Is wide sense stationary iff second order stationary? 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 ?
 A: 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.
A: In the way you've defined the concepts second order stationarity implies wide stationarity as long as variances exist, but not vice versa.  This is because independence implies zero correlation but not the other way around, unless you know your time series to be jointly normal.
