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.
 A: 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 $p > 1$.
A: A non-negative function is not necessarily $L^1$ on the real axis, although that seems to be what one of the answers (the other one) is saying. First of all, one must assume the process is not only stationary, but ergodic (a much stronger assumption) so that its power spectrum at least goes to zero at infinity.  Secondly, even if the power spectrum is, e.g., compactly supported, e.g., a narrow band-limited but flat "pink noise", its Fourier transform need not be $L^1$ and so Fourier inversion will not be true.  In fact the auto-correlation of pink noise is the sinc function $\sin \omega t \over t$ which is not absolutely integrable.  So it is not true to say the the power spectral density is the Fourier transform of the sinc function, it is not true that the power spectral density and the auto-correlation function form a Fourier transform pair (except sometimes in the sense of distributions, e.g., when the process is ergodic).
so we cannot "insist that $R_X(t)$ lies in the domain of the (inverse) Fourier transform".
there are a lot of mistakes floating around the internet, copied from electrical engineering textbooks or handouts which have mistakes in them.
