微积分网课代修|极限理论代写Limit Theory代考|MATH407 Stable Convergence of Random Variables

微积分网课代修|极限理论代写Limit Theory代考|MATH407 Stable Convergence of Random Variables

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微积分网课代修|极限理论代写Limit Theory代考|MATH407 Stable Convergence of Random Variables

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

Based on the notions and results of Chap. 2 we may now introduce and deeply investigate the mode of stable convergence of random variables. Starting from the papers [76-78] expositions can be found in [4,13, 48, 50, 57].

Let $\mathcal{X}$ still be a separable metrizable topological space and fix a metric $d$ that induces the topology on $\mathcal{X}$. For an $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$-valued random variable $X$ and a sub$\sigma$-field $\mathcal{G} \subset \mathcal{F}$ let $P^{X \mid \mathcal{G}}$ denote the conditional distribution which exists, for example, provided that $\mathcal{X}$ is polish. It is a Markov kernel from $(\Omega, \mathcal{G})$ to $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ such that $P^{X \mid \mathcal{G}}(\cdot, B)=P(X \in B \mid \mathcal{G})$ almost surely for all $B \in \mathcal{B}(\mathcal{X})$. The conditional distribution is $P$-almost surely unique by Lemma $2.1$ (b) and characterized by the Radon-Nikodym equations
$$
\int_{G} P^{X \mid \mathcal{G}}(\omega, B) d P(\omega)=P\left(X^{-1}(B) \cap G\right) \quad \text { for every } G \in \mathcal{G}, B \in \mathcal{B}(\mathcal{X}),
$$
or, what is the same,
$$
P \otimes P^{X \mid \mathcal{G}}=P \otimes \delta_{X} \text { on } \mathcal{G} \otimes \mathcal{B}(\mathcal{X})
$$
where $\delta_{X}$ is the Dirac-kernel associated with $X$ given by $\delta_{X}(\omega):=\delta_{X(\omega)}$. If, for example, $X$ is $\mathcal{G}$-measurable, then $P^{X \mid \mathcal{G}}=\delta_{X}$. The distribution of $X$ (under $P$ ) is denoted by $P^{X}$. In the sequel we restrict our attention to sequences of random variables, all defined on the same probability space $(\Omega, \mathcal{F}, P)$.

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

Definition 3.1 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 $K \in \mathcal{K}^{1}(\mathcal{G})$, written as $X{n} \rightarrow K \mathcal{G}$-stably, if $P^{X_{n} \mid \mathcal{G}} \rightarrow K$ weakly as $n \rightarrow \infty$. In case $K$ does not depend on $\omega \in \Omega$ in the sense that $K=\nu P$-almost surely for some $\nu \in \mathcal{M}^{1}(\mathcal{X})$, then $\left(X_{n}\right){n \geq 1}$ is said to converge $\mathcal{G}$-mixing to $\nu$, and we write $X{n} \rightarrow \nu \mathcal{G}$-mixing. Stable and mixing convergence are short for $\mathcal{F}$-stable and $\mathcal{F}$-mixing convergence, respectively.

In Definition $3.1$ and in the sequel we always assume that the conditional distributions involved exist. (Existence is not part of the subsequent assertions.)

Using Fubini’s theorem for Markov kernels (see Lemma $2.1$ (a)) and the fact that $\int h(x) P^{X_{n} \mid \mathcal{G}}(d x)=E\left(h\left(X_{n}\right) \mid \mathcal{G}\right), \mathcal{G}$-stable convergence $X_{n} \rightarrow K$ reads
$$
\lim {n \rightarrow \infty} E\left(f E\left(h\left(X{n}\right) \mid \mathcal{G}\right)\right)=\int f \int h(x) K(\cdot, d x) d P
$$
for every $f \in \mathcal{L}^{1}(P)$ and $h \in C_{b}(\mathcal{X})$. The choice $f=1$ implies $X_{n} \stackrel{d}{\rightarrow} P K$, that is, $P^{X_{n}} \rightarrow P K$ weakly. Here and elsewhere the reference measure for distributional convergence is always $P$. The $\mathcal{G}$-mixing convergence $X_{n} \rightarrow \nu$ means
$$
\lim {n \rightarrow \infty} E\left(f E\left(h\left(X{n}\right) \mid \mathcal{G}\right)\right)=\int f d P \int h d \nu
$$
for every $f \in \mathcal{L}^{1}(P)$ and $h \in C_{b}(\mathcal{X})$, which implies $X_{n} \stackrel{d}{\rightarrow} \nu$. Because $P^{X_{n} \mid \mathcal{G}}=$ $E\left(\delta_{X_{n}} \mid \mathcal{G}\right)$ in the sense of Definition 2.4, $\mathcal{G}$-stable convergence $X_{n} \rightarrow K$ can also be read as $E\left(\delta_{X_{n}} \mid \mathcal{G}\right) \rightarrow K$ weakly. In the extreme case $\mathcal{G}={\emptyset, \Omega}, \mathcal{G}$-stable convergence $X_{n} \rightarrow K$ coincides with distributional convergence $X_{n} \stackrel{d}{\rightarrow} \nu$, because $K=\nu$ for some $\nu \in \mathcal{M}^{1}(\mathcal{X})$ by $\mathcal{G}$-measurability of $K$.

