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)