### 数学代写|随机过程作业代写Stochastic Processes代考|The Central Limit Theorem

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|随机过程作业代写Stochastic Processes代考|The Central Limit Theorem

By far the most important example of weak convergence is the Central Limit Theorem. For its proof we need the following lemma, in which the lack on dependence of the limit on $X$ is of greatest interest; the fact that the second derivative in the limit points out to the normal distribution will become clear in Chapters 7 and 8 (see 8.4.18 in particular).
5.5.1 Lemma Let $X$ be square integrable with $E X=0$ and $E X^{2}=$ 1. Also, let $a_{n}, n \geq 1$ be a sequence of positive numbers such that $\lim {n \rightarrow \infty} a{n}=0$. Then, for any $x \in \mathcal{D}$, the set of twice differentiable functions $x \in C[-\infty, \infty]$ with $x^{\prime \prime} \in C[-\infty, \infty]$, the limit of $\frac{1}{a_{n}^{2}}\left(T_{a_{n}} X x-x\right)$ exists and does not depend on $X$. In fact it equals $\frac{1}{2} x^{\prime \prime}$.

Proof By the Taylor formula, for a twice differentiable $x$, and numbers $\tau$ and $\varsigma$,
$$x(\tau+\varsigma)=x(\tau)+\varsigma x^{\prime}(\tau)+\frac{\varsigma^{2}}{2} x^{\prime \prime}(\tau+\theta \varsigma)$$

where $0 \leq \theta \leq 1$ depends on $\tau$ and $\varsigma$ (and $x$ ). Thus,
\begin{aligned} \frac{1}{a_{n}^{2}}\left[T_{a_{n} X} x(\tau)-x(\tau)\right] &=\frac{1}{a_{n}^{2}} E\left[x\left(\tau+a_{n} X\right)-x(\tau)\right] \ &=\frac{1}{a_{n}} x^{\prime}(\tau) E X+\frac{1}{2} E\left[X^{2} x^{\prime \prime}\left(\tau+\theta a_{n} X\right)\right] \ &=\frac{1}{2} E\left[X^{2} x^{\prime \prime}\left(\tau+\theta a_{n} X\right)\right] \end{aligned}
for $E X=0 . \dagger$ Since $E X^{2}=1$,
$$\left|\frac{1}{a_{n}^{2}}\left(T_{a_{n} X} x-x\right)(\tau)-\frac{1}{2} x^{\prime \prime}(\tau)\right|=\left|\frac{1}{2} E X^{2}\left(x^{\prime \prime}\left(\tau+\theta a_{n} X\right)-x^{\prime \prime}(\tau)\right)\right| .$$
For $x \in \mathcal{D}$, and $\epsilon>0$, one may choose a $\delta$ such that $\left|x^{\prime \prime}(\tau+\varsigma)-x^{\prime \prime}(\tau)\right|<$ $\epsilon$, provided $|\varsigma|<\delta$. Calculating the last expectation on the set where $|X| \geq \frac{\delta}{a_{n}}$ and its complement separately we get the estimate $$\left|\frac{1}{a_{n}^{2}}\left(T_{a_{n}} X x-x\right)-\frac{1}{2} x^{\prime \prime}\right| \leq \frac{1}{2}\left|x^{\prime \prime}\right| E X^{2} 1_{\left{|X| \geq \frac{\delta}{a_{n}}\right}}+\frac{1}{2} \epsilon .$$ Since $\mathbb{P}\left{|X| \geq \frac{\delta}{a_{n}}\right} \rightarrow 0$ as $n \rightarrow \infty$ we are done by the Lebesgue Dominated Convergence Theorem. 5.5.2 The Central Limit Theorem The Central Limit Theorem in its classical form says that if $X_{n}, n \geq 1$ is a sequence of i.i.d. (independent, identically distributed) random variables with expected value $m$ and variance $\sigma^{2}>0$, then
$$\frac{1}{\sqrt{n \sigma^{2}}} \sum_{k=1}^{n}\left(X_{k}-m\right)$$
converges weakly to the standard normal distribution.

