### 统计代写|离散时间鞅理论代写martingale代考|STAT4061

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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|离散时间鞅理论代写martingale代考|A Warming-Up Example

The purpose of this chapter is to present, in the simplest possible context, some of the ideas that will appear recurrently in this book. We assume that the reader is familiar with the basic theory of Markov chains (e.g. Chap. 7 of Breiman 1968 or Chap. 5 of Durrett 1996) and with the spectral theory of bounded symmetric operators (Sect. 107 in Riesz and Sz.-Nagy 1990, Sect. XI.6 in Yosida 1995).

Consider a Markov chain $\left{X_{j}: j \geq 0\right}$ on a countable state space $E$, stationary and ergodic with respect to a probability measure $\pi$. The problem is to find necessary and sufficient conditions on a function $V: E \rightarrow \mathbb{R}$ to guarantee a central limit theorem for
$$\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{j}\right)$$
We assume that $E_{\pi}[V]=0$, where $E_{\pi}$ stands for the expectation with respect to the probability measure $\pi$. The idea is to relate this question to the well-known martingale central limit theorems.

Denote by $P$ the transition probability of the Markov chain and fix a function $V$ in $L^{2}(\pi)$, the space of functions $f: E \rightarrow \mathbb{R}$ square integrable with respect to $\pi$. Assume the existence of a solution of the Poisson equation
$$V=(I-P) f$$
for some function $f$ in $L^{2}(\pi)$, where $I$ stands for the identity. For $j \geq 1$, let
$$Z_{. j}=f\left(X_{j}\right)-(P f)\left(X_{j-1}\right) .$$
It is easy to check that $M_{0}=0, M_{N}=\sum_{1 \leq j \leq N} Z_{j}, N \geq 1$, is a martingale with respect to the filtration $\left{F_{j}: j \geq 0\right}, F_{j}=\sigma\left(X_{0}, \ldots, X_{j}\right)$, and that
$$\sum_{j=0}^{N-1} V\left(X_{j}\right)=M_{N}-f\left(X_{N}\right)+f\left(X_{0}\right)$$

## 统计代写|离散时间鞅理论代写martingale代考|Ergodic Markov Chains

In this section, we present some elementary results on Markov chains. Fix a countable state space $E$ and a transition probability function $P: E \times E \rightarrow \mathbb{R}$ :
$$P(x, y) \geq 0, \quad x, y \in E, \quad \sum_{y \in E} P(x, y)=1, \quad x \in E$$
A sequence of random variables $\left{X_{j}: j \geq 0\right}$ defined on some probability space $(\Omega, \mathscr{F}, \mathbb{P})$ and taking values in $E$ is a time-homogeneous Markov chain on $E$ if
$$\mathbb{P}\left[X_{j+1}=y \mid X_{j}, \ldots, X_{0}\right]=P\left(X_{j}, y\right)$$ for all $j \geq 0, y$ in E. $P(x, y)$ is called the probability of jump from $x$ to $y$ in one step. Notice that it does not depend on time, which explains the terminology of a time-homogeneous chain. The law of $X_{0}$ is called the initial state of the chain.
Assume furthermore that on $(\Omega, \mathscr{F})$ we are given a family of measures $\mathbb{P}{z}$, $z \in E$, each satisfying (1.5) and such that $\mathbb{P}{x}\left[X_{0}=x\right]=1$. We call it a Markov family that corresponds to the transition probabilities $P(\cdot, \cdot)$. For a given probability measure $\mu$ on $E$, let $\mathbb{P}{\mu}=\sum{x \in E} \mu(x) \mathbb{P}{x}$. Observe that $\mu$ is the initial state of the chain under $\mathbb{P}{\mu}$. We shall denote by $\mathbb{E}{\mu}$ the expectation with respect to that measure and by $\mathbb{E}{x}$ the expectation with respect to $\mathbb{P}_{x}$.

