WSS vs SSS vs ergodic

Question

Is this the correct "venn diagram" that related WSS, SSS, and Ergodic process types?

$$\text{all process types}\begin{cases}\text{WSS} \begin{cases}SSS \begin{cases}\text{ergodic} \\ \text{non-ergodic}\end{cases} \\ \text{non-SSS}\end{cases}\\ \text{non-stationary}\end{cases}$$

SSS = Strict Sense Stationary

WSS = Wide Sense Stationary

Just trying to wrap my brain around this one...without getting too confused.

I started out with this diagram:

$$\text{all process types}\begin{cases}\text{stationary}\begin{cases}\text{ergodic} \\ \text{non-ergodic}\end{cases}\\\text{non-stationary}\end{cases}$$

but, i thought I could do better by include WSS in the diagram. Of course, I don't really know 100% if I'm correct... its more of a guess.

Answer 1

There are several questions on this forum dealing with various aspects of strict-sense and wide-sense stationarity and ergodicity, and some of the answers give counterexamples to the Venn diagram that you have constructed, such as

• A strictly stationary process that is not wide-sense-stationary because its mean is undefined
• A wide-sense-stationary process that is not stationary in any sense but is nonetheless ergodic

Answer 2

• 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:

  $\mu_X(t)=\mu=\text{constant}$

  $\sigma^2(t)=\sigma^2=\text{constant}$

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

  $\cdots$

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

• 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:

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

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

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

  $\mu_X(t)=\mu=\text{constant}$

  $\sigma^2(t)=\sigma^2=\text{constant}$

  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:

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

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