### 数学代写|随机过程作业代写Stochastic Processes代考|Preliminaries, notations and conventions

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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|Elements of topology

1.1.1 Basics of topology We assume that the reader is familiar with basic notions of topology. To set notation and refresh our memory, let us recall that a pair $(S, \mathcal{U})$ where $S$ is a set and $\mathcal{U}$ is a collection of subsets of $S$ is said to be a topological space if the empty set and $S$ belong to $\mathcal{U}$, and unions and finite intersections of elements of $\mathcal{U}$ belong to $\mathcal{U}$. The family $\mathcal{U}$ is then said to be the topology in $S$, and its members are called open sets. Their complements are said to be closed. Sometimes, when $\mathcal{U}$ is clear from the context, we say that the set $S$ itself is a topological space. Note that all statements concerning open sets may be translated into statements concerning closed sets. For example, we may equivalently define a topological space to be a pair $(S, \mathcal{C})$ where $\mathcal{C}$ is a collection of sets such that the empty set and $S$ belong to $\mathcal{C}$, and intersections and finite unions of elements of $\mathcal{C}$ belong to $\mathcal{C}$.

An open set containing a point $s \in S$ is said to be a neighborhood of s. A topological space $(S, \mathcal{U})$ is said to be Hausdorff if for all $p_{1}, p_{2} \in S$, there exists $A_{1}, A_{2} \in \mathcal{U}$ such that $p_{i} \in A_{i}, i=1,2$ and $A_{1} \cap A_{2}=\emptyset$. Unless otherwise stated, we assume that all topological spaces considered in this book are Hausdorff.

The closure, $c l(A)$, of a set $A \subset S$ is defined to be the smallest closed set that contains $A$. In other words, $c l(A)$ is the intersection of all closed sets that contain $A$. In particular, $A \subset c l(A) . A$ is said to be dense in $S$ iff $c l(A)=S$.

A family $\mathcal{V}$ is said to be a base of topology $\mathcal{U}$ if every element of $\mathcal{U}$ is a union of elements of $\mathcal{V}$. A family $\mathcal{V}$ is said to be a subbase of $\mathcal{U}$ if the family of finite intersections of elements of $\mathcal{V}$ is a base of $\mathcal{U}$.

If $(S, \mathcal{U})$ and $\left(S^{\prime}, \mathcal{U}^{\prime}\right)$ are two topological spaces, then a map $f: S \rightarrow S^{\prime}$ is said to be continuous if for any open set $A^{\prime}$ in $\mathcal{U}^{\prime}$ its inverse image $f^{-1}\left(A^{\prime}\right)$ is open in $S$.

Let $S$ be a set and let $\left(S^{\prime}, \mathcal{U}^{\prime}\right)$ be a topological space, and let $\left{f_{t}, t \in \mathbb{T}\right}$ be a family of maps from $S$ to $S^{\prime}$ (here $T$ is an abstract indexing set). Note that we may introduce a topology in $S$ such that all maps $f_{t}$ are continuous, a trivial example being the topology consisting of all subsets of $S$. Moreover, an elementary argument shows that intersections of finite or infinite numbers of topologies in $S$ is a topology. Thus, there exists the smallest topology (in the sense of inclusion) under which the $f_{t}$ are continuous. This topology is said to be generated by the family $\left{f_{t}, t \in \mathbb{T}\right}$.

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

1.2.1 Measure spaces and measurable functions Although we assume that the reader is familiar with the rudiments of measure theory as presented, for example, in [103], let us recall the basic notions. A family $\mathcal{F}$ of subsets of an abstract set $\Omega$ is said to be a $\sigma$-algebra if it contains $\Omega$ and complements and countable unions of its elements. The pair $(\Omega, \mathcal{F})$ is then said to be a measurable space. A family $\mathcal{F}$ is said to be an algebra or a field if it contains $\Omega$, complements and finite unions of its elements.

A function $\mu$ that maps a family $\mathcal{F}$ of subsets of $\Omega$ into $\mathbb{R}+$ such that
$$\mu\left(\bigcup_{n \in \mathbb{N}} A_{n}\right)=\sum_{n=1}^{\infty} \mu\left(A_{n}\right)$$

for all pairwise-disjoint elements $A_{n}, n \in \mathbb{N}$ of $\mathcal{F}$ such that the union $\bigcup_{n \in \mathbb{N}} A_{n}$ belongs to $\mathcal{F}$ is called a measure. In most cases $\mathcal{F}$ is a $\sigma$ algebra but there are important situations where it is not, see e.g. $1.2 .8$ below. If $\mathcal{F}$ is a $\sigma$-algebra, the triple $(\Omega, \mathcal{F}, \mu)$ is called a measure space.
Property (1.1) is termed countable additivity. If $\mathcal{F}$ is an algebra and $\mu(S)<\infty$, (1.1) is equivalent to
$$\lim {n \rightarrow \infty} \mu\left(A{n}\right)=0 \text { whenever } A_{n} \in \mathcal{F}, A_{n} \supset A_{n+1}, \bigcap_{n=1}^{\infty} A_{n}=\emptyset .$$
The smallest $\sigma$-algebra containing a given class $\mathcal{F}$ of subsets of a set is denoted $\sigma(\mathcal{F})$. If $\Omega$ is a topological space, then $\mathcal{B}(\Omega)$ denotes the smallest $\sigma$-algebra containing open sets, called the Borel $\sigma$-algebra. A measure $\mu$ on a measurable space $(\Omega, \mathcal{F})$ is said to be finite (or bounded) if $\mu(\Omega)<\infty$. It is said to be $\sigma$-finite if there exist measurable subsets $\Omega_{n}$, $n \in \mathbb{N}$, of $\Omega$ such that $\mu\left(\Omega_{n}\right)<\infty$ and $\Omega=\bigcup_{n \in \mathbb{N}} \Omega_{n}$.

