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# 数学代写|随机过程Stochastic Porcesses代考|STAT507 Introduction and Stationary Time Series

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## 数学代写|随机过程Stochastic Porcesses代考|Introduction and Stationary Time Series

A typical time series can be considered to be the result of four distinct components.
(1) Trend
A smooth long-term movement covering a number of years reflecting the general tendency of the series. Growth curves are good examples of time series exhibiting trend. Naturally, characteristics of the process depend very much on the time of observation. One method of eliminating the trend factor from a time series is the method of moving averages which is well known and we shall not discuss it in some detail in this book.
(2) Cyclical Component (Seasonal Component)
The result of a periodic movement in definite time periods and can be represented by a strictly periodic function of time i.e. $X_t=X_{t+\theta}=X_{t+2 \theta}=\ldots$ (the period is $\theta$ ). Fourier analysis techniques may be employed in the investigation of a known periodic movement like a seasonal variation. Periodogram analysis is a method based on the harmonic analysis of the Fourier representation. We shall consider only sine and cosine functions later on in our studies.
(3) Oscillatory Component
It is the Irregular periodic movement of a time series. Economists and many others go for this type of non seasonal cyclic patterns. The sunspot cycle and outfall of many rivers are good examples.
(4) Random Component
Random disturbance term is the basis for the stochastic Process representation of the time series.

## 数学代写|随机过程Stochastic Porcesses代考|Stationary time series

Denote by $(\Omega, \mathscr{T}, P)$ the probability space and let $T$ be an index set. Then a real valued time series (stochastic process) is a real-valued function $X(t, \omega)$ belonging to the product space $T X \Omega$.

For fixed $t \in T, X(t, \omega)$ is a random variable defined over $\Omega$ and is denoted by $X_t$. The time series $\left{X_t, t \in T\right}$ is therefore a collection of all such random variables.
The series $\left{X_t\right}$ is said to be strictly stationary if

$F_{X_{t_1} \ldots X_{t_n}}\left(x_{t_1}, \ldots, x_{i_n}\right)=: F_{X_{t_1+h \ldots .}} x_{t_n+h}\left(x_{t_1}, \ldots, x_{t_n}\right)$ for $\left(t_1, \ldots, t_n\right)$ and $\left(t_1+h, \ldots, t_n+h\right) \subset T$ and $\left(X_{t_1}, \ldots, X_{t_n}\right)$ lies in the range of the r.v. $X_t$.
Note
(i) The distribution of any point in the index set remains the same,

(ii) From (i) it is clear that the joint distribution of a finite number of points in the index set do not involve the points themselves, instead it is a function of the distance between the points,
(iii) $t_1, \ldots, t_n$ are not necessarily consecutive.
(iv) If the second order moment $E\left(X_t^2\right)<\infty$, then
$\left.\begin{array}{l}E\left(X_t\right)=\text { constant for all } t \ V\left(X_t\right)=\text { constant for all } t\end{array}\right}$ for such a stationary time series.
Recall that $\left{X_t, t \in T\right}$ is covariance stationary if
$$E\left(X_t\right)=\text { constant }$$
and $\operatorname{Cov}\left(X_t, X_{t+h}\right)=\gamma(h)$ for all $t \in T$.

## 数学代写|随机过程Stochastic Porcesses代考|Introduction and Stationary Time Series

(1) 趋势

（2）周期性成分(Seasonal Component)

(3) 振芴分量

(4) 随机分量

## 数学代写|随机过程Stochastic Porcesses代考|Stationary time series

$F_{X_{t_1} \ldots X_{t_n}}\left(x_{t_1}, \ldots, x_{i_n}\right)=: F_{X_{t_1+h} . .} x_{t_n+h}\left(x_{t_1}, \ldots, x_{t_n}\right)$ 为了 $\left(t_1, \ldots, t_n\right)$ 和 $\left(t_1+h, \ldots, t_n+h\right) \subset T$ 和 $\left(X_{t_1}, \ldots, X_{t_n}\right)$ 位于房车范围内 $X_t$.

(i) 索引集中任意点的分布保持不变，
(ii) 从 (i) 可以看出，索引集中有限个点的联合分布不涉及点本身，而是点与点之间距离的函数，
(iii) $t_1, \ldots, t_n$ 不一定是连续的。
(iv) 如果二阶矩 $E\left(X_t^2\right)<\infty$ ， 然后
\right 缺少或无法识别的分隔符

$$E\left(X_t\right)=\text { constant }$$

## MATLAB代写

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