Suppose $Z(t)={\Sigma}_{k=1}^{n}X{e}^{j({\mathit{\omega}}_{0}t+{\mathbf{\Phi}}_{k})}$, $t\in R$ where ${\mathit{\omega}}_{0}$ is a constant, n is a fixed positive integer, ${X}_{1},...,{X}_{n},\u3000{\mathbf{\Phi}}_{1},...,{\mathbf{\Phi}}_{n}$ are mutually independent random variables, and $E{X}_{k}=0,D{X}_{k}={\sigma}_{k}^{2},\mathbf{\Phi}$ , $U[0,2\pi ],k=1,2,...,n$ . Find the mean function and correlation function of $\{Z(t),\text{}t\in R\}$ .

I have tried to solve it.

For mean function,

${m}_{Z}(s)=E\{{Z}_{s}\}=E\{{X}_{s}\}+iE\{{Y}_{t}\}$

$=E\{{\Sigma}_{k=1}^{s}X{e}^{j({\mathit{\omega}}_{0}t+{\mathbf{\Phi}}_{k})}\}$

For correlation function,

${R}_{Z}(s,u)=E\{{Z}_{s},{Z}_{u}\}$

$=E\{Y(s)Y(u)\}$

$=E\{{\Sigma}_{k=1}^{s}X{e}^{j({\mathit{\omega}}_{0}t+{\mathbf{\Phi}}_{k})}{\Sigma}_{k=1}^{u}X{e}^{j({\mathit{\omega}}_{0}t+{\mathbf{\Phi}}_{k})}\}$

$=E\{{\Sigma}_{k=1}^{s}{\Sigma}_{k=1}^{u}X{e}^{j({\mathit{\omega}}_{0}t+{\mathbf{\Phi}}_{k})}X{e}^{j({\mathit{\omega}}_{0}t+{\mathbf{\Phi}}_{k})}\}$

I am stuck here. How to move from here ahead?