# 微积分网课代修|极限理论代写Limit Theory代考|MATH6710 Second Approach

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|极限理论代写Limit Theory代考|Stable Convergence of Random Variables

The limit kernel for $\mathcal{G}$-stable convergence $X_{n} \rightarrow K$ can always be represented as a $\mathcal{G}$ conditional distribution of a further random variable $X$ defined on a suitable extension of the underlying probability space $(\Omega, \mathcal{F}, P)$ : Take $\bar{\Omega}=\Omega \times \mathcal{X}, \overline{\mathcal{F}}=\mathcal{F} \otimes \mathcal{B}(\mathcal{X})$, $\bar{P}=P \otimes K$ and $X(\omega, x)=x$. So, for instance, the Gauss-kernel $N\left(0, \eta^{2}\right)$, where $\eta$ is a $\mathcal{G}$-measurable, nonnegative real random variable, satisfies $N\left(0, \eta^{2}\right)=P^{\eta Z \mid \mathcal{G}}$ assuming the existence of a $N(0,1)$-distributed random variable $Z$ on $(\Omega, \mathcal{F}, P)$ which is independent of $\mathcal{G}$. This motivates the following approach.

Definition 3.15 Let $\mathcal{G} \subset \mathcal{F}$ be a sub- $\sigma$-field. A sequence $\left(X_{n}\right){n \geq 1}$ of $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ valued random variables is said to converge $\mathcal{G}$-stably to an $(\mathcal{X}, \mathcal{B}(\mathcal{X})$ )-valued random variable $X$ if $X{n} \rightarrow P^{X \mid \mathcal{G}} \mathcal{G}$-stably for $n \rightarrow \infty$. Then we write $X_{n} \rightarrow X \mathcal{G}$-stably.
As before, we assume the existence of conditional distributions. By Definition $3.1$ $\mathcal{G}$-stable convergence $X_{n} \rightarrow X$ reads
$$\lim {n \rightarrow \infty} E\left(f E\left(h\left(X{n}\right) \mid \mathcal{G}\right)\right)=E(f E(h(X) \mid \mathcal{G}))$$
for every $f \in \mathcal{L}^{1}(P)$ and $h \in C_{b}(\mathcal{X})$ and implies $X_{n} \stackrel{d}{\rightarrow} X$. The $\mathcal{G}$-mixing convergence $X_{n} \rightarrow X$ corresponds to $P^{X \mid \mathcal{G}}=P^{X} P$-almost surely which is equivalent to the independence of $\sigma(X)$ and $\mathcal{G}$. Thus $X_{n} \rightarrow X \mathcal{G}$-mixing means $X_{n} \rightarrow X \mathcal{G}$-stably and $\sigma(X)$ and $\mathcal{G}$ are independent which is also equivalent to $X_{n} \rightarrow P^{X} \mathcal{G}$-mixing and independence of $\sigma(X)$ and $\mathcal{G}$.

For the formulation of stable limit theorems in subsequent chapters we sometimes use the ” $K$-approach”, sometimes the ” $X$-approach”, and sometimes both.

Example $3.16$ In the situation of Example $3.13$ (b) with $\mathcal{G}=\sigma\left(Z_{n}, n \geq 1\right)$ let $X$ be $N\left(0, \sigma^{2}\right)$-distributed and independent of $\mathcal{G}$. Such an $X$ exists at least after a suitable extension of $(\Omega, \mathcal{F}, P)$. Then Example $3.13$ (b) yields
$$\frac{1}{\sqrt{n}} \sum_{j=1}^{n}\left(Z_{j}-E Z_{1}\right) \rightarrow X \quad \mathcal{G} \text {-mixing. }$$
However, there is nothing special about this $\mathcal{G}$. The above statement holds for any pair $(\mathcal{G}, X)$, where $P^{X}=N\left(0, \sigma^{2}\right)$ and $\sigma(X), \mathcal{G}$ are independent. The random variable $X$ is merely an “artificial” construct to describe the limit kernel. In practice, $\mathcal{G}$ can and will be chosen so large that all random variables of interest are measurable w.r.t. $\mathcal{G}$.
The previous characterizations of $\mathcal{G}$-stable convergence now read as follows.

## 微积分网课代修|极限理论代写Limit Theory代考|Limit Points

In order to describe the fluctuation behavior of stably convergent random variables recall that $x \in \mathcal{X}$ is said to be a limit point of a sequence $\left(x_{n}\right){n \geq 1}$ in $\mathcal{X}$ if it has a subsequence converging to $x$. We denote by $L\left(\left(x{n}\right)\right)$ the set of all limit points of $\left(x_{n}\right){n \geq 1}$. Since $\mathcal{X}$ is first countable (each point has a countable neighborhood basis) the limit points of a sequence are precisely the cluster (or accumulation) points of the sequence, so that $L\left(\left(x{n}\right)\right)=\bigcap_{n \in \mathbb{N}} \overline{\left{x_{k}: k \geq n\right}}$, where $\bar{B}$ denotes the closure of $B \subset \mathcal{X}$. Furthermore, the set $L:=\left{\left(\left(x_{n}\right), x\right) \in \mathcal{X}^{\mathbb{N}} \times \mathcal{X}: x \in L\left(\left(x_{n}\right)\right)\right}$ can be written as $L=\bigcap_{n \in \mathbb{N}} L_{n}$, where

\begin{aligned} L_{n} &:=\left{\left(\left(x_{j}\right), x\right) \in \mathcal{X}^{\mathbb{N}} \times \mathcal{X}: x \in \overline{\left{x_{k}: k \geq n\right}}\right} \ &=\bigcap_{i=1}^{\infty} \bigcup_{k=n}^{\infty}\left{\left(\left(x_{j}\right), x\right) \in \mathcal{X}^{\mathbb{N}} \times \mathcal{X}: d\left(x_{k}, x\right)<\frac{1}{i}\right} \end{aligned}
hence $L_{n}, L \in \mathcal{B}(\mathcal{X})^{\mathbb{N}} \otimes \mathcal{B}(\mathcal{X})$. For $\nu \in \mathcal{M}^{1}(\mathcal{X})$, let $\operatorname{supp}(\nu)$ denote the support of $\nu$ (i.e. the smallest closed set $B$ such that $\nu(B)=1$ ), which exists in our setting ([69], Theorem II.2.1).

## 微积分网课代修|极限理论代写Limit Theory代考|Stable Convergence of Random Variables

$\bar{P}=P \otimes K$ 和 $X(\omega, x)=x$. 因此，例如，高斯核 $N\left(0, \eta^{2}\right)$ ，在哪里 $\eta$ 是一个 $\mathcal{G}-$

$$\lim n \rightarrow \infty E(f E(h(X n) \mid \mathcal{G}))=E(f E(h(X) \mid \mathcal{G}))$$

$$\frac{1}{\sqrt{n}} \sum_{j=1}^{n}\left(Z_{j}-E Z_{1}\right) \rightarrow X \quad \mathcal{G} \text {-mixing. }$$

## 微积分网课代修|极限理论代写Limit Theory代考|Limit Points

$\left(x_{n}\right) n \geq 1$ 在 $\mathcal{X}$ 如果它有一个收敛到的子序列 $x$. 我们表示 $L((x n))$ 的所有极限点的