### 数学代写|随机过程作业代写Stochastic Processes代考|Sequences of independent random variables

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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|Sequences of independent random variables

1.4.1 Definition Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\mathcal{F}{t}, t \in \mathbb{T}$ be a family of classes of measurable subsets ( $\mathbb{T}$ is an abstract set of indexes). The classes are termed mutually independent (to be more precise: mutually $\mathbb{P}$-independent) if for all $n \in N$, all $t{1}, \ldots, t_{n} \in \mathbb{T}$ and all $A_{i} \in \mathcal{F}{t{i}}, i=1, \ldots, n$
$$\mathbb{P}\left(\bigcap_{i=1}^{n} A_{i}\right)=\prod_{i=1}^{n} \mathbb{P}\left(A_{i}\right) .$$
The classes are termed pairwisely independent (to be more precise: pairwisely $\mathbb{P}$-independent) if for all $n \in N$, all $t_{1}, t_{2} \in \mathbb{T}$ and all $A_{i} \in \mathcal{F}{t{1}}, i=1,2$,
$$\mathbb{P}\left(A_{1} \cap A_{2}\right)=\mathbb{P}\left(A_{1}\right) \mathbb{P}\left(A_{2}\right) .$$
It is clear that mutually independent classes are pairwisely independent. Examples proving that pairwise independence does not imply joint independence may be found in many monographs devoted to probability theory. The reader is encouraged to find one.

Random variables $X_{t}, t \in \mathbb{T}$ are said to be mutually (pairwisely) independent if the $\sigma$-algebras $\mathcal{F}{t}=\sigma\left(X{t}\right)$ generated by $X_{t}$ are mutually (pairwisely) independent.

From now on, the phrase “classes (random variables) are independent” should be understood as “classes (random variables) are mutually independent”.
1.4.2 Exercise Suppose that two events, $A$ and $B$, are independent, i.e. $\mathbb{P}(A \cap B)=\mathbb{P}(A) \mathbb{P}(B)$. Show that the $\sigma$-algebras
$$\left{A, A^{\complement}, \Omega, \emptyset\right}, \quad\left{B, B^{\complement}, \Omega, \emptyset\right}$$
are independent.

## 数学代写|随机过程作业代写Stochastic Processes代考|Convex functions. H¨older and Minkowski inequalities

1.5.1 Definition Let $(a, b)$ be an interval (possibly unbounded: $a=$ $-\infty$ and/or $b=\infty)$. A function $\phi$ is termed convex if for all $u, v \in(a, b)$ and all $0 \leq \alpha \leq 1$,
$$\phi(\alpha u+(1-\alpha) v) \leq \alpha \phi(u)+(1-\alpha) \phi(v)$$
1.5.2 Exercise Show that $\phi$ is convex in $(a, b)$ iff for all $a<u_{1} \leq$ $u_{2} \leq u_{3}<b$
$$\phi\left(u_{2}\right) \leq \frac{u_{2}-u_{1}}{u_{3}-u_{1}} \phi\left(u_{3}\right)+\frac{u_{3}-u_{2}}{u_{3}-u_{1}} \phi\left(u_{2}\right)$$

1.5.3 Exercise (a) Assume $\phi$ is convex in $(a, b)$. Define $\tilde{\phi}(u)=\phi(a+$ $b-u$ ). (If $a=-\infty, b=\infty$, put $a+b=0$.) Show that $\tilde{\phi}$ is convex. (b) For convex $\phi$ on the real line and $t \in \mathbb{R}$, define $\bar{\phi}(u)=\phi(2 t-u)$. Prove that $\bar{\phi}$ is convex.
1.5.4 Lemma Suppose $\phi$ is convex in $(a, b)$ and let $u \in(a, b)$. Define
$f(s)=f_{\phi, u}(s)=\frac{\phi(u)-\phi(s)}{u-s}, \quad s \in(a, u)$,
$g(t)=g_{\phi, u}(t)=\frac{\phi(t)-\phi(u)}{t-u}, \quad t \in(u, b) .$
Then (a) $f$ and $g$ are non-decreasing, and (b) $f(s) \leq g(t)$ for any $s$ and $t$ from the domains of $f$ and $g$, respectively.

