In [1, p. 62 exercise 3.15] it is requested to verify the ACF and PSD relationships given in [1, p. 53 eq. (3.50)] and [1, p. 54 eq. (3.51)].
Solution: Starting from the first relationship of [1, p. 53 eq. (3.50)] and the definition of the cross correlation function we can derive

Using the starting assumption that x[n] is WSS we can derive the final result, that the cross correlation is also independent of the observation instance n and depends only on the lag k:

Similar the second relation can be obtained by:

and setting l^{\prime} = -l:
=r_{xx}[k]\star h^{\ast}[-k] =r_{yx}[k]

The third equation can be derived by:
=E\{\sum\limits_{l}h^{\ast}[l]x^{\ast}[n-l]\sum\limits_{\lambda}h[\lambda]x[n+k-\lambda] \}
=\sum\limits_{l}\sum\limits_{\lambda}h^{\ast}[l]h[\lambda]E\{x^{\ast}[n-l]x[n+k-\lambda] \}
=\sum\limits_{l}h^{\ast}[l]\sum\limits_{\lambda} h[\lambda]r_{xx}[k+l-\lambda] (1)

By setting \rho[k+l]=\sum_{\lambda} h[\lambda]r_{xx}[k+l-\lambda] =h[k+l]\ast r_{xx}[k+l] we can rewrite (1) by
r_{yy}[n,k]=\sum\limits_{l}h^{\ast}[l] \rho[k+l]
=h^{\ast}(-k)\star \rho(k)
=h^{\ast}(-k)\star h[k] \ast r_{xx}[k] (2)

which proves the last equation of [1, p. 53 eq. (3.50)] . We can now proceed to prove the relations for [1, p. 54 eq. (3.51)]
=\mathcal\{r_{xx}[k]\star h[k]\}
=H(z)R_{xx}(z) (3)

The second relation P_{yx}(z) can similar be proven by:
=\mathcal{Z}\{h^{\ast}[-k]\star r_{xx}[k]\}

By change of variables z=(u^{-1})^{\ast} and k=-l for the sum of the previous equation can be written as:
=R_{xx}(z) \left[H(u=\frac{1}{z^{\ast}})\right]^{\ast}
=R_{xx}(z) \left[H(\frac{1}{z^{\ast}})\right]^{\ast}

The last equation proves the second part of [1, p. 54 eq. (3.51)]. By similar reasoning the third relation can also be shown to be
=\mathcal{Z}\{h^{\ast}[-k]\star h[k]\star r_{xx}[k]\}
=\mathcal{Z}\{h^{\ast}[-k]\}\mathcal{Z}\{ h[k]\}\mathcal{Z}\{r_{xx}[k]\}

which concludes the proof. QED.

[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.