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.