Harmonizable fractional stable motion: Asymptotically normal estimators for both parameters

Abstract

There are two classical very different extensions of the well-known Gaussian fractional Brownian motion to non-Gaussian frameworks of heavy-tailed stable distributions: the harmonizable fractional stable motion (HFSM) and the linear fractional stable motion (LFSM). As far as we know, while several articles in the literature, some of which appeared a long time ago, have proposed statistical estimators for parameters of LFSM, no estimator has yet been proposed in the framework of HFSM. Among other things, what makes statistical estimation of parameters of HFSM to be a difficult problem is that, in contrast to LFSM, HFSM is not ergodic. The main goal of our work is to propose a new strategy for dealing with this problem and constructing strongly consistent and asymptotically normal statistical estimators for both parameters of HFSM. The keystone of our new strategy consists in the construction of new transforms of HFSM which allow to obtain, at any dyadic level and also at any two consecutive dyadic levels, sequences of independent stable random variables. This new strategy might allow to make statistical inference for more general non-ergodic harmonizable stable processes and fields than HFSM. Moreover, it could turn out to be useful in study of other issues related to them.