Proof To prove the statement for $f$, we take $a<s<s^{\prime}<u$ and do some algebra using $(1.34)$ with $u_{1}=s_{1}, u_{2}=s_{2}$ and $u_{3}=u$. To prove the corresponding statement for $g$ we either proceed similarly, or note that $g_{\phi}(s)=-f_{\bar{\phi}, a+b-u}(a+b-s)$. Indeed,
$$f_{\bar{\phi}, a+b-u}(a+b-s)=\frac{\tilde{\phi}(a+b-u)-\tilde{\phi}(a+b-s)}{a+b-u-(a+b-s)}=\frac{\phi(u)-\phi(s)}{s-u}$$
Finally, (b) follows from $(1.34)$, with $u_{1}=s, u_{2}=u, u_{3}=t$.

## 数学代写|随机过程作业代写Stochastic Processes代考|The Cauchy equation

The content of this section is not needed in what follows but it provides better insight into the results of Chapter 6 and Chapter 7. Plus, it contains a beautiful Theorem of Steinhaus. The casual reader may skip this section on the first reading (and on the second and on the third one as well, if he/she ever reads this book that many times). The main theorem of this section is 1.6.11.
1.6.1 Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a measure space. Show that for all measurable sets $A, B$ and $C$
$$|\mu(A \cap B)-\mu(C \cap B)| \leq \mu(A \div C) .$$
Here $\div$ denotes the symmetric difference of two sets defined as $A \div B=$ $(A \backslash B) \cup(B \backslash A)$.
1.6.2 Lemma If $A \subset \mathbb{R}$ is compact and $B \subset \mathbb{R}$ is Lebesgue measurable, than $x(t)=\operatorname{leb}\left(A_{t} \cap B\right)$ is continuous, where $A_{t}$ is a translation of the set $A$ as defined in (1.4).
Proof By Exercise 1.6.1,
\begin{aligned} \mid \operatorname{leb}\left(A_{t+h} \cap B\right) &-\operatorname{le} b\left(A_{t} \cap B\right) \mid \leq \operatorname{leb}\left(A_{t+h} \div A_{t}\right) \ &=\operatorname{leb}\left(A_{h} \div A\right){t}=\operatorname{leb}\left(A{h} \div A\right), \quad t, h \in \mathbb{R} \end{aligned}
since Lebesgue measure is translation invariant. Therefore it suffices to show that given $\epsilon>0$ there exists a $\delta>0$ such that leb $\left(A_{h} \div A\right)<\epsilon$ provided $|h|<\delta$. To this end let $G$ be an open set such that $\operatorname{le} b(G \backslash A)<$ $\frac{c}{2}$, and take $\delta=\min {a \in A} \min {b \in G^{\mathrm{c}}}|a-b|$. This is a positive number since $A$ is compact, $G^{\mathrm{C}}$ is closed, and $A$ and $G^{\mathrm{C}}$ are disjoint (see Exercise 1.6.3 below). If $|h|<\delta$, then $A_{h} \subset G$. Hence,
$$\operatorname{leb}\left(A_{h} \backslash A\right)<\operatorname{leb}(G \backslash A)<\frac{\epsilon}{2}$$
and
$$\operatorname{leb}\left(A \backslash A_{h}\right)=\operatorname{leb}\left(A \backslash A_{h}\right){-h}=\operatorname{leb}\left(A{-h} \backslash A\right)<\frac{\epsilon}{2}$$
as desired.
1.6.3 Exercise Show that if $A$ and $B$ are disjoint subsets of a metric space $(\mathbb{X}, d), A$ is compact, and $B$ is closed, then $\delta=\min {a \in A} \min {b \in B} \mid a-$ $b \mid$ is positive. Show by example that the statement is not true if $A$ is closed but fails to be compact.

## 数学代写|随机过程作业代写Stochastic Processes代考|Sequences of independent random variables

1.4.1 定义让(Ω,F,磷)是一个概率空间。让F吨,吨∈吨是一系列可测量子集（吨是一组抽象的索引）。这些类被称为相互独立（更准确地说：相互磷-独立的）如果对所有人n∈ñ， 全部吨1,…,吨n∈吨和所有一种一世∈F吨一世,一世=1,…,n

1.4.2 练习假设有两个事件，一种和乙, 是独立的，即磷(一种∩乙)=磷(一种)磷(乙). 表明σ-代数
\left{A, A^{\complement}, \Omega, \emptyset\right}, \quad\left{B, B^{\complement}, \Omega, \emptyset\right}\left{A, A^{\complement}, \Omega, \emptyset\right}, \quad\left{B, B^{\complement}, \Omega, \emptyset\right}

