A "wide-sense stationary process" (WSS) means that its mean is a constant, and its auto-correlation is time-invariant, that is:

$$\begin{aligned}E[x(t)] &= c \\ R_X(t_1, t_2) &= R_X(t_2 - t_1)\end{aligned}$$

And an "nth order Strict sense stationary process" (SSS) means that all moment of order n and below are time invariant, That is:

$$\begin{aligned}E\Big[X(t)\Big] &= E\Big[X(t-\tau)\Big] \\E\Big[X^2(t)\Big] &= E\Big[X^2(t-\tau)\Big] \\E\Big[X^n(t)\Big] &= E\Big[X^n(t-\tau)\Big]\end{aligned}$$

Do these two definitions leave open the possibility that a process can be WSS, and not SSS?

If so, what's an example?

  • first order stationary

    The CDF of any sample is time invariant.

    $F_X(x;~~ t) = F_X(x;~~ t+\tau)~~~\forall \tau$

    Thus, the PDF is time invariant:

    $f_X(x;~~ t) = f_X(x;~~ t+\tau)~~~\forall \tau$

    And Thus, all first order statistics are constant:



    $E\Big[X^2(t)\Big] = \text{constant}~~~~\text{(2nd moment)}$


    $E\Big[X^n(t)\Big] = \text{constant}~~~~\text{(n-th moment)}$

  • second order stationary or WSS

    The joint CDF between any two samples is time-invariant.

    $F_X(x_1, x_2;~~ t_1, t_2) = F_X(x_1, x_2;~~ t_1+\tau,~ t_2+\tau)~~~\forall \tau$

    Thus, all second order statistics depend only on a relative time difference:

    $f_X(x_1, x_2;~~ t_1, t_2) = f_X(x_1, x_2;~~ t_1+\tau,~ t_2+\tau)~~~\forall \tau$

    $R_X(t_1, t_2) = R_X(t_2 - t_2)~~~~~(\text{autocorrelation})$

    $C_X(t_1, t_2) = C_X(t_2 - t_1)~~~~\text{(autocovariance)}$

    And by marginalizing the 2nd order joint CDF distribution, we prove that second order stationary is also first order stationary:

    $F_X(x_1, x_2;~~ t_1, \infty) = F_X(x_1, x_2;~~ t_1+\tau,~ \infty)~~~\forall \tau$

    $F_X(x;~~ t) = F_X(x;~~ t+\tau)~~~\forall \tau$

    $f_X(x;~~ t) = f_X(x;~~ t+\tau)$



    Second Order Station is also called Wide-Sense Stationary

  • 'n' order stationary

    A random process is 'n' order stationary, if the joint CDF distribution for any set of 'n' samples is taken relative to the same time origin, and another set of 'n' samples is taken at a different time origin but having the same time distance to the origin as the first sample set, and the resulting joint CDF in both sample sets is the same, then the 2 sample sets are said to be time-invariant to each because only the relative distance to the time origin matters in determining the resulting distribution and not the absolute time position, that is, 'n' order stationary is defined as:

    $F_X(x_1, x_2, \cdots,~ x_n;~~ t_1, t_2, \cdots,~ t_n) = F_X(x_1, x_2, \cdots,~ x_n;~~ t_1+\tau, t_2+\tau, \cdots,~ t_n+\tau)$

    Thus, the PDF is time invariant:

    $f_X(x_1, x_2, \cdots,~ x_n;~~ t_1, t_2, \cdots,~ t_n) = f_X(x_1, x_2, \cdots,~ x_n;~~ t_1+\tau, t_2+\tau, \cdots,~ t_n+\tau)$

    And all statistics from order '1' to order 'n' are time-invariant. For example, first order statistics such as mean and variance are time-invariant constants, and all second order statistics such as autocorrelation, and autocovariance are time invariant and only rely on a relative time difference, and third order statistics which are not commonly used are time-invariant, and so on and so forth, all the way up to order 'n', which as also time-invariant.

  • A Strict Sense Stationary (SSS) Process is an n-th order stationary process that hold for all values of $n > 0$.

  • For many applications, strict-sense stationarity is too restrictive. Other forms of stationarity such as wide-sense stationarity or 2nd-order stationarity are then employed.

  • A random process is strict sense stationary if the joint distribution of any set of n time samples, for all n > 0, is independent of the placement of the time origin:

    $F_X(x_1, x_2, \cdots,~ x_n;~~ t_1, t_2, \cdots,~ t_n) = F_X(x_1, x_2, \cdots,~ x_n;~~ t_1+\tau, t_2+\tau, \cdots,~ t_n+\tau)$

    $f_X(x_1, x_2, \cdots,~ x_n;~~ t_1, t_2, \cdots,~ t_n) = f_X(x_1, x_2, \cdots,~ x_n;~~ t_1+\tau, t_2+\tau, \cdots,~ t_n+\tau)$

  • $\begingroup$ @Dilip Sarwate does this meet your standard: $\endgroup$ – pico Jul 11 '20 at 15:17
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    $\begingroup$ +1 for your effort and your will to learn. But: it's not true that second-order stationarity and WSS are equivalent. A second-order stationary process is WSS, but not necessarily the other way around. $\endgroup$ – Matt L. Jul 11 '20 at 15:27
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    $\begingroup$ @MattL. In fact, not all second-order stationary processes are WSS processes, either, but all finite-power second-order stationary processes are WSS processes. The existence of a CDF or pdf does not automatically ensure the existence of moments of the random variable, and WSS processes are defined entirely in terms of properties of moments. The finite-power restriction ensures that the mean is defined (in contrast to Cauchy RV for which the mean is undefined) and of course, the variance is finite too. $\endgroup$ – Dilip Sarwate Jul 11 '20 at 15:44
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    $\begingroup$ Much better answer though mostly a rehash of parts of this earlier posting on this forum. Did you get it from the textbook that you were referring to in the comments in the now-deleted earlier self-answer to this question of yours as the basis of the wild claims in the deleted answer?? $\endgroup$ – Dilip Sarwate Jul 11 '20 at 15:53
  • $\begingroup$ @DilipSarwate: Agreed. But am I right to claim that finite power is sufficient but not necessary, i.e., not all non-finite power second-order stationary processes are NOT WSS? $\endgroup$ – Matt L. Jul 11 '20 at 15:53

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