Relation between mean and First order stationarity of a stochastic process

Question

I know that for a stochastic process to be first order -SSS (Strict sense stationary),

$f_X(x,t)=f_X(x,t+c).$ $\forall c \in \mathbb{R}$

This in turn has the implication that

$\mathbb{E}[X(t)]=\mu_x(t)=\mu_x $ is independent of time.

Now, my question is does the converse holds true. i.e.,

Is it true that if $\mathbb{E}[X(t)]$ is independent of time, then the process obeys $f_X(x,t)=f_X(x,t+c).?$

Answer 1

No. Random variables with different distributons may have equal expectations. Say, let

$X(t)\sim \mathcal N(1,1)$ for $0\leq t\leq 1,$

$X(t)\sim Poiss(1)$ for $1< t\leq 2$

$X(t)\sim E(1)$ for $2< t\leq 3$

$X(t)\sim U(0, 2)$ for $3< t\leq 4$

and so on. Then $\mathbb E[X(t)]=1$ for all $t$, and the distributions are not the same.