On the limitations of interchangeability between canonical and microcanonical multiplicative cascade models

Being the most widely used generators of multifractal measures, multiplicative cascade models have been extensively applied in the field of geophysics, and particularly in hydrometeorology. As in any modeling effort, solving the "inverse problem" is essential, and in this case, it can be described as finding the appropriate cascade model that generates a given multifractal measure. Direct measurement of a generated field (e.g., a rainfall field, or a time series thereof) results in an immediate decomposition into breakdown coefficients,  producing a microcanonical (strictly normalized) multiplicative cascade over a limited range of scales. Yet, the canonical (expectation-normalized) phenomenology at underlying scales may generate statistical properties that are non-trivial to reproduce. The present work analyzes such properties for the simplified case of a one-dimensional, beta-lognormal discrete multiplicative cascade.