### 数学代写|随机过程统计代写Stochastic process statistics代考|MTH 3016

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## 数学代写|随机过程统计代写Stochastic process statistics代考|Limit Theorems for Markov Chain

Definition $2.10$ Let $d(\mathrm{i})$ be the greatest common divisor of those $n \geq 1$ for which $p_{i i}^{(n)}>0$. Then $d(i)$ is called the period of the state $i$. If $d(i)=1$, then the state $i$ is called aperiodic.
Note $i \leftrightarrow j$, then $d(i)=d(j)$
There exists $n_{1}$ and $n_{2}$ such that $p_{i j}^{\left(n_{1}\right)}>0$ and $p_{j i}^{\left(n_{2}\right)}>0$.
Now $p_{i i}^{\left(n_{1}+n_{2}\right)} \geq p_{i j}^{\left(n_{1}\right)} p_{j i}^{\left(n_{2}\right)}>0$ and hence $d(i)$ is a divisor of $n_{1}+n_{2}$.
If $p_{j j}^{(n)}>0$, then $p_{i i}^{\left(n_{1}+n+n_{2}\right)} \geq p_{i j}^{\left(n_{1}\right)} p_{j j}^{(n)} p_{j i}^{\left(n_{2}\right)}>0$ (by Chapman Kolmogorov equation).

Hence, $d(i)$ is a divisor of $n_{1}+n+n_{2}$. So $d(i)$ must be a divisor of $n$ if $p_{j i}^{(n)}>0$

Thus $d(i)$ is a divisor of $\left{n \geq 1: p_{j j}^{(n)}>0\right}$. Since $d(j)$ is the largest of such divisors, $d(i) \leq d(j)$. Hence, by symmetry $d(j) \leq d(i)$.
Hence $d(i)=d(j)$. Therefore having a period $d$ is a class property.
Note If $p_{i i}>0$, then $d(i)=1$ and this implies that a sufficient condition for an irreducible M.C. to be aperiodic is that $p_{i i}>0$ for some $i \in S$. Hence a queueing chain is aperiodic.
Theorem $2.7$ Limit Theorem (for diagonal elements)
Let $j$ be any state in a M.C. As $n \rightarrow \infty$.
(i) if $j$ is transient, then $p_{j j}^{(n)} \rightarrow 0$
(ii) if $j$ is null recurrent, then $p_{j j}^{(n)} \rightarrow 0$
(iii) if $j$ is positive (recurrent) and
(a) aperiodic, then $p_{j j}^{(n)} \rightarrow \frac{1}{\sum_{n=1}^{\infty} n f_{j j}^{(n)}}=\frac{1}{\mu_{j}}$ (mean recurrence time of $j$ )(b) periodic with period $d(j)$ then $p_{J \prime}^{(n d(j))} \rightarrow \frac{d(j)}{\mu_{j}}$. Write $d(j) / \mu_{j}=\pi_{j}$.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Special Chains and Foster Type Theorems

