Wide sense stationary (WSS) process is defined by covariance function being independent of time $E[X(t)X(t+\tau)] = g(\tau)$ and mean is a constant $E[X(t)]=\mu$ where $\mu$ is a constant and $g()$ is a finite valued function.
A second order stationary process is defined by $F_{X(t)}=F_{X(t+\tau)}$ for every $t$ and $\tau$, and $F_{X(t_1),X(t_2)}=F_{X(t_1+\tau),X(t_2+\tau)}$ for every $t_1$, $t_2$ and $\tau$, where $F$ is the distribution.
If we assume finite first and second order moments of the process $X(t)$ is it possible that wide sense stationary iff second order stationary ?
Best Answer
Second-order stationarity together with finite variance does imply wide-sense-stationarity but not the other way around.
Consider the process $\{X(t): -\infty < t < \infty\}$ such that $X(t) = A \cos(t) + B \sin(t), -\infty < t < \infty$, with $A$ and $B$ being zero-mean i.i.d. random variable with finite variance $\sigma^2$. Then, $$E[X(t)] = E[A\cos (t) + B \sin (t)] = E[A]\cos (t) + E[B] \sin (t) = 0, -\infty < t < \infty$$ and, since $E[AB] = 0$, $$ E[X(t)X(t+\tau)] = E[A^2]\cos(t)\cos(t+\tau) + E[B^2]\sin(t)\sin(t+\tau) = \sigma^2 \cos(\tau). $$ Thus, the process is wide-sense-stationary but is not necessarily second-order stationary, perhaps not even first-order stationary. For example, can you prove that $X(0) = A$ and $X(\pi/4) = (A+B)/\sqrt{2}$ have the same distribution when, say, $A$ and $B$ are i.i.d $\sim U[-1,1]$ ?
Take a look at this answer of mine on dsp.SE too.