Extension beyond i.i.d. multipliers

Determine whether the hierarchical symmetry axiom A1, defined by the step-k recurrence δ_{p+k} = (1 − β)δ_∞ + βδ_p for incremental exponents, continues to select a log-Poisson cascade multiplier distribution when the multipliers are stationary ergodic or Markovian (i.e., not i.i.d.), in settings where a Lévy–Khintchine representation is not directly available.

Background

The paper proves that within i.i.d. multiplicative cascades, the hierarchical symmetry axiom A1 is both necessary and sufficient for the multiplier to be log-Poisson and provides classification and stability results within the log-infinitely-divisible family.

The authors point out that the i.i.d. hypothesis, while standard, is restrictive in practice. They suggest investigating whether the same classification—that A1 uniquely selects log-Poisson—extends to dependent settings such as stationary ergodic or Markovian multipliers, where the Lévy–Khintchine machinery used in the proofs may not apply.