What is the difference between wide sense and strict sense stationary processes?

Question

What is the difference between wide sense and strict sense stationary processes (SP) ?

According to the definition (by Heinrich Meyr, Marc Moeneclaey, Stefan A. Fechtel in "Synchronization, Channel Estimation, and Signal Processing") :

• strict-sense SP = not time dependent
• wide-sense SP = not dependent on variable t (time)

Could someone please illuminate what this really means ?

Perhaps pointing out a good book in which the difference is explained ?

If an SP is not dependent on variable t (time) then how can it depend on time ( a process which is not strict sense stationary but wide sense stationary)?

Could you give an example of such a process ?

Why and when is making this difference useful/neccessary?

Answer 1

A process is stationary if:

• its mean is a constant value: $\mu_x(t)=\mu x$
• its MSV(mean square value) is a constant value.
• its variance is a constant value. $\sigma^2_x(t)=\sigma^2_x$
• its autocorrelation depends on the time distance between 2 samples : $$R_x(t_1,t_2)=R_x(\tau)$$
• its auto-covariance depends on the time distance between 2 samples at time $t_1$ and $t_2$ : $$K_x(t_1,t_2)=K_x(\tau)$$ So, with respect to information above, we cannot just say that a process is stationary if it does not depend on time, because for mean,MSV and variance a stationary process does not depend on time since its values are constant, but for autocorrelation and auto-covariance it does depend on time actually, but the function itself does not depend on time since it depends only on the time distance between 2 samples.

The process is SSS (strict-sense stationary) if its $N^{\rm th}$ order probability density function is stationary for any given $N$.

The process is WSS (wide/weak-sense-stationary) if its mean value is constant $\mu_x(t)=\mu_x$, and its autocorrelation function depends on time difference between 2 samples, $$R_x(t_1,t_2)=R_x(\tau).$$

Important: Remember that all SSS processes are also WSS, but NOT all WSS processes are SSS.There is only one exception: "Jointly Gaussian Processes". Jointly Gaussian processes have any order of probability density functions(property of SSS). So, even if a jointly gaussian is WSS it is also SSS.And also remember that white gaussian noise process is a jointly gaussian process with 0 mean. So it is WSS and also SSS. Since its mean value is 0, the we can say that its autocorrelation function and autocovariance function are equal and they do depend on time difference between 2 samples at time $t_1$ and $t_2$. $R(\tau)=K(\tau)$ where $\tau=t_1-t_2$.

In conclusion,

• if you see only the mean value is constant (not depend on time), and the autocorrelation function depends on only time difference $\tau= t_1-t_1$. $R(\tau)$. --> WSS
• if you see that the $N$-th order pdf of process is stationary for any $N$ --> SSS
• if the process is jointly gaussian --> WSS and SSS
• if the process is white gaussian noise process --> WSS and SSS with mean=0 and $$R(\tau)=K(\tau).$$

Answer 2

I think an understanding is aided by being playful with the terms.

"Strict sense" stationarity is a strict definition. A "wide sense" SP is "stationary enough" for your practical purposes. It's kind of like a first order estimate. Maybe a higher order fit is more precise, but it could create so much needless complexity.

And maybe you don't have enough observations to prove the process is stationary in the strict sense, but you have enough to say it is stationary in the wide sense.

"If an SP is not dependent on variable t (time) then how can it depend on time ( a process which is not strict sense stationary but wide sense stationary)?"

Yeah, that's confusing isn't it? But I think what it's saying is that the process can be characterized with the observations you have, which includes the variable t (representing time).

But you don't have enough observations to make definitive assertions about the underlying process, which includes time itself (not just the variable t representing time). It's philosophical.