## 数学代写|随机过程作业代写Stochastic Processes代考|Convex functions. H¨older and Minkowski inequalities

1.5.1 定义让(一种,b)是一个区间（可能是无界的：一种= −∞和/或b=∞). 一个函数φ被称为凸如果对于所有在,在∈(一种,b)和所有0≤一种≤1,
φ(一种在+(1−一种)在)≤一种φ(在)+(1−一种)φ(在)
1.5.2 练习 证明φ是凸的(一种,b)对所有人来说一种<在1≤ 在2≤在3<b
φ(在2)≤在2−在1在3−在1φ(在3)+在3−在2在3−在1φ(在2)

1.5.3 练习 (a) 假设φ是凸的(一种,b). 定义φ~(在)=φ(一种+ b−在）。（如果一种=−∞,b=∞， 放一种+b=0。） 显示φ~是凸的。(b) 对于凸φ在实线和吨∈R， 定义φ¯(在)=φ(2吨−在). 证明φ¯是凸的。
1.5.4 引理假设φ是凸的(一种,b)然后让在∈(一种,b). 定义
F(s)=Fφ,在(s)=φ(在)−φ(s)在−s,s∈(一种,在),
G(吨)=Gφ,在(吨)=φ(吨)−φ(在)吨−在,吨∈(在,b).

Fφ¯,一种+b−在(一种+b−s)=φ~(一种+b−在)−φ~(一种+b−s)一种+b−在−(一种+b−s)=φ(在)−φ(s)s−在

## 数学代写|随机过程作业代写Stochastic Processes代考|The Cauchy equation

1.6.1 练习让(Ω,F,μ)成为测度空间。证明对于所有可测集一种,乙和C
|μ(一种∩乙)−μ(C∩乙)|≤μ(一种÷C).

1.6.2 引理如果一种⊂R紧凑且乙⊂R是勒贝格可测的，比X(吨)=勒布⁡(一种吨∩乙)是连续的，其中一种吨是集合的翻译一种如 (1.4) 中所定义。

\begin{aligned} \mid \operatorname{leb}\left(A_{t+h} \cap B\right) &-\operatorname{le} b\left(A_{t } \cap B\right) \mid \leq \operatorname{leb}\left(A_{t+h} \div A_{t}\right) \ &=\operatorname{leb}\left(A_{h} \ div A\right) {t}=\operatorname{leb}\left(A {h} \div A\right), \quad t, h \in \mathbb{R} \end{aligned} s一世nC和大号和b和sG在和米和一种s在r和一世s吨r一种nsl一种吨一世这n一世n在一种r一世一种n吨.吨H和r和F这r和一世吨s在FF一世C和s吨这sH这在吨H一种吨G一世在和nε>0吨H和r和和X一世s吨s一种d>0s在CH吨H一种吨l和b(一种H÷一种)<εpr这在一世d和d|H|<d.吨这吨H一世s和ndl和吨Gb和一种n这p和ns和吨s在CH吨H一种吨乐⁡b(G∖一种)<C2$,一种nd吨一种ķ和$d=分钟一种∈一种分钟b∈GC|一种−b|$.吨H一世s一世s一种p这s一世吨一世在和n在米b和rs一世nC和$一种$一世sC这米p一种C吨,$GC$一世sCl这s和d,一种nd$一种$一种nd$GC$一种r和d一世sj这一世n吨(s和和和X和rC一世s和1.6.3b和l这在).一世F$|H|<d$,吨H和n$一种H⊂G$.H和nC和, \operatorname{leb}\left(A_{h} \backslash A\right)<\operatorname{leb}(G \backslash A)<\frac{\epsilon}{2} 一种nd \operatorname{leb}\left(A \backslash A_{h}\right)=\operatorname{leb}\left(A \backslash A_{h}\right){-h}=\operatorname{leb}\left( A{-h} \backslash A\right)<\frac{\epsilon}{2}$$根据需要。 1.6.3 练习 证明如果一种和乙是度量空间的不相交子集(X,d),一种是紧凑的，并且乙是闭合的，则$\delta=\min {a \in A} \min {b \in B} \mid a-b \中一世sp这s一世吨一世在和.小号H这在b是和X一种米pl和吨H一种吨吨H和s吨一种吨和米和n吨一世sn这吨吨r在和一世FA\$ 已关闭但未能紧凑。

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