# 微积分网课代修|极限理论代写Limit Theory代考|MATH407 Stability of Limit Theorems

• 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|Stability of Limit Theorems

In this chapter we present some first results on the stability of limit theorems taken from [28] (see also [79, 100]). More precisely, we derive simple sufficient conditions for distributional limit theorems to be mixing.

To this end, let $Z_{n}$ be $(\mathcal{Z}, \mathcal{C})$-valued random variables for some measurable space $(\mathcal{Z}, \mathcal{C})$ and $f_{n}:\left(\mathcal{Z}^{n}, \mathcal{C}^{n}\right) \rightarrow(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ measurable maps for every $n \in \mathbb{N}$, where we need a vector space structure for $\mathcal{X}$. So, let $\mathcal{X}$ be a polish topological vector space (like $\mathbb{R}^{d}, C([0, T])$ for $0<T<\infty$ or $\left.C\left(\mathbb{R}{+}\right)\right)$. Then there exists a translation invariant metric $d$ on $\mathcal{X}$ inducing the topology ([86], Theorem 1.6.1) so that $U{n}-V_{n} \rightarrow 0$ in probability for $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$-valued random variables $U_{n}$ and $V_{n}$ means $d\left(U_{n}, V_{n}\right)=$ $d\left(U_{n}-V_{n}, 0\right) \rightarrow 0$ in probability or, what is the same, $E\left(d\left(U_{n}, V_{n}\right) \wedge 1\right) \rightarrow 0$.
Furthermore, let $b_{n} \in \mathcal{X}$ and $a_{n} \in(0, \infty)$. We consider the $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$-valued random variables
$$X_{n}:=\frac{1}{a_{n}}\left(f_{n}\left(Z_{1}, \ldots, Z_{n}\right)-b_{n}\right)$$
for $n \in \mathbb{N}$ and assume $X_{n} \stackrel{d}{\rightarrow} v$ for some $v \in \mathcal{M}^{1}(\mathcal{X})$. The tail $\sigma$-field of $Z=\left(Z_{n}\right)$ is given by
$$\mathcal{T}{Z}=\bigcap{n=1}^{\infty} \sigma\left(Z_{k}, k \geq n\right) .$$

## 微积分网课代修|极限理论代写Limit Theory代考|Martingale Arrays and the Nesting Condition

For every $n \in \mathbb{N}$ let $\left(X_{n k}\right){1 \leq k \leq k{n}}$ be a sequence of real random variables defined on a probability space $(\Omega, \mathcal{F}, P)$, and let $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}}$ be a filtration in $\mathcal{F}$, i.e. $\mathcal{F}{n 0} \subset \mathcal{F}{n 1} \subset \cdots \subset \mathcal{F}{n k{n}} \subset \mathcal{F}$. The sequence $\left(\bar{X}{n k}\right){1 \leq k \leq k_{n}}$ is called adapted to the filtration $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}}$ if $X_{n k}$ is measurable w.r.t. $\mathcal{F}{n k}$ for all $1 \leq k \leq k{n}$. The triangular array $\left(X_{n k}\right){1 \leq k \leq k{n}, n \in \mathbb{N}}$ of random variables is called adapted to the triangular array $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}, n \in \mathbb{N}}$ of $\sigma$-fields if the row $\left(X_{n k}\right){1 \leq k \leq k{n}}$ is adapted to the filtration $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}}$ for every $n \in \mathbb{N}$. Not all of the following results of a more technical nature require the assumption of adaptedness. Therefore, we will always state explicitly where adapted arrays are considered.

An array $\left(X_{n k}\right){1 \leq k \leq k{n}, n \in \mathbb{N}}$ adapted to $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}, n \in \mathbb{N}}$ is called a martingale difference array if $X_{n k} \in \mathcal{L}^{1}(P)$ with $E\left(X_{n k} \mid \mathcal{F}{n, k-1}\right)=0$ for all $1 \leq k \leq k{n}$ and $n \in \mathbb{N}$, which means that for every $n \in \mathbb{N}$ the sequence $\left(X_{n k}\right){1 \leq k \leq k{n}}$ is a martingale difference sequence w.r.t. the filtration $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}}$. A martingale difference array is square integrable if $X_{n k} \in \mathcal{L}^{2}(P)$ for all $1 \leq k \leq k_{n}$ and $n \in \mathbb{N}$. Note that a martingale difference sequence or array is always by definition adapted to the $\sigma$-fields under consideration.

