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# 数学代写|概率论代考Probability Theory代写|MATH6710 Kolmogorov’s 0–1 Law

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## 数学代写|概率论代考Probability Theory代写|Kolmogorov’s 0–1 Law

With the Borel-Cantelli lemma, we have seen a first $0-1$ law for independent events. We now come to another $0-1$ law for independent events and for independent $\sigma$ algebras. To this end, we first introduce the notion of the tail $\sigma$-algebra.

Definition 2.34 (Tail $\sigma$-algebra) Let $I$ be a countably infinite index set and let $\left(\mathcal{A}i\right){i \in I}$ be a family of $\sigma$-algebras. Then
$$\mathcal{T}\left(\left(\mathcal{A}i\right){i \in I}\right):=\bigcap_{\substack{J \subset I \ # J<\infty}} \sigma\left(\bigcup_{j \in I \backslash J} \mathcal{A}j\right)$$ is called the tail $\sigma$-algebra of $\left(\mathcal{A}_i\right){i \in I}$. If $\left(A_i\right){i \in I}$ is a family of events, then we define $$\mathcal{T}\left(\left(A_i\right){i \in I}\right):=\mathcal{T}\left(\left(\left{\emptyset, A_i, A_i^c, \Omega\right}\right){i \in I}\right) .$$ If $\left(X_i\right){i \in I}$ is a family of random variables, then we define $\mathcal{T}\left(\left(X_i\right){i \in I}\right):=$ $\mathcal{T}\left(\left(\sigma\left(X_i\right)\right){i \in I}\right)$
occurrence is independent of any fixed finite subfamily of the $X_i$. To put it differently, for any finite subfamily of the $X_i$, we can change the values of the $X_i$ arbitrarily without changing whether $A$ occurs or not.

## 数学代写|概率论代考Probability Theory代写|Example: Percolation

Consider the $d$-dimensional integer lattice $\mathbb{Z}^d$, where any point is connected to any of its $2 d$ nearest neighbors by an edge. If $x, y \in \mathbb{Z}^d$ are nearest neighbors (that is, $|x-y|_2=1$ ), then we denote by $e=\langle x, y\rangle=\langle y, x\rangle$ the edge that connects $x$ and $y$. Formally, the set of edges is a subset of the set of subsets of $\mathbb{Z}^d$ with two elements:
$$E=\left{{x, y}: x, y \in \mathbb{Z}^d \text { with }|x-y|_2=1\right} .$$
Somewhat more generally, an undirected graph $G$ is a pair $G=(V, E)$, where $V$ is a set (the set of “vertices” or nodes) and $E \subset{{x, y}: x, y \in V, x \neq y}$ is a subset of the set of subsets of $V$ of cardinality two (the set of edges or bonds).
Our intuitive understanding of an edge is a connection between two points $x$ and $y$ and not an (unordered) pair ${x, y}$. To stress this notion of a connection, we use a different symbol from the set brackets. That is, we denote the edge that connects $x$ and $y$ by $\langle x, y\rangle=\langle y, x\rangle$ instead of ${x, y}$.

Our graph $(V, E)$ is the starting point for a stochastic model of a porous medium. We interpret the edges as tubes along which water can flow. However, we want the medium not to have a homogeneous structure, such as $\mathbb{Z}^d$, but an amorphous structure. In order to model this, we randomly destroy a certain fraction $1-p$ of the tubes (with $p \in[0,1]$ a parameter) and keep the others. Water can flow only through the remaining tubes. The destroyed tubes will be called “closed”, the others “open”. The fundamental question is: For which values of $p$ is there a connected infinite system of tubes along which water can flow? The physical interpretation is that if we throw a block of the considered material into a bathtub, then the block will soak up water; that is, it will be wetted inside. If there is no infinite open component, then the water may wet only a thin layer at the surface. See Fig. $2.1$ for a computer simulation of the percolation model.

# 概率论代写

## 数学代写|概率论代考概率论代写|Kolmogorov的0-1定律

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$$\mathcal{T}\left(\left(\mathcal{A}i\right){i \in I}\right):=\bigcap_{\substack{J \subset I \ # J<\infty}} \sigma\left(\bigcup_{j \in I \backslash J} \mathcal{A}j\right)$$ 叫做尾巴 $\sigma$-代数 $\left(\mathcal{A}_i\right){i \in I}$。如果 $\left(A_i\right){i \in I}$ 是一系列事件，然后我们来定义 $$\mathcal{T}\left(\left(A_i\right){i \in I}\right):=\mathcal{T}\left(\left(\left{\emptyset, A_i, A_i^c, \Omega\right}\right){i \in I}\right) .$$ 如果 $\left(X_i\right){i \in I}$ 是一个随机变量族，然后我们定义 $\mathcal{T}\left(\left(X_i\right){i \in I}\right):=$ $\mathcal{T}\left(\left(\sigma\left(X_i\right)\right){i \in I}\right)$的任何固定的有限子族的 $X_i$。换句话说，对于任何有限的子族 $X_i$的值，我们可以更改 $X_i$ 随意而不改变 $A$ 是否发生。

## 数学代写|概率论代考概率论代写|示例:渗滤

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$$E=\left{{x, y}: x, y \in \mathbb{Z}^d \text { with }|x-y|_2=1\right} .$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。