English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Wavelet-type random series representations of the well-known FractionalBrownian Motion (FBM) and many other related stochastic processes and fieldshave started to be introduced since more than two decades. Such representationsprovide natural frameworks for approximating almost surely and uniformly roughsample paths at different scales and for study of various aspects of theircomplex erratic behavior.  Hermite process of an arbitrary integer order $d$, which extends FBM, is aparadigmatic example of a stochastic process belonging to the $d$th Wienerchaos. It was introduced very long time ago, yet many of its properties arestill unknown when $d\ge 3$. In a paper published in 2004, Pipiras raised theproblem to know whether wavelet-type random series representations with awell-localized smooth scaling function, reminiscent to those for FBM due toMeyer, 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 theHermite process is called the Rosenblatt process. Yet, the problem remainsunsolved in the general case $d\ge 3$. The main goal of our article is to solveit, not only for usual Hermite processes but also for generalizations of them.Another important goal of our article is to derive almost sure uniformestimates of the errors related with approximations of such processes byscaling functions parts of their wavelet-type random series representations.

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