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

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 dth
Wiener chaos. It was introduced very long time ago, yet many of its
properties are still unknown when d ≥ 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 ≥ 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.