Instantaneous Frequency Estimation by Hilbert Transform - Theoretical Justification and Proof
See my earlier comments here: Meaning of Hilbert transform  

Common fractal noise isn't an analytic signal (infinitely differentiable).  And a Hilbert transform re-creates the imaginary component of an analytic signal if you have the real component of the complex analytic signal (which one rarely has from real-world data).  

However, sufficiently band-pass filtered data might be similar to a finite length segment of something that looks like an infinite length analytic signal (e.g. is from a source whose behavior can be modeled or estimated by a 2nd order linear differential equation).