An example of a "pathological" power-spectral density function? Suppose that we are given a wide-sense stationary random process $X$ with autocorrelation function $R_X(t)$.  Power spectral density $S_X(f)$ of $X$ is then given by the Fourier transform of $R_X(t)$, i.e. $S_X(f)=\mathcal{F}(R_X(t))$.
I am wondering if there is a valid power spectral density function $S_X(f)$ such that, for a positive integer $n$, the integral over the entire frequency domain of the absolute value of $S_X(f)$ taken to the $n$-th power is not a finite constant.  Formally, is there $S_X(f)$ such that:
$$\int_{-\infty}^{\infty} |S_X(f)|^n df=\infty$$
I know that this is impossible for $n=1$, as $\int_{-\infty}^{\infty} S_X(f) df=R_X(0)=E[X^2]<\infty$, however, I haven't found any result for $n>1$.  Perhaps it's very obvious one way or the other  (though it seems to me that such $S_X(f)$ does not exist, but I can't find a formal proof).  In any case, I would appreciate elucidation.

This is impossible just by definition. 
$S_X(f)$, if it exists, is the Radon-Nikodym derivative of the spectral measure $\mu$ with respect to the Lebesgue measure and, being nonnegative, necessarily $L^1$ as you noted. By definition, the Fourier transform of $S_X(f)$ (or $\mu$ in general) is $R_X(t)$. If one insists that $R_X(t)$ lies in the domain of the (inverse) Fourier transform, then Fourier inversion theorem implies that $S_X(f)$ is in fact continuous almost everywhere. A continuous function in $L^1$ lies also in $L^p$ for any $p > 1$.
