### 金融代写|金融数学代写Financial Mathematics代考|MATHS 1009

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• Statistical Inference 统计推断
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• (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

## 金融代写|金融数学代写Financial Mathematics代考|σ-algebra generated by a random variable

We now define the $\sigma$-algebra generated by a random variable. This concept is important for several reasons. For instance, it can make it possible to define the independence of random variables. It is also at the heart of the definition of conditional expectations; see Chapter 2 .

PROPOSITION 1.6. – Let $X$ be a real random variable, defined on $(\Omega, \mathcal{F}, \mathbb{P})$ taking values in $(E, \mathcal{E})$. Then, $\mathcal{F}{X}=X^{-1}(\mathcal{E})=\left{X^{-1}(A) ; A \in \mathcal{E}\right}$ is a sub- $\sigma$-algebra of $\mathcal{F}$ on $\Omega$. This is called the $\sigma$-algebra generated by the random variable $X$. It is written as $\sigma(X)$. It is the smallest $\sigma$-algebra on $\Omega$ that makes $X$ measurable: $$\sigma(X)=X^{-1}(\mathcal{B}(\mathbb{R}))=\left{X^{-1}(B) ; B \in \mathcal{B}(\mathbb{R})\right}={(X \in B) ; B \in \mathcal{B}(\mathbb{R})}$$ EXAMPLE 1.19.- Let $\mathcal{F}{0}={\emptyset, \Omega}$ and $X=c \in \mathbb{R}$ be a constant. Then, for any Borel set $B$ in $\mathbb{R},(X \in B)$ has the value $\emptyset$ if $c \notin B$ and $\Omega$ if $c \in B$. Thus, the $\sigma$-algebra generated by $X$ is $\mathcal{F}{0}$. Reciprocally, it can be demonstrated that the only $\mathcal{F}{0}$-measurable random variables are the constants. Indeed, let $X$ be a $\mathcal{F}{0}$-measurable random variable. Assume that it takes at least two different values, $x$ and $y$. It may be assumed that $y \geq x$ without loss of generality. Therefore, let $B=\left[x, \frac{x+y}{2}\right]$. We have that $(X \in B)$ is non-empty because $x \in B$ but is not $\Omega$ since $y \notin B$. Therefore, $X$ is not $\mathcal{F}{0}$-measurable.

PROPOSITION 1.7.-Let $X$ be a random variable on $(\Omega, \mathcal{F}, \mathbb{P})$ taking values in $(E, \mathcal{E})$ and let $\sigma(X)$ be the $\sigma$-algebra generated by $X$. Thus, a random variable $Y$ is $\sigma(X)$-measurable if and only if there exists a measurable function $f$ such that $Y=f(X)$.

This technical result will be useful in certain demonstrations further on in the text. In general, if it is known that $Y$ is $\sigma(X)$-measurable, we cannot (and do not need to) make explicit the function $f$. Reciprocally, if $Y$ can be written as a measurable function of $X$, it automatically follows that $Y$ is $\sigma(X)$-measurable.

EXAMPLE 1.20.- A die is rolled 2 times. This experiment is modeled by $\Omega={1,2,3,4,5,6}^{2}$ endowed with the $\sigma$-algebra of its subsets and the uniform distribution. Consider the mappings $X_{1}, X_{2}$ and $Y$ from $\Omega$ onto $\mathbb{R}$ defined by
\begin{aligned} &X_{1}\left(\omega_{1}, \omega_{2}\right)=\omega_{1} \ &X_{2}\left(\omega_{1}, \omega_{2}\right)=\omega_{2} \ &Y\left(\omega_{1}, \omega_{2}\right)=\mathbb{1}{{2,4,6}}\left(\omega{1}\right) \end{aligned}
thus, $X_{i}$ is the result of the ith roll and $Y$ is the parity indicator of the first roll. Therefore, $Y=\mathbb{1}{{2,4,6}}\left(X{1}\right)$; thus, $Y$ is $\sigma\left(X_{1}\right)$-measurable. On the other hand, $Y$ cannot be written as a function of $X_{2}$.

The $\sigma$-algebra generated by $X$ represents all the events that can be observed by drawing $X$. It represents the information revealed by $X$.
DEFINITION 1.14.-Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space.

• Let $X$ and $Y$ be two random variables on $(\Omega, \mathcal{F}, \mathbb{P})$ taking values in $\left(E_{1}, \mathcal{E}{1}\right)$ and $\left(E{2}, \mathcal{E}_{2}\right)$. Then, $X$ and $Y$ are said to be independent if the $\sigma$-algebras $\sigma(X)$ and $\sigma(Y)$ are independent.
• Any family $\left(X_{i}\right){i \in I}$ of random variables is independent if the $\sigma$-algebras $\sigma\left(X{i}\right)$ are independent.
• Let $\mathcal{G}$ be a sub- $\sigma$-algebra of $\mathcal{F}$, and let $X$ be a random variable. Then, $X$ is said to be independent of $\mathcal{G}$ if $\sigma(X)$ is independent of $\mathcal{G}$ or, in other words, $\forall A \in \mathcal{G}, X$ and $\mathbb{1}_{A}$ are independent.

