Question: Let ({X(t), t in(-infty,+infty)}) and ({Y(t), t in(-infty,+infty)}) be two independent, weakly stationary stochastic processes, whose trend functions are identically 0 and which have the

Let $\{X(t), t \in(-\infty,+\infty)\}$ and $\{Y(t), t \in(-\infty,+\infty)\}$ be two independent, weakly stationary stochastic processes, whose trend functions are identically 0 and which have the same covariance function $C(\tau)$.

Verify: The stochastic process $\{Z(t), t \in(-\infty,+\infty)\}$ with

\[Z(t)=X(t) \cos \omega t-Y(t) \sin \omega t\]

is weakly stationary.

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