## 数学代写|随机过程作业代写Stochastic Processes代考|Weak convergence in metric spaces

The assumption that the space $S$ where our probability measures are defined is compact (or locally compact) is quite restrictive and is not fulfilled in many important cases of interest. On the other hand, assuming just that $S$ is a topological space leads to an unnecessarily general class. The golden mean for probability seems to lie in separable metric spaces, or perhaps, Polish spaces. A Polish space is by definition a separable, complete metric space. We start with general metric spaces to specialize to Polish spaces later when needed. As an application of the theory developed here, in the next section we will give another proof of the existence of Brownian motion.

5.6.1 Definition Let $(S, d)$ be a metric space, and let $B C(S)$ be the space of continuous (with respect to the metric $d$, of course) functions on $S$. A sequence $\mathbb{P}{n}$ of Borel probability measures on $S$ is said to converge weakly to a Borel probability measure $\mathbb{P}$ on $S$ iff, for all $x \in B C(S)$, $$\lim {n \rightarrow \infty} \int_{S} x \mathrm{dP}{n}=\int{S} x \mathrm{dP} .$$
It is clear that this definition agrees with the one introduced in the previous section, as in the case where $S$ is both metric and compact, $B C(S)$ coincides with $C(S)$.
We will sometimes write $E_{n} x$ for $\int_{S} x \mathbb{d P}{n}$ and $E x$ for $\int{S} x \mathrm{dP}$.
5.6.2 Corollary Suppose $\mathbb{P}{n}, n \geq 1$ is a sequence of Borel probability measures on $(S, d)$ and $f: S \rightarrow S^{\prime}$, where $\left(S^{\prime}, d^{\prime}\right)$ is another metric space, is a continuous map. Then the transport measures $\left(\mathbb{P}{n}\right){f}, n \geq 1$ on $S^{\prime}$ converge weakly. The proof is immediate by the change of variables formula $(1.6)$. 5.6.3 Portmanteau Theorem Let $\mathbb{P}$ and $\mathbb{P}{n}, n \geq 1$ be probability measures on a metric space $(S, d)$. The following are equivalent:
(a) $\mathbb{P}{n}$ converge weakly to $\mathbb{P}$, (b) condition (5.25) holds for Lipschitz continuous $x$ with values in $[0,1]$, (c) limsup $\operatorname{sun}{n \rightarrow \infty} \mathbb{P}{n}(F) \leq \mathbb{P}(F)$, for closed $F \subset S$, (d) $\liminf {n \rightarrow \infty} \mathbb{P}{n}(G) \geq \mathbb{P}(G)$, for open $G \subset S$, (e) $\lim {n \rightarrow \infty} \mathbb{P}_{n}(B)=\mathbb{P}(B)$, for Borel $B$ with $\mu(\partial B)=0$.

## 数学代写|随机过程作业代写Stochastic Processes代考|Compactness everywhere

I believe saying that the notion of compactness is one of the most important ones in topology and the whole of mathematics is not an exaggeration. Therefore, it is not surprising that it comes into play in a crucial way in a number of theorems of probability theory as well (see e.g. $5.4 .18$ or $6.6 .12$ ). To be sure, Helly’s principle, so familiar to all students of probability, is simply saying that any sequence of probability measures on $\mathbb{R}$ is relatively compact; in functional analysis this theorem finds its important generalization in Alaoglu’s Theorem. We will discuss compactness of probability measures on separable metric spaces, as well (Prohorov’s Theorem), and apply the results to give another proof of existence of Brownian motion (Donsker’s Theorem). On our way to Brownian motion we will prove the Arzela-Ascoli Theorem, too.

We start by looking once again at the results of Section 3.7, to continue with Alexandrov’s Lemma and Tichonov’s Theorem that will lead directly to Alaoglu’s Theorem mentioned above.
5.7.1 Compactness and convergence of martingales As we have seen in 3.7.7, a martingale converges in $L^{1}$ iff it is uniformly integrable. Moreover, in $3.7 .15$ we proved that a martingale converges in $L^{p}, p>1$ iff it is bounded. Consulting [32] p. 294 we see that uniform integrability is necessary and sufficient for a sequence to be relatively compact in the weak topology of $L^{1}$. Similarly, in [32] p. 289 it is shown that a sequence in $L^{p}, p>1$ is weakly relatively compact iff it is bounded. Hence, the results of $3.7 .7$ and $3.7 .15$ may be summarized by saying that a martingale in $L^{p}, p \geq 1$, converges iff it is weakly relatively compact. However, my attempts to give a universal proof that would work in both cases covered in $3.7 .7$ and $3.7 .15$ have failed. I was not able to find such a proof in the literature, either.

