# 微积分网课代修|极限理论代写Limit Theory代考|MATH7710 Stable Functional Martingale Central 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|Stable Functional Martingale Central Limit Theorems

This chapter is devoted to stable functional central limit theorems for partial sum processes based on martingale differences which correspond to the results for partial sums presented in Sects. 6.1,6.3 and 6.4. As in Chap. 6 it is convenient to consider arrays of martingale differences, but to keep technicalities as simple as possible, we consider a fixed filtration $\mathbb{F}=\left(\mathcal{F}{k}\right){k \geq 0}$ on the basic probability space $(\Omega, \mathcal{F}, P)$. As usual, $\mathcal{F}{\infty}=\sigma\left(\bigcup{k=0}^{\infty} \mathcal{F}{k}\right)$. For every $n \in \mathbb{N}$, let $\left(X{n k}\right){k \geq 1}$ be a martingale difference sequence w.r.t. $\mathbb{F}$, and for every $n \in \mathbb{N}$ and $t \in[0, \infty)$ set $$S{(n)}(t):=\sum_{k=1}^{[n t]} X_{n k}+(n t-[n t]) X_{n,[n t]+1} .$$
Then $\left(S_{(n)}(t)\right){t \in[0, \infty)}$ is a random process with sample paths in $C\left(\mathbb{R}{+}\right)$. Note that the array $\left(\mathcal{F}{n, k}\right){k \geq 0, n \in \mathbb{N}}$ with $\mathcal{F}{n, k}:=\mathcal{F}{k}$ is obviously nested.

For a nonnegative stochastic process $(\eta(t)){t \in[0, \infty)}$ with paths in $C\left(\mathbb{R}{+}\right)$and square integrable $X_{n k}$ we introduce the conditions
$\left(\mathrm{N}{t}\right) \quad \sum{k=1}^{[n t]} E\left(X_{n k}^{2} \mid \mathcal{F}{k-1}\right) \rightarrow \eta^{2}(t)$ in probability as $n \rightarrow \infty$ for all $t \in[0, \infty)$ and $\left(\mathrm{CLB}{t}\right) \quad \sum_{k=1}^{[n t]} E\left(X_{n k}^{2} 1_{\left{\left|X_{n k}\right| \geq \varepsilon\right}} \mid \mathcal{F}_{k-1}\right) \rightarrow 0$ in probability as $n \rightarrow \infty$ for all $\varepsilon>0$ and all $t \in[0, \infty)$.

## 微积分网课代修|极限理论代写Limit Theory代考|A Stable Limit Theorem with Exponential Rate

In this chapter we establish a stable limit theorem for “explosive” processes with exponential rates. The increments of these processes are not asymptotically negligible and thus do not satisfy the conditional Lindeberg condition. A simple example is given by an independent sequence $\left(Z_{n}\right){n \geq 1}$ with $P^{Z{n}}=N\left(0,2^{n-1}\right), X_{0}:=0$, $X_{n}:=\sum_{i=1}^{n} Z_{i}$ and rate $a_{n}:=2^{n / 2}$. The subsequent limit theorem is suitable for such situations. In order to formulate this limit theorem we need the following observation.