If the Markov Chain is infinite, the number of equations given by $\pi(P-I)=0$ will be infinite involving an infinite number of unknowns. In some particular cases we can solve these equations. The following examples will illustrate this point.
Example $2.5$ Birth and.Death Chain (Non-Homogeneous Random Walk) Consider a birth and death chain on ${0,1,2, \ldots, d}$ or a set of non-negative integers i.e. where $d=\infty$. Assume that the chain is irreducible i.e. $p_{j}>0$ and $q_{j}>0$ in case $0 \leq j \leq d$ (i.e. when $d$ is finite) $p_{j}>0$ for $0 \leq j<\infty$ and $q_{j}>0$ for $0<j<\infty$ if $d$ is infinite. Consider the transition matrix
$$\left(\begin{array}{cccccc} r_{0} & p_{0} & 0 & \cdots & \cdot & \ q_{1} & r_{1} & p_{1} & 0 & \cdots & \ 0 & q_{2} & r_{2} & p_{2} & 0 & . \ 0 & 0 & q_{3} & r_{3} & p_{3} & 0 \ \cdot & \cdot & \cdot & \cdot & \cdots & \end{array}\right)$$
when $d<\infty$ we assume that $r_{i}=0$ for $i \geq 0$ and $p_{0}=1$.
Particular Case: First consider that $d$ is still infinite and $r_{1}=0$ for $i \geq 0$, $p_{0}=1$. The stationary distribution is given by or $X=X P$. Let $x_{0} \neq 0$. Then
\begin{aligned} &x_{0}=x_{1} q_{1}, \ &x_{1}=x_{0}+x_{2} q_{2}, \ &x_{3}=x_{2} p_{2}+x_{4} q_{4}, \ &x_{4}=\ldots \ &\ldots \end{aligned}
Define
$$y_{i}=\frac{x_{i}}{x_{0}}, y_{0}=1, i=1,2,3, \ldots$$
Then
\begin{aligned} &y_{1}=1 / q_{1}, y_{1}=1+y_{2} q_{2} \text { or } y_{2}=\frac{y_{1}-1}{q_{2}}=\frac{1-q_{1}}{q_{1} q_{2}}=\frac{p_{1}}{q_{1} q_{2}} \ &y_{3}=\frac{p_{1} p_{2}}{q_{1} q_{2} q_{3}}, \ldots, y_{n}=\frac{p_{1} p_{2} \ldots p_{n-1}}{q_{1} q_{2} \ldots q_{n}}>0 \quad \text { for all } n=1,2, \ldots \end{aligned}
(by assumption that all $p, q$ ‘s are $>0$ ).

## 数学代写|随机过程统计代写Stochastic process statistics代考|Foster type theorems

The following theorems, associated with Foster, give criteria for transient and recurrent chains in terms of solution of certain equations. Assume that the M.C. is irreducible.

Theorem 2.11 (Foster, 1953) Let the Markov chain be irreducible. Assume that there exists $x_{k} . k \in S$ such that $x_{k}=\sum_{k \in S} x_{i} p_{i k}$ and $0<\sum_{k \in S}\left|x_{k}\right|<\infty$. Then the Markóv Chain is positive recurrent (this is a soort of converse of Theoremem $2.9$ ). Proof Since $y_{k}=\frac{1}{\sum_{k \in S}\left|x_{k}\right|}>0, \sum_{k \in S} y_{k}=1$.

Without loss of generality $\left{x_{k}, k \in S\right}$ is a stationary distribution of a M.C. Then

$$x_{k}=\sum_{k \in S} x_{i} p_{i k}^{(n)} \text { for all } n=1,2, \ldots$$
Suppose that there is no positive state.
Since the M.C. is irreducible, then all the states are either transient or null. In that case $p_{i k}^{(n)} \rightarrow 0$ as $n \rightarrow \infty$ for all $i, k \in S$. By Lebesgue Dominated Convergence Theorem, taking $n \rightarrow \infty$ in (2.19)
$$x_{k}=\sum_{i \in S}\left(x_{i}\right), 0=0 \text { for all } k \in S$$
But $0<\sum_{k \in S} x_{k}<\infty$ is a contradiction to $(2.20)$.
Hence, there is at least one positive recurrent state. Since M.C. is irreducible, by Solidarity Theorem the M.C. must be positive recurrent.

Conclusion An ireducible aperiodic M.C. has a stationary distribution iff all states are positive recurrent.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Limit Theorems for Markov Chain

(i) 如果j是瞬态的，那么pjj(n)→0
(ii) 如果j是零循环的，那么pjj(n)→0
(iii) 如果j是正的（经常性的）和
(a) 非周期性的，那么pjj(n)→1∑n=1∞nFjj(n)=1μj（平均复发时间j)(b) 有周期的周期性d(j)然后pĴ′(nd(j))→d(j)μj. 写d(j)/μj=圆周率j.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Special Chains and Foster Type Theorems

(r0p00⋯⋅ q1r1p10⋯ 0q2r2p20. 00q3r3p30 ⋅⋅⋅⋅⋯)

X0=X1q1, X1=X0+X2q2, X3=X2p2+X4q4, X4=… …

（假设所有p,q是>0 ).

## 数学代写|随机过程统计代写Stochastic process statistics代考|Foster type theorems

Xķ=∑ķ∈小号X一世p一世ķ(n) 对所有人 n=1,2,…

Xķ=∑一世∈小号(X一世),0=0 对所有人 ķ∈小号

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