### 统计代写|蒙特卡洛方法代写monte carlo method代考| Classification of States

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• Statistical Inference 统计推断
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• (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|蒙特卡洛方法代写monte carlo method代考|Classification of States

Let $X$ be a Markov chain with discrete state space $E$ 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.

## 统计代写|蒙特卡洛方法代写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.

## 统计代写|蒙特卡洛方法代写monte carlo method代考|Random Walk on the Positive Integers

This is a slightly different random walk than the one in Example 1.10. Let $X$ be a random walk on $\mathscr{E}={0,1,2, \ldots}$ with transition matrix
$$P=\left(\begin{array}{cccccc} q & p & 0 & \ldots & & \ q & 0 & p & 0 & \ldots & \ 0 & q & 0 & p & 0 & \cdots \ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots \end{array}\right)$$
where $0<p<1$ and $q=1-p . X_{l}$ could represent, for example, the number of customers who are waiting in a queue at time $t$.

All states can be reached from each other, so the chain is irreducible and every state is either recurrent or transient. The equation $\pi=\pi P$ becomes
\begin{aligned} &\pi_{0}=q \pi_{0}+q \pi_{1} \ &\pi_{1}=p \pi_{0}+q \pi_{2} \ &\pi_{2}=p \pi_{1}+q \pi_{3} \ &\pi_{3}=p \pi_{2}+q \pi_{4} \end{aligned}
and so on. We can solve this set of equation sequentially. If we let $r=p / q$, then we can express the $\pi_{1}, \pi_{2}, \ldots$ in terms of $\pi_{0}$ and $r$ as
$$\pi_{j}=r^{j} \pi_{0}, j=0,1,2, \ldots$$

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