A Ferguson–Klass–LePage series representation of multistable multifractional motions and related processes

Abstract

The study of non-stationary processes whose local form has controlled properties is a fruitful and important area of research, both in theory and applications. In (J. Theoret. Probab. 22 (2009) 375–401), a particular way of constructing such processes was investigated, leading in particular to multifractional multistable processes, which were built using sums over Poisson processes. We present here a different construction of these processes, based on the Ferguson–Klass–LePage series representation of stable processes. We consider various particular cases of interest, including multistable Lévy motion, multistable reverse Ornstein–Uhlenbeck process, log-fractional multistable motion and linear multistable multifractional motion. We also show that the processes defined here have the same finite dimensional distributions as the corresponding processes built in (J. Theoret. Probab. 22 (2009) 375–401). Finally, we display numerical experiments showing graphs of synthesized paths of such processes.