Is it right, my understanding about (order and sense) stationarity?

Question

Before writing this post, I want to say thank you for every people who answered and commented to me. Thanks to mathematicians being active in this site, I think that I finally understood a little more than before about stationarity of random process.

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In first-order stationarity,

A continuous-time random process $\{X(t), t\in\mathbb{R}\}$ is strict-sense stationary if $$ \forall t_1\in\mathbb{R}, \forall \Delta\in\mathbb{R},\\ F_{X(t_1)}(x_1) = F_{X(t_1+\Delta)}(x_1) $$

A continuous-time random process $\{X(t), t\in\mathbb{R}\}$ is wide-sense stationary if $$ E[X(t_1)]=E[X(t_1+\Delta)]\\ $$

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In second-order stationarity,

A continuous-time random process $\{X(t), t\in\mathbb{R}\}$ is strict-sense stationary if $$ \forall t_1, t_2\in\mathbb{R}, \forall \Delta\in\mathbb{R},\\ F_{X(t_1)X(t_2)}(x_1, x_2) = F_{X(t_1+\Delta)X(t_2+\Delta)}(x_1, x_2) $$

A continuous-time random process $\{X(t), t\in\mathbb{R}\}$ is wide-sense stationary if $$ E[X(t_1)]=E[X(t_2)]\\ E[X(t_1)X(t_2)]=E[X(t_1+\Delta)X(t_2+\Delta)] $$

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In n-th-order stationarity,

A continuous-time random process $\{X(t), t\in\mathbb{R}\}$ is strict-sense stationary if $$ \forall t_1, t_2, \cdots, t_n\in\mathbb{R}, \forall \Delta\in\mathbb{R},\\ F_{X(t_1)X(t_2)\cdots X(t_n)}(x_1, x_2, \cdots, x_n) = F_{X(t_1+\Delta)X(t_2+\Delta)\cdots X(t_n+\Delta)}(x_1, x_2, \cdots, x_n) $$

A continuous-time random process $\{X(t), t\in\mathbb{R}\}$ is wide-sense stationary if $$ E[X(t_1)]=E[X(t_2)]\\ E[X(t_1)X(t_2)]=E[X(t_1+\Delta)X(t_2+\Delta)]\\ E[X(t_1)X(t_2)X(t_3)]=E[X(t_1+\Delta)X(t_2+\Delta)X(t_3+\Delta)]\\ \cdots\\ E[X(t_1)X(t_2)\cdots X(t_n)]=E[X(t_1+\Delta)X(t_2+\Delta)\cdots X(t_n+\Delta)] $$

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Just order is how many JOINT there exist.

Strict-sense is a perspective on PDF (CDF).

Wide-sense is a perspective on Expectation.

Right? I do hope I'm right.

Answer 1

No i dont think you are right.

I will mainly focus on wide-sense stationarity and i will use a different notation, from Probability and Random Processes by Grimmet and Stirzaker which is a commonly used book. But i think you will understand.

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Def:

The process $X = \{X(t):\: t \geq 0\}$, taking values in $\mathbb{R}$, is called strictly stationary if the families

$$\{X(t_1), X(t_2),..., X(t_n)\}\:\: \text{and} \:\: \{X(t_1+h), X(t_2+h),..., X(t_n+h)\}\:\:\:\: (*)$$

have the same joint distribution for all $t_1, t_2,...,t_n$ and $h > 0$.

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Def:

The process $X = \{X(t):\: t \geq 0\}$ is called wide-sense (or second order) stationary if, for all $t_1, t_2$ and $h > 0$

$$E(X(t_1)) = E(X(t_2))\:\: \text{and} \:\: \text{cov}(X(t_1),X(t_2)) = \text{cov}(X(t_1 + h),X(t_2 + h))$$

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If the stationary condition $(*)$ of a random process $X(t)$ does not hold for all $n$ but holds for $n \leq k$, then we say that the process $X(t)$ is stationary to order $k$. If $X(t)$ is stationary to order $2$, then $X(t)$ is said to be wide-sense stationary (WSS). If $X(t)$ is a WSS random process, then we have

$1.$ $E[X(t)] = \mu$ (constant)

$2.$ $R_x(t_1, t_2) = E[X(t_1)X(t_2)] = R_X(|t_2 - t_1|) = R_X(\tau)$

WSS processes are completely characterized by only the first- and second-order distributions.

$\mu$ does not depend on $t$ since $F_{X(t)}(x)$ does not depend on $t$ and $R_X(t, t+\tau)$ does not depend on $t$ since $F_{X(t), X(t + \tau)}(x_1, x_2)$ does not depend on $t$.

And strict sense stationary $\Rightarrow$ WSS obviously.