# A special case of 2 jointly Weak-Sense Stationary (WSS) stochastic processes

I know that 3 conditions must be met in order a pair of stochastic processes $X(t)$ and $Y(t)$ to be characterized as jointly WSS:

1. $X(t)\;\; WSS$

2. $Y(t)\;\; WSS$

3. $R_{xy}(t_1,t_2) = R_{xy}(t_1 - t_2) = R_{xy}(t_2 - t_1)$ which means that their cross-correlation function should depend only on the difference $T = t_1 - t_2$.

My question is: if the first 2 conditions are met but the cross-correlation function does not depend on any variable (I mean that it's a constant e.g. $R_{xy}(t_1,t_2) = 0$) can we assume that the two stochastic processes $X(t)$ and $Y(t)$ are jointly WSS?

Yes, when $R_{X,Y}(t_1, t_2) = E[X(t_1)Y(t_2)] = 0~ \forall ~t_1, t_2$, $\{X(t)\}$ and $\{Y(t)\}$ are often called uncorrelated WSS processes, and they are considered to be jointly WSS processes. More nitpicky folks agree that the processes are indeed jointly WSS when $R_{X,Y}(t_1, t_2) = 0~ \forall ~t_1, t_2$, but insist on reserving the adjective uncorrelated for processes for which the cross-covariance function
$$C_{X,Y}(t_1,t_2) = E\left[\left(X(t_1)-E[X(t_1)]\right)\left(Y(t_2)-E[Y(t_2]\right)\right] =E[X(t_1)Y(t_2)]-E[X(t_1)]E[Y(t_2)] = R_{X,Y}(t_1,t_2)-E[X(t_1)]E[Y(t_2)]$$ equals $0$ for all $t_1, t_2$, that is, every pair of random variables $X(t_1)$, $Y(t_2)$ from the processes is a pair of uncorrelated random variables. Of course, when at least one of the processes has zero mean (as is usually the case in signal processing applications), then $C_{X,Y}(t_1,t_2) = R_{X,Y}(t_1,t_2)$ and so the nit gets picked without requiring any further consideration.