Typical limit kernels for $\mathcal{G}$-stable convergence $X_{n} \rightarrow K$ are of the type $K(\omega, \cdot)=\mu^{\varphi(\omega, \cdot)}$, where $\mu \in \mathcal{M}^{1}(\mathcal{Y}), \mathcal{Y}$ is a separable metrizable space and $\varphi:(\Omega \times \mathcal{Y}, \mathcal{G} \otimes \mathcal{B}(\mathcal{Y})) \rightarrow(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ is some “concrete” measurable map. Here $\mu^{\varphi(\omega, \cdot)}$ is the image measure of $\mu$ under the map $\varphi(\omega, \cdot)$ so that $K(\omega, B)=$ $\mu({y \in \mathcal{Y}: \varphi(\omega, y) \in B})$. In fact, every kernel has such a representation provided $\mathcal{X}$ is polish; see [51], Lemma 3.22)). In particular, if $\mathcal{X}=\mathcal{Y}=\mathbb{R}, \mu=N(0,1)$ and $\varphi(\omega, x):=\eta(\omega) x$ for some $\mathcal{G}$-measurable and nonnegative real random variable $\eta$, we obtain the Gauss-kernel $K(\omega, \cdot)=N(0,1)^{\varphi(\omega, \cdot)}=N\left(0, \eta^{2}(\omega)\right)$.

微积分网课代修|极限理论代写Limit Theory代考|MATH407 Stable Convergence of Random Variables

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


基于第一章的概念和结果。2 我们现在可以介绍并深入研究随机变量的稳定收敛模式。 从论文 [76-78] 开始,可以在 $[4,13,48,50,57]$ 中找到说明。
让 $\mathcal{X}$ 仍然是一个可分离的可度量拓扑空间并固定一个度量 $d$ 导致拓扑 $\mathcal{X}$. 为 $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ 值随机变量 $X$ 和一个子 $\sigma$-场地 $\mathcal{G} \subset \mathcal{F}$ 让 $P^{X \mid \mathcal{G}}$ 表示存在的条件分布,例如,假设 $\mathcal{X}$ 是 波兰语。它是一个马尔可夫核 $(\Omega, \mathcal{G})$ 至 $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ 这样
$P^{X \mid \mathcal{G}}(\cdot, B)=P(X \in B \mid \mathcal{G})$ 几乎可以肯定的是 $B \in \mathcal{B}(\mathcal{X})$. 条件分布是 $P-$
Lemma 几平肯定是独一无二的 $2.1$ (b) 并以 Radon-Nikodym 方程为特征
$\int_{G} P^{X \mid \mathcal{G}}(\omega, B) d P(\omega)=P\left(X^{-1}(B) \cap G\right) \quad$ for every $G \in \mathcal{G}, B \in \mathcal{B}(\mathcal{X})$,
或者,有什么相同之处,
$$
P \otimes P^{X \mid \mathcal{G}}=P \otimes \delta_{X} \text { on } \mathcal{G} \otimes \mathcal{B}(\mathcal{X})
$$
在哪里 $\delta_{X}$ 是与相关的狄拉克核 $X$ 由 $\delta_{X}(\omega):=\delta_{X(\omega)}$. 例如,如果, $X$ 是 $\mathcal{G}$ – 可测量 的,那么 $P^{X \mid \mathcal{G}}=\delta_{X}$. 的分布 $X$ (在下面 $P$ ) 表示为 $P^{X}$. 在续集中,我们将注意力限 制在随机变量序列上,所有变量都定义在同一个概率空间上 $(\Omega, \mathcal{F}, P)$.


