### 数学代写|随机过程作业代写Stochastic Processes代考|Dual spaces and convergence of probability

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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
• 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

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

Limit theorems of probability theory constitute an integral, and beautiful, part of this theory and of mathematics as a whole. They involve, of course, the notion of convergence of random variables and the reader has already noticed there are many modes of convergence, including almost sure convergence, convergence in $L^{1}$, and convergence in probability. By far the most important mode of convergence is so-called weak convergence. Strictly speaking, this is not a mode of convergence of random variables themselves but of their distributions, i.e. measures on $\mathbb{R}$. The famous Riesz Theorem, to be discussed in 5.2.9, says that the space $\mathbb{B M}(S)$ of Borel measures on a locally compact topological space $S$ is isometrically isomorphic to the dual of $C_{0}(S)$. This gives natural ways of defining new topologies in $\mathbb{B M}(S)$ (see Section $5.3$ ). It is almost magical, though in fact not accidental at all, that one of these topologies is exactly “what the doctor prescribes” and what is needed in probability. This particular topology is, furthermore, very interesting in itself. As one of the treats, the reader will probably enjoy looking at Helly’s principle, so important in probability, from the broader perspective of Alaoglu’s Theorem.

We start this chapter by learning more on linear functionals. An important step in this direction is the famous Hahn-Banach Theorem on extending linear functionals; as an application we will introduce the notion of a Banach limit. Then, in Section $5.2$ we will study examples of dual spaces, and in Section $5.3$ some topologies in the dual of a Banach space. Finally, we will study compact sets in the weak topology and approach the problem of existence of Brownian motion from this perspective.

## 数学代写|随机过程作业代写Stochastic Processes代考|The Hahn–Banach Theorem

5.1.1 Definition If $X$ is a linear normed space, then the space of linear maps from $X$ to $\mathbb{R}$ is called the space of linear functionals. Its algebraic subspace composed of bounded linear functionals is termed the dual space and denoted $X^{}$. The elements of $X^{}$ will be denoted $F, G$, etc. The value of a functional $F$ on a vector $x$ will be denoted $F x$ or $\langle F, x\rangle$. In some contexts, the letter notation is especially useful showing the duality between $x$ and $F$ (see below).

Let us recall that boundedness of a linear functional $F$ means that there exists an $M>0$ such that
$$|F x| \leq M|x|, \quad x \in \mathbb{X}$$
Note that $F x$ is a number, so that we write $|F x|$ and not $|F x|$.
$5.1 .2$ Theorem Let $F$ be a linear functional in a normed space $X$, and let $L={x \in X: F x=0}$. The following are equivalent:
(a) $F$ is bounded.
(b) $F$ is continuous,
(c) $L$ is closed,
(d) either $L=X$ or there exists a $y \in X$ and a number $r>0$ such that $F x \neq 0$ whenever $|x-y|<r$.

Proof Implications $(a) \Rightarrow(b) \Rightarrow(c)$ are immediate (see $2.3 .3$ ). If $L \neq \mathrm{X}$, then there exists a $y \in X \backslash L$, and if (c) holds then $X \backslash L$ is open, so that (d) holds also.

To prove that $(\mathrm{d})$ implies $(\mathrm{a})$, let $B(y, r)={x:|x-y|0$ for some $x, x^{\prime} \in B(y, r)$, then the convex combination $x_{c}=\frac{F x^{\prime}}{F x^{\prime}-F x} x+\frac{-F x}{F x^{\prime}-F x} x^{\prime}$ satisfies $F x_{c}=\frac{F x^{\prime} F x-F x F x^{\prime}}{F x^{\prime}-F x}=0$, contrary to our assumption (note that $B(y, r)$ is convex). Hence, without loss of generality we may assume that $F x>0$ for all $x \in B(y, r)$. Let $z \neq 0$ be an arbitrary element of $X$, and set $x_{+}=y+\frac{r}{| z} z \in B(y, r)$ and $x_{-}=y-\frac{r}{|z|} z \in B(y, r)$. Since $F x_{+}>0,-F z<\frac{F_{y}}{r}|z|$. Analogously, $F x_{-}>0$ implies $F z<\frac{F_{y}}{r}|z|$. Thus, (a) follows with $M=\frac{F_{y}}{r}$.

## 数学代写|随机过程作业代写Stochastic Processes代考|Form of linear functionals in specific Banach spaces

In 3.1.28 we saw that all bounded linear functionals on a Hilbert space $\mathbb{H}$ are of the form $F x=(x, y)$ where $y$ is an element of $\mathbb{H}$. In this section we will provide forms of linear functionals in some other Banach spaces. It will be convenient to agree that from now on the phrase “a linear functional” means “a bounded linear functional”, unless stated otherwise.

