Let \(\left(\varepsilon_{k}, \varepsilon_{k}^{\prime}ight)_{k \geq 0}\) be independent and identically distributed standard Gaussian variables. For real coefficients \(\sigma_{k}\), show that the random function

\[
\eta(t)=\sum_{k=0}^{\infty} \sigma_{k} \varepsilon_{k} \cos (k t)+\sigma_{k} \varepsilon_{k}^{\prime} \sin (k t)
\]

is stationary with covariance

\[
\operatorname{cov}\left(\eta(t), \eta\left(t^{\prime}ight)ight)=\sum_{k=0}^{\infty} \sigma_{k}^{2} \cos \left(k\left(t-t^{\prime}ight)ight)
\]

provided that the series converges in a suitable sense.