The transition probability $P$ can be considered as an operator on $C_{b}(E)$, the space of (continuous) bounded functions on $E$. In this case, for $f$ in $C_{b}(E)$, $P f: E \rightarrow E$ is defined by
$$(P f)(x)=\sum_{y \in E} P(x, y) f(y)=\mathbb{E}\left[f\left(X_{1}\right) \mid X_{0}=x\right] .$$

## 统计代写|离散时间鞅理论代写martingale代考|Almost Sure Central Limit Theorem for Ergodic Markov Chains

Consider a time-homogeneous irreducible (or indecomposable in the terminology of Breiman 1968) Markov chain $\left{X_{j}: j \geq 0\right}$ on a countable state space $E$ with transition probability function $P: E \times E \rightarrow \mathbb{R}{+}$. Assume that there exists a stationary probability measure, denoted by $\pi$. By (Breiman, 1968 , Theorem $7.16$ ), $\pi$ is unique and ergodic. In particular, for any bounded function $g: E \rightarrow \mathbb{R}$ and any $x$ in $E$, $$\lim {N \rightarrow \infty} \frac{1}{N} \sum_{j=0}^{N-1}\left(P^{j} g\right)(x)=E_{\pi}\lfloor g\rfloor .$$
Fix a function $V: E \rightarrow \mathbb{R}$ in $L^{2}(\pi)$ which has mean zero with respect to $\pi$. In this section, we prove a central limit theorem for the sequence $N^{-1 / 2} \sum_{j=0}^{N-1} V\left(X_{j}\right)$ assuming that the solution of the Poisson equation (1.2) belongs to $L^{2}(\pi)$. Under this hypothesis we obtain a central limit theorem which holds $\pi$-a.s. with respect to the initial state.

Theorem 1.1 Fix a function $V: E \rightarrow \mathbb{R}$ in $L^{2}(\pi)$ which has mean zero with respect to $\pi$. Assume that there exists a solution $f$ in $L^{2}(\pi)$ of the Poisson equation (1.2).

Then, for all $x$ in $E$, as $N \uparrow \infty$,
$$\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{j}\right)$$
converges in $\mathbb{I}{X}$ distribution to a mean zero Gaussian random variable with variance $\sigma^{2}(V)=E{\pi}\left[f^{2}\right]-E_{\pi}\left[(P f)^{2}\right]$

Proof Fix a mean zero function $V$ in $L^{2}(\pi)$ and an initial state $x$ in $E$. By assumption, there exists a solution $f$ in $L^{2}(\pi)$ of the Poisson equation (1.2). Consider the sequence $\left{Z_{j}: j \geq 1\right}$ of random variables defined by
$$Z_{j}=f\left(X_{j}\right)-P f\left(X_{j-1}\right)$$

## 统计代写|离散时间鞅理论代写martingale代考|A Warming-Up Example

$$\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{j}\right)$$

$$V=(I-P) f$$

$$Z_{. j}=f\left(X_{j}\right)-(P f)\left(X_{j-1}\right) .$$

$\mathrm{~ L e f t { F _ { j } : ~ j ~ l g e q ~ O \ r i g h t } , ~ F _ { j } = I s i g m a l l e f t ( X _ { 0 } , ~ I d o t s , ~ X _ { j }}$
$$\sum_{j=0}^{N-1} V\left(X_{j}\right)=M_{N}-f\left(X_{N}\right)+f\left(X_{0}\right)$$

## 统计代写|离散时间鞅理论代写martingale代考|Ergodic Markov Chains

$$P(x, y) \geq 0, \quad x, y \in E, \quad \sum_{y \in E} P(x, y)=1, \quad x \in E$$

$$\mathbb{P}\left[X_{j+1}=y \mid X_{j}, \ldots, X_{0}\right]=P\left(X_{j}, y\right)$$

$$(P f)(x)=\sum_{y \in E} P(x, y) f(y)=\mathbb{E}\left[f\left(X_{1}\right) \mid X_{0}=x\right]$$

## 统计代写|离散时间鞅理论代写martingale代考|Almost Sure Central Limit Theorem for Ergodic Markov Chains

$$\lim N \rightarrow \infty \frac{1}{N} \sum_{j=0}^{N-1}\left(P^{j} g\right)(x)=E_{\pi}\lfloor g\rfloor .$$

$$\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{j}\right)$$

$$Z_{j}=f\left(X_{j}\right)-\operatorname{Pf}\left(X_{j-1}\right)$$

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