On Estimation Theory for Multiplicative Cascades

The notion of multiplicative cascade was introduced into the statistical theory of turbulence by A.N. Kolmogorov as a phenomenological framework intended to accommodate the intermittency and large fluctuations observed in turbulent fluid flows. The basic idea is that energy is redistributed from larger to smaller scales via a splitting mechanism involving random multiplicative factors known as cascade generators. Primarily owing to the scaling structure of this class of models, applications have been extended to a wide variety of other naturally occurring phenomena such as rainfall, internet packet traffic, market prices, etc. which exhibit intermittent and highly variable behavior in space and time. The probability distribution of the cascade generators represents a hidden parameter which is reflected in the fine scale limiting behavior of certain scaling exponents calculated from a single sample realization. In this paper we describe the underlying statistical theory for estimation of the distribution of the generators, discuss examples, and provide a number of open problems in the general theory. Some new results involving estimation of an important intermittency parameter, the Hausdorff dimension of the support set, are also included. We then proceed to identify an outstanding open statistical problem from turbulence data.