A measure space $(\Omega, \mathcal{F}, \mu)$ is said to be complete if for any set $A \subset \Omega$ and any measurable $B$ conditions $A \subset B$ and $\mu(B)=0$ imply that $A$ is measurable (and $\mu(A)=0$, too). When $\Omega$ and $\mathcal{F}$ are clear from the context, we often say that the measure $\mu$ itself is complete. In Exercise $1.2 .10$ we provide a procedure that may be used to construct a complete measure from an arbitrary measure. Exercises 1.2.4 and 1.2.5 prove that properties of complete measure spaces are different from those of measure spaces that are not complete.

## 数学代写|随机过程作业代写Stochastic Processes代考|Functions of bounded variation

1.3.1 Functions of bounded variation A function $y$ defined on a closed interval $[a, b]$ is said to be of bounded variation if there exists a number $K$ such that for every natural $n$ and every partition $a=t_{1} \leq t_{2} \leq \cdots \leq$ $t_{n}=b$,
$$\sum_{i=2}^{n}\left|y\left(t_{i}\right)-y\left(t_{i-1}\right)\right| \leq K$$
The infimum over all such $K$ is then denoted var $[y, a, b]$. We do not exclude the case where $a=-\infty$ or $b=\infty$. In such a case we understand that $y$ is of bounded variation on finite subintervals of $[a, b]$ and that $\operatorname{var}[y,-\infty, b]=\lim {c \rightarrow-\infty} \operatorname{var}[y, c, b]$ is finite and/or that $$\operatorname{var}[y, a, \infty]=\lim {c \rightarrow \infty} \operatorname{var}[y, a, c]$$
is finite. It is clear that $\operatorname{var}[y, a, b] \geq 0$, and that it equals $|y(b)-y(a)|$ if $y$ is monotone. If $y$ is of bounded variation on $[a, b]$ and $a \leq c \leq b$, then $y$ is of bounded variation on $[a, c]$ and $[c, b]$, and
$$\operatorname{var}[y, a, b]=\operatorname{var}[y, a, c]+\operatorname{var}[y, c, b]$$
Indeed, if $a=t_{1} \leq t_{2} \leq \cdots \leq t_{n}=c$ and $c=s_{1} \leq s_{2} \leq \cdots \leq s_{m}=b$, then $u_{i}=t_{i}, i=1, \ldots, n-1, u_{n}=t_{n}=s_{1}$ and $u_{n+i}=s_{i+1}, i=$ $1, \ldots, m-1$, is a partition of $[a, b]$, and
$$\sum_{i=2}^{m+n-1}\left|y\left(u_{i}\right)-y\left(u_{i-1}\right)\right|=\sum_{i=2}^{n}\left|y\left(t_{i}\right)-y\left(t_{i-1}\right)\right|+\sum_{i=2}^{m}\left|y\left(s_{i}\right)-y\left(s_{i-1}\right)\right|$$

## 数学代写|随机过程作业代写Stochastic Processes代考|Elements of topology

1.1.1 拓扑基础 我们假设读者熟悉拓扑的基本概念。为了设置符号并刷新我们的记忆，让我们回忆一下(小号,在)在哪里小号是一个集合并且在是子集的集合小号如果空集和小号属于在, 和元素的并集和有限交集在属于在. 家庭在则称其为拓扑小号，其成员称为开集。据说它们的补码是封闭的。有时，当在从上下文很清楚，我们说集合小号本身就是一个拓扑空间。请注意，所有关于开集的陈述都可以翻译成关于闭集的陈述。例如，我们可以等价地定义一个拓扑空间是一对(小号,C)在哪里C是集合的集合，使得空集和小号属于C, 和元素的交集和有限并集C属于C.

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

1.2.1 测度空间和可测函数 虽然我们假设读者熟悉例如在[103]中介绍的测度论的基础知识，但让我们回顾一下基本概念。一个家庭F抽象集的子集Ω据说是一个σ-代数如果它包含Ω及其元素的补数和可数并集。这对(Ω,F)则称其为可测空间。一个家庭F如果它包含，则称其为代数或域Ω，其元素的补码和有限并集。

μ(⋃n∈ñ一种n)=∑n=1∞μ(一种n)

## 数学代写|随机过程作业代写Stochastic Processes代考|Functions of bounded variation

1.3.1 有界变分函数 A 函数是在闭区间上定义[一种,b]如果存在一个数，则称其为有界变化ķ这样对于每一个自然n和每个分区一种=吨1≤吨2≤⋯≤ 吨n=b,
∑一世=2n|是(吨一世)−是(吨一世−1)|≤ķ

∑一世=2米+n−1|是(在一世)−是(在一世−1)|=∑一世=2n|是(吨一世)−是(吨一世−1)|+∑一世=2米|是(s一世)−是(s一世−1)|

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