Slow, ordinary and rapid points for Gaussian Wavelets Series and application to Fractional Brownian Motions

Abstract

We study the Hölderian regularity of Gaussian wavelets series and show that they display, almost surely, three types of points: slow, ordinary and rapid. In particular, this fact holds for the Fractional Brownian Motion. Finally, we remark that the existence of slow points is specific to these functions.

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

“…In particular, we use different bases depending EJP 27 (2022), paper 152. on whether we deal with the finiteness of the limits in Theorem 1.2 or with their strict positiveness. This is very different from [18] where the authors always work with the same wavelet. The reason is that the expression (3.3) in Theorem 3.3 below is not a wavelet series: it involves additional quantities.…”

“…As in [18], for all j P N, we denote by k j ptq the unique integer such that t P rk j ptq2 ´j , pk j ptq `1q2 ´j q. In other words, k j ptq " spλ j ptqq.…”

“…Wavelet methods to study the pointwise regularity of the generalized Rosenblatt process In what follows, we show how to modify Lemmata 3.7 to 3.19 from the previous subsection, using L k1,k2 j1,j2 ptq instead of L k1,k2 j1,j2 . Before all, we need the following Lemma which is inspired by results from [18] that can be extended in our case. Lemma 3.22.…”

“…In [25], Kahane described a procedure to insure the existence of slow points for the Brownian motion. This procedure was then generalized in [18] to fit for any arbitrary fractional Brownian motion. It consists in showing that for any m ą 0, almost surely, there exist µ ą 0 and t P p0, 1q such that, if one sets Λ 0 j ptq " tλ P Λ j : |spλptqq ´spλq| ď 1u…”

“…In [18], Esser and Loosveldt undertook a systematic study of Gaussian wavelet series. Thanks to (1.4), it applies in particular to the fractional Brownian motion and leads to the following theorem.…”