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# 数学代写| 随机过程代考|The law of the iterated logarithm

A population starts with one individual at time $n=0: Z_{0}=1$.

After one unit of time (at time $n=1$ ) the sole individual produces $Z_{1}$ identical clones of itself and dies. $Z_{1}$ is an $\mathbb{N}_{0}$-valued random variable.

(a) If $Z_{1}$ happens to be equal to 0 the population is dead and nothing happens at any future time $n \geq 2$.

(b) If $Z_{1}>0$, a unit of time later, each of $Z_{1}$ individuals gives birth to a random number of children and dies. The first one has $Z_{1,1}$ children, the second one $Z_{1,2}$ children, etc. The last, $Z_{1}^{\text {th }}$ one, gives birth to $Z_{1, Z_{1}}$ children. We assume that the distribution of the number of children is the same for each individual in every generation and independent of either the number of individuals in the generation and of the number of children the others have. This distribution, shared by all $Z_{n, i}$ and $Z_{1}$, is called the offspring distribution. The total number of individuals in the second generation is now
$$Z_{2}=\sum_{k=1}^{Z_{1}} Z_{1, k}$$
(c) The third, fourth, etc. generations are produced in the same way. If it ever happens that $Z_{n}=0$, for some $n$, then $Z_{m}=0$ for all $m \geq n$ – the population is extinct. Otherwise,
$$Z_{n+1}=\sum_{k=1}^{Z_{n}} Z_{n, k}$$

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## 数学代写| 随机过程代考|growth rate

Let $S_{n}=X_{1}+X_{2}+\cdots+X_{n}$ be the partial sum of independent identically distributed variables, as usual, and suppose further that $\mathbb{E}\left(X_{i}\right)=0$ and $\operatorname{var}\left(X_{i}\right)=1$ for all $i$. To date, we have two results about the growth rate of $\left{S_{n}\right}$.
Law of large numbers: $\frac{1}{n} S_{n} \rightarrow 0$ a.s. and in mean square.
Central limit theorem: $\frac{1}{\sqrt{n}} S_{n} \stackrel{\mathrm{D}}{\rightarrow} N(0,1)$
Thus the sequence $U_{n}=S_{n} / \sqrt{n}$ enjoys a random fluctuation which is asymptotically regularly distributed. Apart from this long-term trend towards the normal distribution, the sequence $\left{U_{n}\right}$ may suffer some large but rare fluctuations. The law of the iterated logarithm is an extraordinary result which tells us exactly how large these fluctuations are. First note that,
$$U=\limsup {n \rightarrow \infty} \frac{U{n}}{\sqrt{2 \log \log n}}$$

## 数学代写| 随机过程代考|iterated logarithm

Theorem. Law of the iterated logarithm. If $X_{1}, X_{2}, \ldots$ are independent identically distributed random variables with mean 0 and variance 1 then
$$\mathbb{P}\left(\limsup {n \rightarrow \infty} \frac{S{n}}{\sqrt{2 n \log \log n}}=1\right)=1$$
The proof is long and difficult and is omitted (but see the discussion in Billingsley (1995) or Laha and Rohatgi (1979)). The theorem amounts to the assertion that
$$A_{n}=\left{S_{n} \geq c \sqrt{2 n \log \log n}\right}$$
occurs for infinitely many values of $n$ if $c<1$ and for only finitely many values of $n$ if $c>1$, with probability 1 . It is an immediate corollary of (1) that
$$\mathbb{P}\left(\liminf {n \rightarrow \infty} \frac{S{n}}{\sqrt{2 n \log \log n}}=-1\right)=1$$
just apply (1) to the sequence $-X_{1},-X_{2}, \ldots$

## 数学代写| 随机过程代考|GROWTH RATE

$$U=\limsup {n \rightarrow \infty} \frac{U {n}}{\sqrt{2 \log \log n}}$$

## 数学代写| 随机过程代考|ITERATED LOGARITHM

$$\mathbb{P}\left(\limsup {n \rightarrow \infty} \frac{S {n}}{\sqrt{2 n \log \log n}}=1\右）=1 吨H和pr○○Fs一○nG A_{n}=\left{S_{n} \geq c \sqrt{2 n \log \log n}\right} ○CC \mathbb{P}\left(\liminf {n \rightarrow \infty} \frac{S {n}}{\sqrt{2 n \log \log n}}=-1\right)=1$$

## 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。