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


定义 $3.1$ 让 $\mathcal{G} \subset \mathcal{F}$ 成为一个子 $\sigma$-场地。一个序列 $\left(X_{n}\right) n \geq 1$ 的 $(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ 据哾有值 的随机变量会收敛 $\mathcal{G}$-稳定地 $K \in \mathcal{K}^{1}(\mathcal{G})$ ,写为 $X n \rightarrow K \mathcal{G}$-䅎定,如果 $P^{X_{n} \mid \mathcal{G}} \rightarrow K$ 弱如 $n \rightarrow \infty$. 如果 $K$ 不依赖于 $\omega \in \Omega$ 在某种意义上说 $K=\nu P$-对于某 些人来说几平可以肯定 $\nu \in \mathcal{M}^{1}(\mathcal{X})$ ,然后 $\left(X_{n}\right) n \geq 1$ 据说收敛 $\mathcal{G}$-混合到 $\nu$ ,我们 写 $X n \rightarrow \nu \mathcal{G}$-混合。稳定和混合收敛的简称 $\mathcal{F}$-稳定且 $\mathcal{F}$-混合收敛,分别。
在定义中 $3.1$ 在续集中,我们总是假设所涉及的条件分布存在。(存在不是后续断言的 一部分。)
对马尔可夫核使用 Fubini 定理 (参见引理 $2.1(\mathrm{a})$ ) 以及以下事实 $\int h(x) P^{X_{n} \mid \mathcal{G}}(d x)=E\left(h\left(X_{n}\right) \mid \mathcal{G}\right), \mathcal{G}$-稳定收敛 $X_{n} \rightarrow K_{\text {读 }}$
$$
\lim n \rightarrow \infty E(f E(h(X n) \mid \mathcal{G}))=\int f \int h(x) K(\cdot, d x) d P
$$
对于每个 $f \in \mathcal{L}^{1}(P)$ 和 $h \in C_{b}(\mathcal{X})$. 选择 $f=1$ 暗示 $X_{n} \stackrel{d}{\rightarrow} P K$ ,那是, $P^{X_{n}} \rightarrow P K$ 弱。在这里和其他地方,分布收敛的参考度量总是 $P$. 这 $\mathcal{G}$-混合收敛 $X_{n} \rightarrow \nu$ 方法
$$
\lim n \rightarrow \infty E(f E(h(X n) \mid \mathcal{G}))=\int f d P \int h d \nu
$$
对于每个 $f \in \mathcal{L}^{1}(P)$ 和 $h \in C_{b}(\mathcal{X})$ ,这意味着 $X_{n} \stackrel{d}{\rightarrow} \nu$. 因为 $P^{X_{n} \mid \mathcal{G}}=$ $E\left(\delta_{X_{n}} \mid \mathcal{G}\right)$ 在定义 $2.4$ 的意义上, $\mathcal{G}$-稳定收敛 $X_{n} \rightarrow K$ 也可以读作 $E\left(\delta_{X_{n}} \mid \mathcal{G}\right) \rightarrow K$ 弱。在极端情况下 $\mathcal{G}=\emptyset, \Omega, \mathcal{G}$-稳定收玫 $X_{n} \rightarrow K$ 与分布收敛一 致 $X_{n} \stackrel{d}{\rightarrow} \nu$ , 因为 $K=\nu$ 对于一些 $\nu \in \mathcal{M}^{1}(\mathcal{X})$ 经过 $\mathcal{G}$ – 可测量性 $K$.
典型的极限核 $\mathcal{G}$-稳定收敛 $X_{n} \rightarrow K$ 是类型 $K(\omega, \cdot)=\mu^{\varphi(\omega,)}$ ,在哪里 $\mu \in \mathcal{M}^{1}(\mathcal{Y}), \mathcal{Y}$ 是一个可分的可度量空间,并且
$\varphi:(\Omega \times \mathcal{Y}, \mathcal{G} \otimes \mathcal{B}(\mathcal{Y})) \rightarrow(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ 是一些“具体的”可测量的地图。这里 $\mu^{\varphi(\omega, \cdot)}$ 是图像度量 $\mu$ 在地图下 $\varphi(\omega, \cdot)$ 以便 $K(\omega, B)=\mu(y \in \mathcal{Y}: \varphi(\omega, y) \in B)$. 事实上, 每个内核都提供了这样的表示 $\mathcal{X}$ 是波兰语; 见 [51],引理 3.22))。特别是,如果 $\mathcal{X}=\mathcal{Y}=\mathbb{R}, \mu=N(0,1)$ 和 $\varphi(\omega, x):=\eta(\omega) x$ 对于一些 $\mathcal{G}$ – 可测量且非负的实数 随机变量 $\eta$ ,我们得到高斯核 $K(\omega, \cdot)=N(0,1)^{\varphi(\omega, \cdot)}=N\left(0, \eta^{2}(\omega)\right)$.

微积分网课代修|极限理论代写Limit Theory代考|MATH407 Stable Convergence of Random Variables
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