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## 金融代写|风险估值理论代写Uncertainty Quantification代考|Modes of Convergence of Random Variables

In the section, we discuss notions of convergence for random variables.
Definition 2.5.1 Let $H$ be a real Hilbert space and $\left{X_n\right}$ be a sequence of $H$-valued random variables. The sequence $\left{X_n\right}$ is said to converge to $X \in H$

1. almost surely, if $X_n(\omega) \rightarrow X(\omega)$ for almost all $\omega \in \Omega$. That is, if $\mathbb{P}\left(\lim _{n \rightarrow \infty}\left|X_n-X\right|=0\right)=1$
2. in probability if $\mathbb{P}\left(\left|X_n-X\right|>\epsilon\right) \rightarrow 0$ as $n \rightarrow \infty$ for any $\epsilon>0$.
3. in $p$-th mean or in $L^p(\Omega, H)$, for $1 \leq p<\infty$, if $\mathbb{E}\left[\left|X_n-X\right|^p\right] \rightarrow 0$ as $n \rightarrow \infty$. For $p=2$, this convergence is referred to as the mean-square convergence.

Remark 2.5.1 For $q>p$, the convergence in $q$-th mean implies the convergence in $p$-th mean which further implies convergence in probability. Almost surely convergence implies convergence in probability. Furthermore, convergence almost surely implies convergence in $p$-th mean if the random variables are uniformly bounded.
Definition 2.5.2 A set of random variables $\left{X_1, X_2, \ldots\right}$ is called independent and identically distributed (iid, for short) if each $X_j$ has identical distribution and the distinct random variables $X_i, X_j$, for $i \neq j$, are independent.

Theorem 2.5.1 (Strong Law of Large Numbers.) Let $\left{X_1, X_2, \ldots, X_M\right}$ be $M$ iid real-valued random variable with mean $\mu$ such that $\mathbb{E}\left[\left|X_j\right|\right]<\infty$. Then, the sample mean $\bar{X}_M$ of $X_1, X_2, \ldots, X_M$, defined by
$$\bar{X}_M=\frac{X_1+X_2+\cdots+X_M}{M},$$
converges to $\mu$, almost surely, as $M \rightarrow \infty$.

## 金融代写|风险估值理论代写Uncertainty Quantification代考|Projections on Convex Sets in Hilbert Spaces

Let $H$ be a Hilbert space with the inner product $\langle\cdot, \cdot\rangle$ and norm $|\cdot|=\sqrt{\langle\cdot, \cdot}$. Let $H^*$ be the dual of $H$ which, by the Riesz isomorphism, will be identified with $H$. For simplicity, we only consider real Hilbert/Banach spaces. Our focus is on the projection map which is formally defined as follows:

Definition 3.1.1 Let $H$ be a Hilbert space and $K$ be a nonempty, closed, and convex subset of $H$. The projection map $P_K: H \rightarrow K$ assigns to any $x \in H$, the unique point $P_K x$ in $K$ which is closest to $x$. The point $P_K x$ is called the projection of $x$ onto K. That is,
$$\left|x-P_K x\right| \leq|x-z|, \text { for every } z \in K .$$
In the following result, besides showing that the projection map is well-defined, we give its important equivalent variational characterization:

Theorem 3.1.1 Let $H$ be a Hilbert space, and $K$ be a nonempty, closed, and convex subset of H. Let $x \in H$. Then there exists a unique $P_K x \in K$ such that
$$\left|x-P_K x\right| \leq|x-z|, \text { for every } z \in K .$$
Moreover, $P_K x$ is the projection of $x$, if and only if, it satisfies
$$\left\langle x-P_K x, z-P_K x\right\rangle \leq 0, \text { for every } z \in K .$$
Proof. Since
$$\delta:=\inf {z \in K}|x-z| \geq 0,$$ there is a minimizing sequence $\left{y_n\right}$ in $K$ with $\left|x-y_n\right| \rightarrow \delta$, as $n \rightarrow \infty$. Evidently, the sequence $\left{y_n\right}$ is bounded in $H$, and by the reflexivity of $H$, there exists a weakly convergent subsequence. By keeping the same notation for subsequences as well, let $\left{y_n\right}$ be the subsequence that converges weakly to some $y \in H$. The set $K$ being closed and convex is also weakly closed, and hence $y \in K$. Since any norm is a weakly lower semi-continuous function, we obtain $$|x-y| \leq \liminf {n \rightarrow \infty}\left|x-y_n\right|=\delta,$$
which in view of the inequality $\delta \leq|x-y|$ confirms that indeed $|x-y|=\delta$. This proves the existence of an element $y:=P_K x \in K$ that satisfies (3.1).

## 五融代写|风险估值理论代写Uncertainty Quantification代考|Modes of Convergence of Random Variables

\left 的分隔符缺失或无法识别 是一个序列 $H$ 值随机㚆量。序列

$$\bar{X}_M=\frac{X_1+X_2+\cdots+X_M}{M},$$

## 金融代写|风险估值理论代与写certainty Quantification代考|Projections on

Convex Sets in Hilbert Spaces 考慮真正的 Hilbert/Banach 空间。我们的重点是投影图，其正式定义如下:

$$\left|x-P_K x\right| \leq|x-z| \text {, for every } z \in K \text {. }$$

$$\left|x-P_K x\right| \leq|x-z| \text {, for every } z \in K \text {. }$$

$$\left\langle x-P_K x, z-P_K x\right\rangle \leq 0 \text {, for every } z \in K .$$

$$\delta:=\inf z \in K|x-z| \geq 0,$$

\left 的分隔符缺失或无法识别 有界 $H$ ，并且通过自反性 $H$ ，存在弱收玫子序列。通过对子序列也保持相同的 符号，让 left 的分隔符缺失或无法识别 是弱收敛到某个的子序列 $y \in H$. 套装 $K$ 封闭和凸也也是弱封闭的，因 此 $y \in K$. 由于任何范数都是弱下半连续函数，我们得到
$$|x-y| \leq \liminf n \rightarrow \infty\left|x-y_n\right|=\delta,$$

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