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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

## 统计代写|应用随机过程代写Stochastic process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 Markov Chain is positive recurrent (this is a sort of converse of Theorem 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.

## 统计代写|应用随机过程代写Stochastic process代考|Theorems Regarding Finite Markov Chain

Theorem 2(a). In a M.C. with a finite number of states, there is no null state and not all states can be transient.

Proof Suppose the chain has $N<\infty$ states. If all states are transient, then letting $n \rightarrow \infty$ in the relation $\sum_{j=0}^{N} p_{i j}^{(n)}=1$ we get $0=1$ (since by Theorem $2.8$, $\lim {n \rightarrow \infty} p{i j}^{(n)}=0$ for each $j$, which is absured and hence not all states in a finite M.C. are transient. Consider the subchain $C_{1}$ formed by a closed set of null recurrent states. Then $\sum_{j \in C_{1}} p_{i j}^{(n)}=\alpha$ (say) $>0$. Letting $n \rightarrow \infty, 0=\alpha>0$ which is also absurd. So there cannot be any null recurrent state in a finite M.C.
Theorem 2 (b). An irreducible M.C. having a finite number of states is positive recurrent.

Proof By previous theorem, there is no null recurrent state and not all states are transient. Suppose there is one transient state. Then all states are transient by Solidarity Theorem. Hence, all states are positive recurrent.

Exercise 2.6 If a finite M.C. is irreducible, aperiodic and has doubly stochastic transition matrix, then show that $\lim {n \rightarrow \infty} p{i j}^{(n)}=1 / k$, where $k$ is the number of states in the chain.

## 统计代写|应用随机过程代写Stochastic process代考|Foster type theorems

$$x_{k}=\sum_{k \in S} x_{i} p_{i k}^{(n)} \text { for all } n=1,2, \ldots$$

$$x_{k}=\sum_{i \in S}\left(x_{i}\right) \cdot 0=0 \text { for all } k \in S$$

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assignmentutor™您的专属作业导师
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