### 统计代写|离散时间鞅理论代写martingale代考|STAT4528

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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|Central Limit Theorem for Martingales

Fix a probability space $(\Omega, \mathscr{F}, \mathbb{P})$ and an increasing filtration $\left{\mathscr{F}{j}: j \geq 0\right}$. Denote by $\mathbb{E}$ the expectation with respect to the probability measure $\mathbb{P}$. Let $\left{Z{j}: j \geq 1\right}$ be a stationary and ergodic sequence of random variables adapted to the filtration $\left{\mathscr{F}{j}\right}$ and such that $$\mathbb{E}\left[Z{1}^{2}\right]<\infty, \quad \mathbb{E}\left[Z_{j+1} \mid \mathscr{F}{j}\right]=0, \quad j \geq 0 .$$ The variables $\left{Z{j}: j \geq 1\right}$ are usually called martingale differences because the process $\left{M_{j}: j \geq 0\right}$ defined as $M_{0}:=0, M_{j}:=\sum_{1 \leq k \leq j} Z_{k}, j \geq 1$, is a zero-mean, square integrable martingale with respect to the filtration $\left{\mathscr{F}_{j}: j \geq 0\right}$.

Theorem 1.2 Let $\left{Z_{j}: j \geq 1\right}$ be a sequence of stationary, ergodic random variables satisfying (1.10). Then, $N^{-1 / 2} \sum_{1 \leq j \leq N} Z_{j}$ converges in distribution, as $N \uparrow \infty$, to a Gaussian law with zero mean and variance $\sigma^{2}=\mathbb{E}\left[Z_{1}^{2}\right]$.

Proof If one assumes that the martingale differences $\left{Z_{j}\right}$ are bounded, the proof is elementary and follows from the ergodic assumption. Suppose therefore that $\left|Z_{1}\right| \leq$ $C_{0}, \mathbb{P}$-a.s. for some finite constant $C_{0}$.

We first build exponential martingales. Since $\left{Z_{j}\right}$ are martingale differences, $\mathbb{E}\left[\sum_{j+1 \leq k \leq j+K} Z_{k} \mid \mathscr{F}{j}\right]=0$ for all $j \geq 0, K \geq 1$. Therefore, since $\left|e^{i x}-1-i x\right| \leq$ $x^{2} / 2, x \in \mathbb{R}$, subtracting $\mathbb{E}\left[i \theta \sum{j+1 \leq k \leq j+K} Z_{k} \mid \mathscr{F}{j}\right]$ from the expression on the lefthand side in the next formula we obtain that $$\left|\mathbb{E}\left[\exp \left{i \theta \sum{k=j+1}^{j+K} Z_{k}\right} \mid \mathscr{F}{j}\right]-1\right| \leq \frac{\theta^{2}}{2} \mathbb{E}\left[\left(\sum{k=j+1}^{j+K} Z_{k}\right)^{2} \mid \mathscr{F}_{j}\right]$$

## 统计代写|离散时间鞅理论代写martingale代考|Time-Variance in Reversible Markov Chains

In this section, we examine the asymptotic behavior of the variance of
$$\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{j}\right)$$
for square integrable functions $V$ in the context of reversible Markov chains. Reversibility with respect to $\pi$ means that $P$ is a symmetric operator in $L^{2}(\pi)$ :
$$\langle P f, g\rangle_{\pi}=\langle f, P g\rangle_{\pi}$$
for all $f, g$ in $L^{2}(\pi)$. It is easy to check that a probability measure $\pi$ is reversible if and only if it satisfies the detailed balance condition:
$$\pi(x) P(x, y)=\pi(y) P(y, x)$$
for all $x, y$ in $E$, which means that
$$\mathbb{P}{\pi}\left[X{n}=x, X_{n+1}=y\right]=\mathbb{P}{\pi}\left[X{n}=y, X_{n+1}=x\right]$$
A reversible measure is necessarily invariant since
$$(\pi P)(x)=\sum_{y \in E} \pi(y) P(y, x)=\sum_{y \in E} \pi(x) P(x, y)=\pi(x) .$$
In this section, we prove that the following limit exists:
$$\sigma^{2}(V)=\lim {N \rightarrow \infty} \mathbb{E}{\pi}\left[\left(\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{J}\right)\right)^{2}\right]$$
where we admit $+\infty$ as a possible value, and we find necessary and sufficient conditions for $\sigma^{2}(V)$ to be finite. We also introduce Hilbert spaces associated to the transition operator $P$ which will play a central role in the following chapters.