Lemma 8.1 Let $\left(Z_{n}\right){n \geq 0}$ be an independent and identically distributed sequence of real random variables and $t \in \mathbb{R}$ with $|t|>1$. Then (i) $t^{-n} Z{n} \rightarrow 0$ a.s.,
(ii) $\sum_{n=0}^{\infty} t^{-n} Z_{n}$ converges a.s. in $\mathbb{R}$,
(iii) $\sum_{n=0}^{\infty}|t|^{-n}\left|Z_{n}\right|<\infty$ a.s., (iv) $E \log ^{+}\left|Z_{0}\right|<\infty$ are equivalent assertions. Proof (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i) are obvious. (i) $\Rightarrow$ (iv). We have $P\left(\lim \sup {n \rightarrow \infty}\left{\left|t^{-n} Z{n}\right|>1\right}\right)=0$, implying by the Borel-Cantelli lemma
$$\infty>\sum_{n=0}^{\infty} P\left(|t|^{-n}\left|Z_{n}\right|>1\right)=\sum_{n=0}^{\infty} P\left(\left|Z_{0}\right|>|t|^{n}\right)=\sum_{n=0}^{\infty} P\left(\log ^{+}\left|Z_{0}\right|>n \log |t|\right),$$
hence (iv).
(iv) $\Rightarrow$ (iii). Choose \$1s^{n}\right)=\sum_{n=0}^{\infty} P\left(\log ^{+}\left|Z_{0}\right|>n \log s\right)<\infty
$$## 微积分网课代修|极限理论代写Limit Theory代考| Stable Functional Martingale Central Limit Theorems 本章专门讨论基于马丁格尔差分的部分和过程的稳定泛函中心极限定理，这些定理对应 于 Sects 中提出的部分和的结果。6.1、6.3 和 6.4。与第6章一样，考虑马丁格尔差异 数组很方便，但为了保持技术细节尽可能简单，我们考虑固定过滤 \mathbb{F}=(\mathcal{F} k) k \geq 0 在 基本概率空间上 (\Omega, \mathcal{F}, P). 岹常 \mathcal{F} \infty=\sigma\left(\bigcup k=0^{\infty} \mathcal{F} k\right). 对于每个 n \in \mathbb{N} 让 (X n k) k \geq 1 是马丁格尔差分序列 w.r.t. \mathbb{F} ，并且对于每个 n \in \mathbb{N} 和 t \in[0, \infty) 设置$$
S(n)(t):=\sum_{k=1}^{[n t]} X_{n k}+(n t-[n t]) X_{n,[n t]+1} .
$$然后 \left(S_{(n)}(t)\right) t \in[0, \infty) 是一个随机过程，其中的样本路径位于 C(\mathbb{R}+). 请注意， 数组 (\mathcal{F} n, k) k \geq 0, n \in \mathbb{N} 跟 \mathcal{F} n, k:=\mathcal{F} k 显然是嵌套的。 对于非负随机过程 (\eta(t)) t \in[0, \infty) 路径位于 C(\mathbb{R}+) 和平方可积 X_{n k} 我们介绍条件 ( \mathrm{N} t) \quad \sum k=1^{[n t]} E\left(X_{n k}^{2} \mid \mathcal{F} k-1\right) \rightarrow \eta^{2}(t) 在概率中作为 n \rightarrow \infty 面向所有 人 t \in[0, \infty) 和 在概率中作为 n \rightarrow \infty 面向所有人 \varepsilon>0 和所有 t \in[0, \infty). ## 微积分网课代修|极限理论代写Limit Theory代考| A Stable Limit Theorem with Exponential Rate 在本章中，我们为具有指数速率的“爆炸性”过程建立了一个稳定的极限定理。这些过程 的增量不是渐近可忽略的，因此不满足条件林德伯格条件。一个简单的例子是由一个独 立的序列给出的 \left(Z_{n}\right) n \geq 1 1跟 P^{Z n}=N\left(0,2^{n-1}\right), X_{0}:=0, X_{n}:=\sum_{i=1}^{n} Z_{i} 和速率 a_{n}:=2^{n / 2}. 随后的极限定理适用于这种情况。为了制定这个极限定理，我们需 要以下观察。 引理 8.1 让 \left(Z_{n}\right) n \geq 0 是真实随机变量的独立且分布相同的序列，并且 t \in \mathbb{R} 跟 |t|>1.然后 (i) t^{-n} Z n \rightarrow 0 a.s., (ii) \sum_{n=0}^{\infty} t^{-n} Z_{n} 收敛于 \mathbb{R} ， (三) \sum_{n=0}^{\infty}|t|^{-n}\left|Z_{n}\right|<\infty a.s., (iv) E \log ^{+}\left|Z_{0}\right|<\infty 是等效的断言。证明 (三) \Rightarrow( (二) \Rightarrow(-) 是显而易见的。 (-) \Rightarrow (iv) . 我们有 Borel-Cantelli 引理暗示$$ \infty>\sum_{n=0}^{\infty} P\left(|t|^{-n}\left|Z_{n}\right|>1\right)=\sum_{n=0}^{\infty} P\left(\left|Z_{0}\right|>|t|^{n}\right)=\sum_{n=0}^{\infty} P\left(\log ^{+}\left|Z_{0}\right|>n \log |t|\right),
$$因此（四）。 (四) \Rightarrow (三) 选择 1 \mathrm{~s}^{\wedge}{\mathrm{n}} \backslash right )=\mid sum_ {\mathrm{n}=0}^{\wedge}{\backslash \operatorname{linfty}} \mathrm{P} \backslash \operatorname{left}\left(\backslash \log { }^{\wedge}{+} \backslash|\operatorname{lft}| \mathrm{Z}_{-}{0} \mid\right. right |>\mathrm{n} \backslash \log s| right )<\langle infty \$$