### 统计代写|应用随机过程代写Stochastic process代考|MATH477

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|应用随机过程代写Stochastic process代考|An Introduction to Stationary Processes

A stochastic process $\left{X_{t}, t \in T\right}$ with $E X_{t}^{2}<\infty$ for all $t \in T$ is called covariance stationary or stationary in the wide-sense or weakly stationary if its covariance function $C_{s, t}=E\left(X_{t} X_{s}\right)$ depends only on the difference $|t-s|$ for all $t, s \in T$. Note that in our definition we have taken a zero mean stochastic process.

(a) Electrical pulses in communication theory are often postulated to describe a stationary process. Of course, in any physical system there is a transient period at the beginning of a signal. Since typically this has a short duration compared to the signal length, a stationary model may be appropriate. In electrical communication theory, often both the electrical potential and the current are represented as complex variables. Here we may encounter complex-valued stationary processes.
(b) The spatial and/or planar distributions of stars of galaxies, plants and animals, are often stationary. Time parameter set $T$ might be Euclidean space, the surface of a sphere or the plane.

A stationary distribution may be postulated for the height of a wave and $T$ is taken to be a set of longitudes and latitudes, again two dimensional.
(c) Economic time series, such as unemployment, gross national product, national income etc., are often assumed to correspond to a stationary process, at least after some correction for long-term growth has been made.

## 统计代写|应用随机过程代写Stochastic process代考|Ergodicity

The behavior in which sample averages formed from a process converge to some underlying parameter of the process is termed ergodic. To make inference about the underlying laws governing an ergodic process, one need not observe separate independent replications of entire processes or sample paths. Instead, one need only observe a single realization of the process, but over a sufficiently long span of time. Thus, it is an important practical problem to determine conditions that lead to a stationary process being ergodic. The theory of stationary processes has a prime goal the clarification of ergodic behavior and the prediction problem for processes falling in the wide range of extremeties.

In covariance stationary process usually the added condition that $E\left(X_{t}\right)$ does not depend on $t$ is imposed. But it should be noted that in order for a stochastic process with $E\left(X_{t}^{2}\right)<\infty$ to be covariance stationary it is not necessary that its mean function $m(t)=E\left(X_{t}\right)$ be a constant. Consider the example: $X(t)=$ $\cos \left(\frac{2 \pi t}{L}\right)+Y(t)$, where $Y(t)=N(t+L)-N(t),{N(t), t \geq 0}$ be a Poisson process with intensity parameter $\lambda$ (to be defined in Chapter 7 ) and $L$ is a positive constant. Its mean function $m(t)=E\left(X_{t}\right)=\lambda(t+L)-\lambda(t)+\cos \left(\frac{2 \pi t}{L}\right)$ is functionally dependent on $t$. But \begin{aligned} \operatorname{Cov}(X(t), X(s)) &=\operatorname{Cov}(Y(t), Y(s)) \ &=\left{\begin{aligned} \lambda(L-|t-s|) & \text { if }|t-s| \leq L \ 0 & \text { if }|t-s|>L \end{aligned}\right. \end{aligned}
depends on $t-s$ only.

## 统计代写|应用随机过程代写Stochastic process代考|An Introduction to Stationary Processes

(a) 通信理论中的电脉中通常被假设为描述一个平稳的过程。当然，在任何物理系统中，信号开始时都有一个瞬态 周期。由于与信号长度相比，这通常具有较短的持续时间，因此固定模型可能是合适的。在电通信理论中，电势和 电流通常都表示为复变量。在这里，我们可能会遇到复值平稳过程。
(b) 星系、植物和动物的恒星的空间和/或平面分布通常是静止的。时间参数集 $T$ 可能是欧几里得空间、球面或平 面。

(c) 经济时间序列，例如失业、国民生产总值、国民收入等，通常被假定为对应于一个平稳的过程，至少在对长期 增长进行了一些修正之后是这样。

## 统计代写|应用随机过程代写Stochastic process代考|Ergodicity

lend{aligned $}$
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## MATLAB代写

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