微积分网课代修|极限理论代写Limit Theory代考|MATH7710 Why Stable Convergence?

微积分网课代修|极限理论代写Limit Theory代考|MATH7710 Why Stable Convergence?

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微积分网课代修|极限理论代写Limit Theory代考|MATH7710 Why Stable Convergence?

微积分网课代修|极限理论代写Limit Theory代考|Why Stable Convergence?

This chapter is of an introductory nature. We make the motivation for the study of stable convergence more precise and present an exposition of some of its features. With the exception of Example 1.2, no proofs are given, only references to later chapters where proofs may be found.

Our starting point is the classical central limit theorem. For this, let $\left(Z_{k}\right){k \geq 1}$ be a sequence of independent and identically distributed real random variables, defined on some probability space $(\Omega, \mathcal{F}, P)$. Assume $Z{1} \in \mathcal{L}^{2}(P)$ and set $\mu=E Z_{1}$ and $\sigma^{2}=\operatorname{Var} Z_{1}$. To exclude the trivial case of almost surely constant variables, assume also $\sigma^{2}>0$. Then the classical central limit theorem says that
$$
\lim {n \rightarrow \infty} P\left(\frac{1}{n^{1 / 2}} \sum{k=1}^{n} \frac{Z_{k}-\mu}{\sigma} \leq x\right)=\Phi(x)=\int_{-\infty}^{x} \varphi(u) d u \text { for all } x \in \mathbb{R},
$$
where $\varphi(u)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{1}{2} u^{2}\right), u \in \mathbb{R}$, denotes the density of the standard normal distribution. It is customary to write this convergence of probabilities in a somewhat more abstract way as convergence in distribution of random variables, i.e. as
$$
\frac{1}{n^{1 / 2}} \sum_{k=1}^{n} \frac{Z_{k}-\mu}{\sigma} \stackrel{d}{\rightarrow} N(0,1) \quad \text { as } n \rightarrow \infty
$$
where $N(0,1)$ denotes the standard normal distribution, or as
$$
\frac{1}{n^{1 / 2}} \sum_{k=1}^{n} \frac{Z_{k}-\mu}{\sigma} \stackrel{d}{\rightarrow} N \quad \text { as } n \rightarrow \infty
$$

微积分网课代修|极限理论代写Limit Theory代考|Weak Convergence of Markov Kernels

As indicated in the previous chapter, stable convergence of random variables can be seen as suitable convergence of Markov kernels given by conditional distributions. Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\mathcal{X}$ be a separable metrizable topological space equipped with its Borel $\sigma$-field $\mathcal{B}(\mathcal{X})$. In this chapter we briefly describe the weak topology on the set of Markov kernels (transition kernels) from $(\Omega, \mathcal{F})$ to $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$

Let us first recall the weak topology on the set $\mathcal{M}^{1}(\mathcal{X})$ of all probability measures on $\mathcal{B}(\mathcal{X})$. It is the topology generated by the functions
$$
\nu \mapsto \int h d \nu, \quad h \in C_{b}(\mathcal{X}),
$$
where $C_{b}(\mathcal{X})$ denotes the space of all continuous, bounded functions $h: \mathcal{X} \rightarrow \mathbb{R}$ equipped with the sup-norm $|h|_{\text {sup }}:=\sup {x \in \mathcal{X}}|h(x)|$. The weak topology on $\mathcal{M}^{1}(\mathcal{X})$ is thus the weakest topology for which each function $\nu \mapsto \int h d \nu$ is continuous. Consequently, weak convergence of a net $\left(\nu{\alpha}\right){\alpha}$ in $\mathcal{M}^{1}(\mathcal{X})$ to $\nu \in \mathcal{M}^{1}(\mathcal{X})$ means $$ \lim {\alpha} \int h d \nu_{\alpha}=\int h d \nu
$$
for every $h \in C_{b}(\mathcal{X})$ (here and elsewhere we omit the directed set on which a net is defined from the notation). Because $\int h d \nu_{1}=\int h d \nu_{2}$ for $\nu_{1}, \nu_{2} \in \mathcal{M}^{1}(\mathcal{X})$ and every $h \in C_{b}(\mathcal{X})$ implies that $\nu_{1}=\nu_{2}$, this topology is Hausdorff and the limit is unique. Moreover, the weak topology is separable metrizable e.g. by the Prohorov metric, see e.g. [69], Theorem II.6.2, and polish if $\mathcal{X}$ is polish; see e.g. [69], Theorem II.6.5, [26], Corollary 11.5.5. The relatively compact subsets of $\mathcal{M}^{1}(\mathcal{X})$ are exactly the tight ones, provided $\mathcal{X}$ is polish, where $\Gamma \subset \mathcal{M}^{1}(\mathcal{X})$ is called tight if for every $\varepsilon>0$ there exists a compact set $A \subset \mathcal{X}$ such that $\sup _{\nu \in \Gamma} \nu(\mathcal{X} \backslash A) \leq \varepsilon$; see e.g. [69], Theorem II.6.7, [26], Theorem 11.5.4.

微积分网课代修|极限理论代写Limit Theory代考|MATH7710 Why Stable Convergence?

