an irreducible Markov chain, one state being recurrent implies that all other states are also recurrent. And if one state is transient, then so are all the others.

## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|Limiting Behavior

The limiting or “steady-state” behavior of Markov chains as $t \rightarrow \infty$ is of considerable interest and importance, and this type of behavior is often simpler to describe and analyze than the “transient” behavior of the chain for fixed $t$. It can be shown (see, for example, [3]) that in an irreducible, aperiodic Markov chain with transition matrix $P$ the $t$-step probabilities converge to a constant that does not depend on the initial state. More specifically,
$$\lim _{t \rightarrow \infty} P^t(i, j)=\pi_j$$
for some number $0 \leqslant \pi_j \leqslant 1$. Moreover, $\pi_j>0$ if $j$ is positive recurrent and $\pi_j=0$ otherwise. The intuitive reason behind this result is that the process “forgets” where it was initially if it goes on long enough. This is true for both finite and countably infinite Markov chains. The numbers $\left{\pi_j, j \in \mathscr{E}\right}$ form the limiting distribution of the Markov chain, provided that $\pi_j \geqslant 0$ and $\sum_j \pi_j=1$. Note that these conditions are not always satisfied: they are clearly not satisfied if the Markov chain is transient, and they may not be satisfied if the Markov chain is recurrent (i.e., when the states are null-recurrent). The following theorem gives a method for obtaining limiting distributions. Here we assume for simplicity that $\mathscr{E}={0,1,2, \ldots}$. The limiting distribution is identified with the row vector $\pi=$ $\left(\pi_0, \pi_1, \ldots\right)$

Theorem 1.13.2 For an irreducible, aperiodic Markov chain with transition matrix $P$, if the limiting distribution $\pi$ exists, then it is uniquely determined by the solution of
$$\pi=\pi P,$$
with $\pi_j \geqslant 0$ and $\sum_j \pi_j=1$. Conversely, if there exists a positive row vector $\pi$ satisfying (1.35) and summing up to 1 , then $\pi$ is the limiting distribution of the Markov chain. Moreover, in that case, $\pi_j>0$ for all $j$ and all states are positive recurrent.

Proof: (Sketch) For the case where $\mathscr{E}$ is finite, the result is simply a consequence of (1.33). Namely, with $\pi^{(0)}$ being the $i$-th unit vector, we have
$$P^{t+1}(i, j)=\left(\pi^{(0)} P^t P\right)(j)=\sum_{k \in \mathcal{E}} P^t(i, k) P(k, j) .$$
Letting $t \rightarrow \infty$, we obtain (1.35) from (1.34), provided that we can change the order of the limit and the summation. To show uniqueness, suppose that another vector $\mathbf{y}$, with $y_j \geqslant 0$ and $\sum_j y_j=1$, satisfies $\mathbf{y}=\mathbf{y} P$. Then it is easy to show by induction that $\mathbf{y}=\mathbf{y} P^t$, for every $t$. Hence, letting $t \rightarrow \infty$, we obtain for every $j$
$$y_j=\sum_i y_i \pi_j=\pi_j,$$
since the $\left{y_j\right}$ sum up to unity. We omit the proof of the converse statement.

# 模拟和蒙特卡洛方法代写

## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte-carlo-method代考| limit – Behavior

$$\lim _{t \rightarrow \infty} P^t(i, j)=\pi_j$$
。此外，如果$j$是正循环则为$\pi_j>0$，否则为$\pi_j=0$。这个结果背后的直观原因是，如果过程持续的时间足够长，它就会“忘记”最初的位置。这对于有限和可数无限的马尔可夫链都是成立的。数字$\left{\pi_j, j \in \mathscr{E}\right}$构成马尔可夫链的极限分布，前提是$\pi_j \geqslant 0$和$\sum_j \pi_j=1$。注意，这些条件并不总是被满足:如果马尔可夫链是瞬态的，它们显然是不被满足的，如果马尔可夫链是周期性的(即，当状态为零周期性时)，它们可能是不被满足的。下面的定理给出了求极限分布的一种方法。为了简单起见，我们在这里假设$\mathscr{E}={0,1,2, \ldots}$。极限分布用行向量$\pi=$$\left(\pi_0, \pi_1, \ldots\right) 来标识 1.13.2对于具有跃迁矩阵P的不可约非周期马尔可夫链，如果极限分布\pi存在，则由$$ \pi=\pi P, $$具有\pi_j \geqslant 0和\sum_j \pi_j=1的解唯一确定。反之，如果存在满足(1.35)且和为1的正行向量\pi，则\pi为马尔可夫链的极限分布。此外，在这种情况下，\pi_j>0对所有j和所有状态都是正循环 证明:(Sketch)对于\mathscr{E}是有限的情况，结果只是(1.33)的一个结果。即，\pi^{(0)}是i -th单位向量，我们有$$ P^{t+1}(i, j)=\left(\pi^{(0)} P^t P\right)(j)=\sum_{k \in \mathcal{E}} P^t(i, k) P(k, j) . $$令t \rightarrow \infty，我们从(1.34)得到(1.35)，前提是我们可以改变极限和和的顺序。为了显示唯一性，假设另一个向量\mathbf{y}，包含y_j \geqslant 0和\sum_j y_j=1，满足\mathbf{y}=\mathbf{y} P。然后很容易用归纳法表明，\mathbf{y}=\mathbf{y} P^t对应每一个t。因此，让t \rightarrow \infty，我们得到每个j$$ y_j=\sum_i y_i \pi_j=\pi_j,$$，因为$\left{y_j\right}\$的和为单位。

## 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。

assignmentutor™您的专属作业导师
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