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

In this section, we prove a central limit theorem for additive functionals of reversible Markov chains. Fix a zero-mean function $V$ in $L^{2}(\pi)$. We have seen in the beginning of this chapter that a central limit theorem for the additive functional $N^{-1 / 2} \sum_{0 \leq j<N} V\left(X_{j}\right)$ follows easily from a central limit theorem for martingales if $V$ belongs to the range of $I-P$, i.e., if there is a solution in $L^{2}(\pi)$ of the Poisson equation $(I-P) f=V$. This assumption is too strong and should be relaxed. A natural condition to impose on $V$ is to require that its time-variance $\sigma^{2}(V)$ is finite. In this case we may try to repeat the approach presented in the beginning of the chapter replacing the solution of the Poisson equation $(I-P) f=V$, which may not exist, by the solution $f_{\lambda}$ of the resolvent equation $\lambda f_{\lambda}+(I-P) f_{\lambda}=V$ which always exists.

Fix therefore a zero-mean function $V$ and assume that its variance $\sigma^{2}(V)$ is finite. Let $f_{\lambda}$ be the solution of the resolvent equation (1.16). For $N \geq 1$,
\begin{aligned} \sum_{j=0}^{N-1} V\left(X_{j}\right) &=\lambda \sum_{j=0}^{N-1} f_{\lambda}\left(X_{j}\right)+\sum_{j=0}^{N-1}\left{f_{\lambda}\left(X_{j}\right)-\left(P f_{\lambda}\right)\left(X_{j}\right)\right} \ &=M_{N}^{\lambda}+f_{\lambda}\left(X_{0}\right)-f_{\lambda}\left(X_{N}\right)+\lambda \sum_{j=0}^{N-1} f_{\lambda}\left(X_{j}\right) \end{aligned}
where $\left{M_{N}^{\lambda}: N \geq 0\right}$ is the martingale with respect to the filtration $\left{\mathscr{F}{j}: j \geq 0\right}$, $\mathscr{F}{j}=\sigma\left(X_{0}, \ldots, X_{j}\right)$, defined by $M_{0}^{\lambda}:=0$,
$$M_{N}^{\lambda}:=\sum_{j=1}^{N} Z_{j}^{\lambda}$$
for $Z_{j}^{\lambda}=f_{\lambda}\left(X_{j}\right)-\left(P f_{\lambda}\right)\left(X_{j-1}\right)$ for $j \geq 1$

## 统计代写|离散时间鞅理论代写martingale代考|Central Limit Theorem for Martingales

$\mathrm{~ 让 ~ U l e f t { Z { j } : ~ j g e q ~ 1 | r i g h t } ~ 是 适 应 过 滤 的 随 机 变 量 的 平 稳 和 遍 历 序 列 【 V e f t {}$
$$\mathbb{E}\left[Z 1^{2}\right]<\infty, \quad \mathbb{E}\left[Z_{j+1} \mid \mathscr{F} j\right]=0, \quad j \geq 0 .$$

$M_{0}:=0, M_{j}:=\sum_{1 \leq k \leq j} Z_{k}, j \geq 1 \mathrm{~ , ~ 是 关 于 过 滤 的 零 均 值 平 方 可 积 䩗 祥 ⿰}$

## 统计代写|离散时间鞅理论代写martingale代考|Time-Variance in Reversible Markov Chains

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

$$\langle P f, g\rangle_{\pi}=\langle f, P g\rangle_{\pi}$$

$$\pi(x) P(x, y)=\pi(y) P(y, x)$$

$$\mathbb{P} \pi\left[X n=x, X_{n+1}=y\right]=\mathbb{P} \pi\left[X n=y, X_{n+1}=x\right]$$

$$(\pi P)(x)=\sum_{y \in E} \pi(y) P(y, x)=\sum_{y \in E} \pi(x) P(x, y)=\pi(x)$$

$$\sigma^{2}(V)=\lim N \rightarrow \infty \mathbb{E} \pi\left[\left(\frac{1}{\sqrt{N}} \sum_{j=0}^{N-1} V\left(X_{J}\right)\right)^{2}\right]$$

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

\begin{对斉 } } \mathrm { ~ \ s u m _ { j = 0 } ^ { N – 1 } ~ V
$\mathrm{~ 在 哪 里 ~ \ l e f t { M _ { N } ^ { N l a m b d a } : ~ N ~ I g e q ~ O \ r i g h t ~}$
$\mathscr{F} j=\sigma\left(X_{0}, \ldots, X_{j}\right)$ ， 被定义为 $M_{0}^{\lambda}:=0$ ，
$$M_{N}^{\lambda}:=\sum_{j=1}^{N} Z_{j}^{\lambda}$$

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