Wavelet-Type Expansion of the Generalized Rosenblatt Process and Its Rate of Convergence

Abstract

Pipiras introduced in the early 2000s an almost surely and uniformly convergent (on compact intervals) wavelet-type expansion of classical Rosenblatt process. Yet, the issue of estimating, almost surely, its uniform rate of convergence remained an open question. The main goal of our present article is to provide an answer to it in the more general framework of generalized Rosenblatt process, under the assumption that the underlying wavelet basis belongs to the class due to Meyer. The main ingredient of our strategy consists in expressing in a non-classical (new) way the approximation errors related with the approximation spaces of a multiresolution analysis of L 2 (R 2). Such a non-classical expression may also be of interest in its own right.

Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process

“…Then, in Section 4, we give lower bounds for the so-called wavelet-leaders, see Section 2, of the generalized Rosenblatt process on a given compactly supported wavelet basis. This will prove the positiveness of the limits (5), (6). In particular, we use different bases depending on whether we deal with the finiteness of the limits in Theorem 1.2 or with their strict positiveness.…”

“…In fact, for that, we would need to find an almost-sure uniform lower modulus of continuity for the generalized Rosenblatt process and to be able to judge its optimality, which seems to be a difficult task. This is discussed in details in Remark 5.1 below, where we give an almost-sure uniform lower modulus of continuity using the techniques we use to prove the positiveness of the limits in ( 5) and (6).…”

“…First, in Section 3 we derive upper-bounds for the oscillations |R H1,H2 pt, ωq ´RH1,H2 ps, ωq| that are sharp enough to imply the finiteness of the limits ( 5), ( 6) and (7). This is done by means of the wavelet-type expansion given in [6], see Theorem 3.3 below. Then, in Section 4, we give lower bounds for the so-called wavelet-leaders, see Section 2, of the generalized Rosenblatt process on a given compactly supported wavelet basis.…”

“…In [6], Ayache and Esmili presented a wavelet-type representation of the generalized Rosenblatt process, very similar to the one given in [36] for fractional Brownian motion, excepted for the use of integrals of two-dimensional wavelet bases. This representation is the starting point of this paper.…”

Fractal dimensions of the Rosenblatt process

“…There are three integral representations: in terms of time, the spectrum and on finite intervals, see [46]. There is also a wavelet representation [36] (see also the recent article [3] for the wavelet representation of the generalized Rosenblatt process and its rate of convergence). From a statistical point of view, the value of the Hurst index H is important for practical applications and various estimators exist, see [5,53].…”