Statistical estimation for multiplicative cascades

The probability distribution of the cascade generators in a random multipli cative cascade represents a hidden parameter which is reflected in the fine scale limiting behavior of the scaling exponents (sample moments) of a sin gle sample cascade realization as a.s. constants. We identify a large class of cascade generators uniquely determined by these scaling exponents. For this class we provide both asymptotic consistency and confidence intervals for two different estimators of the cumulant generating function (log Lapla ce transform) of the cascade generator distribution. These results are deri ved from investigation of the convergence properties of the fine scale samp le moments of a single cascade realization.