The Markov-switching multifractal as an estimable propagating-fluctuations model

Abstract

The MSM cascade reproduces, in closed form, the two marginal-law signatures that motivate propagating-fluctuations models of accretion variability: a lognormal flux marginal and a linear rms–flux relation. This paper derives those correspondences, measures how far each survives the observation process (a binned Cox photon channel), and asks the estimability question directly: at what photon rate does the exact 2^k-state filter stop recovering the injected cascade contrast? The answer is a metrology result rather than an astrophysical one. Recovery of the true contrast holds down to ~0.06 counts/bin (~500 photons), well below the rate at which the signal has already vanished from the moments — the closed-form overdispersion available for detection falls under the sampling floor near 0.05 counts/bin — and the failure mode at lower rates is an upward blow-up of the estimate, never a quiet collapse toward a flat cascade. The operational reading: an apparent likelihood collapse on real streams is emission-model misspecification, not photon starvation, and the diagnostic for it is a failed resynthesis (a statistical Turing test), not a flat likelihood.