This is mistaking two kinds of power laws. OP is considering power laws in the frequency domain. You are considering power laws in the heavy tail of a distrib

This is mistaking two kinds of power laws. OP is considering power laws in the frequency domain. You are considering power laws in the heavy tail of a distribution. Different things! But there are confounders that make discussion seem similar: - questions about moments and convergence (OP: do we have finite energy in the Fourier domain; your comment: do the tails of the distribution fall off fast enough to have finite moments of order 1 or 2) - questions about averaging (OP: in the time domain; your comment: as an expectation obtained by integrating a distribution, or as a closure property of the stable class of distributions)