Stochastic integration and the spectral representation

Let $S$ be the spectral process of a stationary process $X$ with zero mean and unit variance. Show that the increments of $S$ have zero means.

Let $X$ be a discrete-time stationary process having zero means, continuous strictly positive spectral density function $f$, and with spectral process $S$. Let

Show that $\ldots , Y_{-1}, Y_{0}, Y_{1}, \ldots$ is a sequence of uncorrelated random variables with zero means and unit variances.

Show that $X_{n}$ may be represented as a moving average $X_{n} = \sum_{j = -\infty }^{\infty } a_{j} Y_{n-j}$ where the $a_{j}$ are constants satisfying

Let $X$ be a discrete-time stationary sequence with zero mean and unit variance, and whose fdds are of the multivariate-normal type. Show that the spectral process of $X$ has independent increments having normal distributions.