From now on, we assume that the sequence $\left(k_{n}\right){n \in \mathbb{N}}$ is nondecreasing with $k{n} \geq n$ for all $n \in \mathbb{N}$. We always set $\mathcal{F}{\infty}=\sigma\left(\bigcup{n=1}^{\infty} \mathcal{F}{n k{n}}\right)$. The array $\left(\mathcal{F}{n k}\right){0 \leq k \leq k_{n}, n \in \mathbb{N}}$ is called nested if $\mathcal{F}{n k} \subset \mathcal{F}{n+1, k}$ holds for all $n \in \mathbb{N}$ and $0 \leq k \leq k_{n}$. The subtle role of this property of the $\sigma$-fields in stable martingale central limit theorems will become evident in the sequel.
Our basic stable martingale central limit theorem reads as follows.

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

$f_{n}:\left(\mathcal{Z}^{n}, \mathcal{C}^{n}\right) \rightarrow(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ 每个可测量的地图 $n \in \mathbb{N}$ ，其中我们需要一个向量空 间结构 $\mathcal{X}$. 所以，让 $\mathcal{X}$ 是一个抛光拓扑向量空间（如 $\mathbb{R}^{d}, C([0, T])$ 为 $0<T<\infty$ 或 $C(\mathbb{R}+))$. 然后存在一个平移不变度量 $d$ 上 $\mathcal{X}$ 诱导拓扑（[86]，定理1.6.1），以便
$U n-V_{n} \rightarrow 0$ 在概率中 $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$-值随机变量 $U_{n}$ 和 $V_{n}$ 方法 $d\left(U_{n}, V_{n}\right)=$
$d\left(U_{n}-V_{n}, 0\right) \rightarrow 0$ 在概率上，或者，什么是相同的， $E\left(d\left(U_{n}, V_{n}\right) \wedge 1\right) \rightarrow 0$.

$$X_{n}:=\frac{1}{a_{n}}\left(f_{n}\left(Z_{1}, \ldots, Z_{n}\right)-b_{n}\right)$$

$$\mathcal{T} Z=\bigcap n=1^{\infty} \sigma\left(Z_{k}, k \geq n\right) .$$

## 微积分网课代修|极限理论代写Limit Theory代考| Martingale Arrays and the Nesting Condition

$\mathcal{F} n 0 \subset \mathcal{F} n 1 \subset \cdots \subset \mathcal{F} n k n \subset \mathcal{F}$. 序列 $(\bar{X} n k) 1 \leq k \leq k_{n}$ 称为适应过滤
$(\mathcal{F} n k) 0 \leq k \leq k_{n}$ 如果 $X_{n k}$ 是可测量的 w.r.t. $\mathcal{F} n k$ 面向所有人 $1 \leq k \leq k n$.三角形 阵列 $\left(X_{n k}\right) 1 \leq k \leq k n, n \in \mathbb{N}$ 的随机变量称为适应三角形数组
$(\mathcal{F} n k) 0 \leq k \leq k_{n}, n \in \mathbb{N}$ 之 $\sigma$-字段，如果行 $\left(X_{n k}\right) 1 \leq k \leq k n$ 适应过滤
$(\mathcal{F} n k) 0 \leq k \leq k_{n}$ 对于每个 $n \in \mathbb{N}$. 并非所有技术性更强的以下结果都需要假设适应 性。因此，我们将始终明确说明考虑自适应数组的位置。

$X_{n k} \in \mathcal{L}^{2}(P)$ 面向所有人 $1 \leq k \leq k_{n}$ 和 $n \in \mathbb{N}$. 请注意，根据定义，马丁格尔差分 序列或数组始终适应于 $\sigma$-正在考虑中的字段。