Proposition 1.8.- If $X$ and $Y$ are two integrable and independent random variables, then their product $X Y$ is integrable and $\mathbb{E}[X Y]=\mathbb{E}[X] \mathbb{E}[Y]$.

## 金融代写|金融数学代写Financial Mathematics代考|Random vectors

We will now more closely study random variables taking values in $\mathbb{R}^{d}$, with $d \geq 2$. This concept has already been defined in Definition 1.9. We will now look at the

relations between the random vector and its coordinates. When $d=2$, we then speak of a random couple.

PROPOSITION 1.9.-Let $X$ be a real random vector on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, taking values in $\mathbb{R}^{d}$. Then,
$$X(w)=\left(\begin{array}{c} X_{1}(w) \ \vdots \ X_{n}(w) \end{array}\right)$$
is such that for any $i \in{1, \ldots, d}, X_{i}$ is a real random variable.
DEFINITION 1.15.-A random vector is said to be discrete if each of its components, $X_{i}$, is a discrete random variable.
DEFINITION 1.16.- Let $X=\left(\begin{array}{c}X_{1} \ X_{2}\end{array}\right)$ be a discrete random couple such that
$$X_{1}(\Omega)=\left{x_{1 j}, j \in I_{1}\right} \text { et } X_{2}(\Omega)=\left{x_{2 k}, k \in I_{2}\right}$$
The conjoint distribution (or joint distribution or, simply, the distribution) of $X$ is given by the family
$$\left{\mathbb{P}\left(X_{1}=x_{1 j}, X_{2}=x_{2 k}\right) ;(j, k) \in I_{1} \times I_{2}\right} .$$
The marginal distributions of $X$ are the distributions of $X_{1}$ and $X_{2}$. These distributions may be derived from the conjoint distribution of $X$ through:
$$\forall j \in I_{1}, \quad \mathbb{P}\left(X_{1}=x_{1 j}\right)=\sum_{k \in I_{2}} \mathbb{P}\left(X_{1}=x_{1 j}, X_{2}=x_{2 k}\right)$$
and
$$\forall k \in I_{2}, \quad \mathbb{P}\left(X_{2}=x_{2 k}\right)=\sum_{j \in I_{1}} \mathbb{P}\left(X_{1}=x_{1 j}, X_{2}=x_{2 k}\right)$$
The concept of joint distributions and marginal distributions can naturally be extended to vectors with dimension larger than 2 .

EXAMPLE 1.21.- A coin is tossed 3 times, and the result is noted. The universe of possible outcomes is $\Omega={T, H}^{3}$. Let $X$ denote the total number of tails obtained and $Y$ denote the number of tails obtained at the first toss. Then,
$$X(\Omega)={0,1,2,3} \text { and } Y(\Omega)={0,1} .$$

## 金融代写|金融数学代写Financial Mathematics代考|Convergence of sequences of random variables

To conclude this section on random variables, we will review some classic results of convergence for sequences of random variables. Throughout the rest of this book, the abbreviation $r v$. signifies random variable.
DEFINITION 1.17.- Let $\left(X_{n}\right){n \geq 1}$ and $X$ be rv.s defined on $(\Omega, \mathcal{F}, \mathbb{P})$. 1) It is assumed that there exists $p>0$ such that, for any $n \geq 0, \mathbb{E}\left[\left|X{n}\right|^{p}\right]<\infty$, and $\mathbb{E}\left[|X|^{p}\right]<\infty$. It is said that the sequence of random variables $\left(X_{n}\right){n \geq 1}$ converges on the average of the order $p$ or converges in $L^{p}$ towards $X$, if $$\lim {n \rightarrow \infty} \mathbb{E}\left[\left|X_{n}-X\right|^{p}\right]=0$$
We then write $X_{n} \stackrel{L^{P}}{\longrightarrow} X$. In the specific case where $p=2$, we say there is $a$ convergence in quadratic mean.

