Wavelet-Type Expansion of Generalized Hermite Processes with rate of convergence

Abstract

Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and many other related stochastic processes and fields have started to be introduced since more than two decades. Such representations provide natural frameworks for approximating almost surely and uniformly rough sample paths at different scales and for study of various aspects of their complex erratic behavior. Hermite process of an arbitrary integer order $d$, which extends FBM, is a paradigmatic example of a stochastic process belonging to the $d$th Wiener chaos. It was introduced very long time ago, yet many of its properties are still unknown when $d\ge 3$. In a paper published in 2004, Pipiras raised the problem to know whether wavelet-type random series representations with a well-localized smooth scaling function, reminiscent to those for FBM due to Meyer, Sellan and Taqqu, can be obtained for a Hermite process of any order $d$. He solved it in this same paper in the particular case $d=2$ in which the Hermite process is called the Rosenblatt process. Yet, the problem remains unsolved in the general case $d\ge 3$. The main goal of our article is to solve it, not only for usual Hermite processes but also for generalizations of them. Another important goal of our article is to derive almost sure uniform estimates of the errors related with approximations of such processes by scaling functions parts of their wavelet-type random series representations.