# 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|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|Stable Convergence of Random Variables

$P^{X \mid \mathcal{G}}(\cdot, B)=P(X \in B \mid \mathcal{G})$ 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 $G \in \mathcal{G}, B \in \mathcal{B}(\mathcal{X})$,

$$P \otimes P^{X \mid \mathcal{G}}=P \otimes \delta_{X} \text { on } \mathcal{G} \otimes \mathcal{B}(\mathcal{X})$$

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

$$\lim n \rightarrow \infty E(f E(h(X n) \mid \mathcal{G}))=\int f \int h(x) K(\cdot, d x) d P$$

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

$\varphi:(\Omega \times \mathcal{Y}, \mathcal{G} \otimes \mathcal{B}(\mathcal{Y})) \rightarrow(\mathcal{X}, \mathcal{B}(\mathcal{X}))$ 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 $K(\omega, \cdot)=N(0,1)^{\varphi(\omega, \cdot)}=N\left(0, \eta^{2}(\omega)\right)$.