## 数学代写|随机过程作业代写Stochastic Processes代考|The Central Limit Theorem

5.5.1 引理让X平方可积和X=0和和X2=1.另外，让一种n,n≥1是一个正数序列，使得 $\lim {n \rightarrow \infty} a {n}=0.吨H和n,F这r一种n是x \in \mathcal{D},吨H和s和吨这F吨在一世C和d一世FF和r和n吨一世一种bl和F在nC吨一世这nsx \in C[-\infty, \infty]在一世吨Hx^{\prime \prime} \in C[-\infty, \infty],吨H和l一世米一世吨这F\frac{1}{a_{n}^{2}}\left(T_{a_{n}} X xx\right)和X一世s吨s一种ndd这和sn这吨d和p和nd这nX.一世nF一种C吨一世吨和q在一种ls\frac{1}{2} x^{\prime \prime}$。

X(τ+ε)=X(τ)+εX′(τ)+ε22X′′(τ+θε)

1一种n2[吨一种nXX(τ)−X(τ)]=1一种n2和[X(τ+一种nX)−X(τ)] =1一种nX′(τ)和X+12和[X2X′′(τ+θ一种nX)] =12和[X2X′′(τ+θ一种nX)]

|1一种n2(吨一种nXX−X)(τ)−12X′′(τ)|=|12和X2(X′′(τ+θ一种nX)−X′′(τ))|.

1nσ2∑ķ=1n(Xķ−米)

## 数学代写|随机过程作业代写Stochastic Processes代考|Weak convergence in metric spaces

5.6.1 定义让(小号,d)为度量空间，令乙C(小号)是连续的空间（关于度量d, 当然) 函数小号. 一个序列磷n的 Borel 概率测度小号据说弱收敛到 Borel 概率测度磷在小号如果，对所有人X∈乙C(小号),林n→∞∫小号Xd磷n=∫小号Xd磷.

5.6.2 推论假设磷n,n≥1是一系列 Borel 概率测度(小号,d)和F:小号→小号′， 在哪里(小号′,d′)是另一个度量空间，是一个连续映射。然后运输措施(磷n)F,n≥1在小号′弱收敛。通过变量公式的变化立即证明(1.6). 5.6.3 Portmanteau 定理让磷和磷n,n≥1是度量空间上的概率测度(小号,d). 以下是等效的：
(a)磷n弱收敛到磷, (b) 条件 (5.25) 适用于 Lipschitz 连续X与值[0,1], (c) 限制太阳⁡n→∞磷n(F)≤磷(F), 对于封闭F⊂小号, (d)林infn→∞磷n(G)≥磷(G), 对于开G⊂小号， （和）林n→∞磷n(乙)=磷(乙), 对于博雷尔乙和μ(∂乙)=0.

## 数学代写|随机过程作业代写Stochastic Processes代考|Compactness everywhere

5.7.1 鞅的紧致性和收敛性 正如我们在 3.7.7 中看到的，鞅收敛于大号1当且仅当它是一致可积的​​。此外，在3.7.15我们证明了鞅收敛于大号p,p>1当且仅当它是有界的。咨询 [32] 页。294 我们看到一致可积性对于序列在弱拓扑中相对紧凑是必要和充分的大号1. 同样，在 [32] p。第289章大号p,p>1当它是有界的时，它是弱相对紧致的。因此，结果3.7.7和3.7.15可以概括为一个鞅大号p,p≥1, 收敛当它是弱相对紧凑。然而，我试图给出一个普遍的证明，在这两种情况下都适用3.7.7和3.7.15失败了。我也无法在文献中找到这样的证据。

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