$5.2 .1$ Theorem Let $X=c_{0}$ be the space of sequences $x=\left(\xi_{n}\right){n \geq 1}$ such that $\lim {n \rightarrow \infty} \xi_{n}=0$, equipped with the supremum norm. $F$ is a functional on $X$ if and only if there exists a unique sequence $\left(\alpha_{n}\right){n \geq 1} \in l^{1}$ such that $$F x=\sum{n=1}^{\infty} \xi_{n} \alpha_{n}$$
where the last series converges uniformly. Also, $|F|_{c_{0}^{}}=\left|\left(\alpha_{n}\right){n \geq 1}\right|{l^{11}}$ In words: $c_{0}^{}$ is isometrically isomorphic to $l^{1}$.

Proof Define $e_{i}=\left(\delta_{i, n}\right){n \geq 1}$. Since $\left|\sum{i=1}^{n} \xi_{i} e_{i}-x\right|=\sup {i \geq n+1}\left|\xi{i}\right|$ which tends to zero as $n \rightarrow \infty$, we may write $x=\lim {n \rightarrow \infty} \sum{i=1}^{n} \xi_{i} e_{i}=$ $\sum_{i=1}^{\infty} \xi_{i} e_{i}$. In particular, $\left(e_{n}\right){n \geq 1}$ is linearly dense in $c{0}$. This is crucial for the proof.
If $\left(\alpha_{n}\right){n \geq 1}$ belongs to $l^{1}$, then $$\sum{n=1}^{\infty}\left|\xi_{n} \alpha_{n}\right| \leq|x|_{c_{0}} \sum_{n=1}^{\infty}\left|\alpha_{n}\right|=|x|_{c_{0}}\left|\left(\alpha_{n}\right){n \geq 1}\right|{l^{1}}$$
and the formula $(5.7)$ defines a bounded linear functional on $c_{0}$.
Conversely, suppose that $F$ is a linear functional on $c_{0}$. Define $\alpha_{n}=$ $F e_{n}$, and $x_{n}=\sum_{i=1}^{n}\left(\operatorname{sgn} \alpha_{i}\right) e_{i} \in c_{0}$. We have $\left|x_{n}\right|_{c_{0}} \leq 1$, and $F x_{n}=$ $\sum_{i=1}^{n}\left|\alpha_{i}\right|$. Since $|F x| \leq|F|$, if $|x| \leq 1,\left(\alpha_{n}\right){n \geq 1}$ belongs to $l^{1}$ and its norm in this space is does not exceed $|F|$. Using continuity and linearity of $F$, for any $x \in c{0}$,
\begin{aligned} F x &=F \lim {n \rightarrow \infty} \sum{i=1}^{n} \xi_{i} e_{i}=\lim {n \rightarrow \infty} F \sum{i=1}^{n} \xi_{i} e_{i} \ &=\lim {n \rightarrow \infty} \sum{i=1}^{n} \xi_{i} F e_{i}=\lim {n \rightarrow \infty} \sum{i=1}^{n} \xi_{i} \alpha_{i}=\sum_{n=1}^{\infty} \xi_{i} \alpha_{i} \end{aligned}
Estimate (5.8) proves that the last series converges absolutely and that $|F| \leq\left|\left(\alpha_{n}\right){n \geq 1}\right|{l^{2}}$. Combining this with (5.8) we obtain $|F|_{c_{0}^{*}}=$ $\left|\left(\alpha_{n}\right){n \geq 1}\right|{l^{1}}$

## 数学代写|随机过程作业代写Stochastic Processes代考|The Hahn–Banach Theorem

5.1.1 定义如果X是一个线性范数空间，那么线性映射的空间来自X到R称为线性泛函空间。它由有界线性泛函组成的代数子空间称为对偶空间，记为$X^{ }.吨H和和l和米和n吨s这FX^{ }在一世llb和d和n这吨和dF, G,和吨C.吨H和在一种l在和这F一种F在nC吨一世这n一种lF这n一种在和C吨这rX在一世llb和d和n这吨和dFx这r\角度 F, x\角度.一世ns这米和C这n吨和X吨s,吨H和l和吨吨和rn这吨一种吨一世这n一世s和sp和C一世一种ll是在s和F在lsH这在一世nG吨H和d在一种l一世吨是b和吨在和和nX一种ndF$（见下文）。

|FX|≤米|X|,X∈X

5.1.2定理让F是范数空间中的线性泛函X， 然后让大号=X∈X:FX=0. 以下是等效的：
(a)F是有界的。
(二)F是连续的，
(c)大号已关闭，
(d) 要么大号=X或者存在一个是∈X和一个数字r>0这样FX≠0每当|X−是|<r.

## 数学代写|随机过程作业代写Stochastic Processes代考|Form of linear functionals in specific Banach spaces

5.2.1定理让X=C0是序列的空间X=(Xn)n≥1这样林n→∞Xn=0，配备最高范数。F是一个功能上X当且仅当存在唯一序列(一种n)n≥1∈l1这样FX=∑n=1∞Xn一种n

FX=F林n→∞∑一世=1nX一世和一世=林n→∞F∑一世=1nX一世和一世 =林n→∞∑一世=1nX一世F和一世=林n→∞∑一世=1nX一世一种一世=∑n=1∞X一世一种一世

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