2) The sequence of $r v .\left(X_{n}\right){n \geq 1}$ is called almost surely (a.s.) convergent towards $X$, if $$\lim {n \rightarrow \infty} \mathbb{P}\left(w \in \Omega ; \lim {n \rightarrow \infty} X{n}(w)=X(w)\right)=1 .$$
We then write $X_{n} \stackrel{a . s .}{\longrightarrow} X$.
THEOREM $1.1$ (Monotone convergence theorem).-Let $\left(X_{n}\right){n \geq 1}$ be a sequence of positive and non-decreasing random variables and let $X$ be an integrable random variable, all of these defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. If $\left(X{n}\right)$ converges almost surely to $X$, then
$$\lim {n \rightarrow+\infty} \mathbb{E}\left[X{n}\right]=\mathbb{E}[X]$$
THEOREM $1.2$ (Dominated convergence theorem).-Let $\left(X_{n}\right){n \geq 1}$ be a sequence of random variables and let $X$ be another random variable, all defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. If the sequence $\left(X{n}\right)$ converges to $X$ a.s., and for any $n \geq 1,\left|X_{n}\right| \leq Z$, where $Z$ is an integrable random variable, then $X_{n} \stackrel{L^{1}}{\longrightarrow} X$ and, in particular,
$$\lim {n \rightarrow+\infty} \mathbb{E}\left[X{n}\right]=\mathbb{E}[X]$$
THEOREM $1.3$ (Strong law of large numbers).-Let $\left(X_{n}\right){n \geq 1}$ be a sequence of integrable, independent random variables from the same distribution. Then, $$\frac{X{1}+X_{2}+\cdots+X_{n}}{n} \underset{n \rightarrow \infty}{\stackrel{\text { a.s. }}{\longrightarrow}} \mathbb{E}\left[X_{1}\right] .$$

## 金融代写|金融数学代写Financial Mathematics代考|σ-algebra generated by a random variable

\sigma(X)=X^{-1}(\mathcal{B}(\mathbb{R}))=\left{X^{-1}(B) ; B \in \mathcal{B}(\mathbb{R})\right}={(X \in B) ; B \in \mathcal{B}(\mathbb{R})}\sigma(X)=X^{-1}(\mathcal{B}(\mathbb{R}))=\left{X^{-1}(B) ; B \in \mathcal{B}(\mathbb{R})\right}={(X \in B) ; B \in \mathcal{B}(\mathbb{R})}例 1.19.- 让F0=∅,Ω和X=C∈R成为一个常数。那么，对于任何 Borel 集乙在R,(X∈乙)有价值∅如果C∉乙和Ω如果C∈乙. 就这样σ- 代数由X是F0. 反过来，可以证明只有F0- 可测量的随机变量是常数。确实，让X做一个F0- 可测量的随机变量。假设它至少需要两个不同的值，X和是. 可以假设是≥X不失一般性。因此，让乙=[X,X+是2]. 我们有那个(X∈乙)是非空的，因为X∈乙但不是Ω自从是∉乙. 所以，X不是F0- 可测量的。

X1(ω1,ω2)=ω1 X2(ω1,ω2)=ω2 是(ω1,ω2)=12,4,6(ω1)

• 让X和是是两个随机变量(Ω,F,磷)取值(和1,和1)和(和2,和2). 然后，X和是如果σ-代数σ(X)和σ(是)是独立的。
• 任何家庭(X一世)一世∈我的随机变量是独立的，如果σ-代数σ(X一世)是独立的。
• 让G成为一个子σ- 代数F， 然后让X是一个随机变量。然后，X据说独立于G如果σ(X)独立于G或者，换句话说，∀一个∈G,X和1一个是独立的。

## 金融代写|金融数学代写Financial Mathematics代考|Random vectors

X(在)=(X1(在) ⋮ Xn(在))

X_{1}(\Omega)=\left{x_{1 j}, j \in I_{1}\right} \text { et } X_{2}(\Omega)=\left{x_{2 k} , k \in I_{2}\right}X_{1}(\Omega)=\left{x_{1 j}, j \in I_{1}\right} \text { et } X_{2}(\Omega)=\left{x_{2 k} , k \in I_{2}\right}

\left{\mathbb{P}\left(X_{1}=x_{1 j}, X_{2}=x_{2 k}\right) ;(j, k) \in I_{1} \times I_ {2}\右} 。\left{\mathbb{P}\left(X_{1}=x_{1 j}, X_{2}=x_{2 k}\right) ;(j, k) \in I_{1} \times I_ {2}\右} 。

∀j∈我1,磷(X1=X1j)=∑ķ∈我2磷(X1=X1j,X2=X2ķ)

∀ķ∈我2,磷(X2=X2ķ)=∑j∈我1磷(X1=X1j,X2=X2ķ)

X(Ω)=0,1,2,3 和 是(Ω)=0,1.

## 金融代写|金融数学代写Financial Mathematics代考|Convergence of sequences of random variables

2) 顺序r在.(Xn)n≥1几乎可以肯定地被称为 (as) 收敛于X， 如果

X1+X2+⋯+Xnn⟶ 作为 n→∞和[X1].

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