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# 数学代写|概率论代考Probability Theory代写|Zero-or-one laws

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## 数学代写|概率论代考Probability Theory代写|Zero-or-one laws

In this chapter we adopt the notation $N$ for the set of strictly positive integers, and $N^0$ for the set of positive integers; used as an index set, each is endowed with the natural ordering and interpreted as a discrete time parameter. Similarly, for each $n \in N, N$ denotes the ordered set of integers from 1 to $n$ (both inclusive); $N_n^0$ that of integers from 0 to $n$ (both inclusive); and $N_n^{\prime}$ that of integers beginning with $n+1$.

On the probability triple $(\Omega, \mathscr{F}, \mathscr{P})$, a sequence $\left{X_n, n \in N\right}$ where each $X_n$ is an r.v. (defined on $\Omega$ and finite a.e.), will be called a (discrete parameter) stochastic process. Various Borel fields connected with such a process will now be introduced. For any sub-B.F. $\mathscr{G}$ of $\bar{H}$, we shall write
$$X \in G$$
and use the expression ” $X$ belongs to $\mathscr{G}$ ” or ” $G$ contains $X$ ” to mean that $X^{-1}(\bar{B}) \subset \mathscr{G}$ (see Sec. 3.1 for notation): in the customary language $X$ is said as follows:

$=$ the augmented B.F. generated by the family of r.v.’s $\left{X_k, k \in N_n\right}$; that is, $\bar{F}_n$ is the smallest B.F.G containing all $X_k$ in the family and all null sets;
$=$ the augmented B.F. generated by the family of r.v.’s $\left{X_k, k \in N_n^{\prime}\right}$.

## 数学代写|概率论代考Probability Theory代写|Basic notions

From now on we consider only a stationary independent process $\left{X_n, n \in\right.$ $N}$ on the concrete probability triple specified in the preceding section. The common distribution of $X_n$ will be denoted by $\mu$ (p.m.) or $F$ (d.f.); when only this is involved, we shall write $X$ for a representative $X_n$, thus $\mathscr{E}(X)$ for $\mathscr{E}\left(X_n\right)$.
Our interest in such a process derives mainly from the fact that it underlies another process of richer content. This is obtained by forming the successive partial sums as follows:
$$S_n=\sum_{j=1}^n X_j, \quad n \in N .$$
An initial r.v. $S_0 \equiv 0$ is adjoincd whenever this serves notational convenience, as in $X_n=S_n-S_{n-1}$ for $n \in N$. The sequence $\left{S_n, n \in N\right}$ is then a very familiar object in this book, but now we wish to find a proper name for it. An officially correct one would be “stochastic process with stationary independent differences”; the name “homogeneous additive process” can also be used. We have, however, decided to call it a “random walk (process)”, although the use of this term is frequently restricted to the case when $\mu$ is of the integer lattice type or even more narrowly a Bernoullian distribution.

DETINTIIO Of RANDOM WALK. A random walk is the process $\left{S_n, n \in N\right}$ defined in (1) where $\left{X_n, n \in N\right}$ is a stationary independent process. By convention we set also $S_0 \equiv 0$.

A similar definition applies in a Euclidean space of any dimension, but we shall be concerned only with $\mathscr{R}^1$ except in some exercises later.

Let us observe that even for an independent process $\left{X_n, n \in N\right}$, its remote field is in general different from the remote field of $\left{S_n, n \in N\right}$, where $S_n=\sum_{j=1}^n X_j$. They are almost the same, being both almost trivial, for a stationary independent process by virtue of Theorem 8.1.4, since the remote field of the random walk is clearly contained in the permutable field of the corresponding stationary independent process.

We add that, while the notion of remoteness applies to any process, “(shift)-invariant” and “permutable” will be used here only for the underlying “coordinate process” $\left{\omega_n, n \in N\right}$ or $\left{X_n, n \in N\right}$.
The following relation will be much used below, for $m<n$ :
$$S_{n-m}\left(\tau^m \omega\right)=\sum_{j=1}^{n-m} X_j\left(\tau^m \omega\right)=\sum_{j=1}^{n-m} X_{j+m}(\omega)=S_n(\omega)-S_m(\omega)$$

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|Zero-or-one laws

$$X \in G$$

$=$ 由r.v.家族产生的增广B.F.。’s $\left{X_k, k \in N_n\right}$;即$\bar{F}_n$是包含族中所有$X_k$和所有空集的最小B.F.G;
$=$由r.v.家族产生的增强B.F.。网址是$\left{X_k, k \in N_n^{\prime}\right}$。

## 数学代写|概率论代考Probability Theory代写|Basic notions

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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。