微积分网课代修|极限理论代写Limit Theory代考|Why Stable Convergence?


本章是介绍性的。我们使研究稳定收敛的动机更加精确,并阐述了它的一些特征。除了 例 $1.2$ 之外,没有给出任何证明,只引用了后面可以找到证明的章节。
我们的出发点是经典的中心极限定理。为此,让 $\left(Z_{k}\right) k \geq 1$ 是一系列独立同分布的实 随机变量,定义在某个概率空间上 $(\Omega, \mathcal{F}, P)$. 认为 $Z 1 \in \mathcal{L}^{2}(P)$ 并设置 $\mu=E Z_{1}$ 和 $\sigma^{2}=\operatorname{Var} Z_{1}$. 为了排除几乎可以肯定是恒定变量的琐碎情况,还假设 $\sigma^{2}>0$. 然后 经典中心极限定理说
$$
\lim n \rightarrow \infty P\left(\frac{1}{n^{1 / 2}} \sum k=1^{n} \frac{Z_{k}-\mu}{\sigma} \leq x\right)=\Phi(x)=\int_{-\infty}^{x} \varphi(u) d u \text { for all } x \in \mathbb{R}
$$
在哪里 $\varphi(u)=\frac{1}{\sqrt{2 \pi}} \exp \left(-\frac{1}{2} u^{2}\right), u \in \mathbb{R}$ ,表示标准正态分布的密度。通常将这种概 率的收敛以一种更抽象的方式写成随机变量分布的收敛,即
$$
\frac{1}{n^{1 / 2}} \sum_{k=1}^{n} \frac{Z_{k}-\mu}{\sigma} \stackrel{d}{\rightarrow} N(0,1) \quad \text { as } n \rightarrow \infty
$$
在哪里 $N(0,1)$ 表示标准正态分布,或为
$$
\frac{1}{n^{1 / 2}} \sum_{k=1}^{n} \frac{Z_{k}-\mu}{\sigma} \stackrel{d}{\rightarrow} N \quad \text { as } n \rightarrow \infty
$$


微积分网课代修|极限理论代写Limit Theory代考|Weak Convergence of Markov Kernels


如前一章所述,随机变量的稳定收敛可以看作是条件分布给出的马尔可夫核的合适收 敛。让 $(\Omega, \mathcal{F}, P)$ 是一个概率空间,让 $\mathcal{X}$ 是一个可分离的可度量拓扑空间,带有它的 Borel $\sigma-$-场地 $\mathcal{B}(\mathcal{X})$. 在本章中,我们简要描述了一组马尔可夫核 (转换核) 上的弱拓 扑,来自 $(\Omega, \mathcal{F})$ 至 $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$
让我们首先回顾一下集合上的弱拓扑 $\mathcal{M}^{1}(\mathcal{X})$ 在所有概率测度中 $\mathcal{B}(\mathcal{X})$. 是函数生成的 拓扑
$$
\nu \mapsto \int h d \nu, \quad h \in C_{b}(\mathcal{X})
$$
在哪里 $C_{b}(\mathcal{X})$ 表示所有连续有界函数的空间 $h: \mathcal{X} \rightarrow \mathbb{R}$ 配备超规范
$|h|{\text {sup }}:=\sup x \in \mathcal{X}|h(x)|$. 弱拓扑 $\mathcal{M}^{1}(\mathcal{X})$ 因此是每个函数的最弱拓扑 $\nu \mapsto \int h d \nu$ 是连续的。因此,网络的弱收敛 $(\nu \alpha) \alpha$ 在 $\mathcal{M}^{1}(\mathcal{X})$ 至 $\nu \in \mathcal{M}^{1}(\mathcal{X})$ 方法 $$ \lim \alpha \int h d \nu{\alpha}=\int h d \nu
$$
对于每个 $h \in C_{b}(\mathcal{X})$ (在这里和其他地方,我们省略了从符号中定义网络的有向
集)。因为 $\int h d \nu_{1}=\int h d \nu_{2}$ 为了 $\nu_{1}, \nu_{2} \in \mathcal{M}^{1}(\mathcal{X})$ 和每一个 $h \in C_{b}(\mathcal{X})$ 暗示
$\nu_{1}=\nu_{2}$ ,这个拓扑是豪斯多夫,极限是唯一的。此外,弱拓扑是可分离的,例如通过
Prohorov 度量可度量,参见例如 [69],定理 II.6.2,如果 $\mathcal{X}$ 是波兰语;参见例如
[69],定理 II.6.5,[26],推论 11.5.5。相对紧凑的子集 $\mathcal{M}^{1}(\mathcal{X})$ 正好是紧的,前提是
$\mathcal{X}$ 是波兰语,在哪里 $\Gamma \subset \mathcal{M}^{1}(\mathcal{X})$ 被称为紧,如果对于每个 $\varepsilon>0$ 存在一个紧集
$A \subset \mathcal{X}$ 这样 $\sup _{\nu \in \Gamma} \nu(\mathcal{X} \backslash A) \leq \varepsilon$; 参见例如 [69]、定理 II.6.7、 [26]、定理 11.5.4。

微积分网课代修|极限理论代写Limit Theory代考|MATH7710 Why Stable Convergence?
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