Lacunary Fractional Brownian Motion

Abstract

Abstract. In this paper, a new class of Gaussian field is introduced called Lacunary FractionalBrownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.Mathematics Subject Classification. 42C40, 26B35.

Generalized dimensions of images of measures under Gaussian processes

“…We recall a class of Gaussian random fields with α * < α * , due to Clausel [7]. The approach is similar to the method for constructing Lévy processes with different upper and lower Blumenthal-Getoor indices [5].…”

A Wavelet Characterization for the Upper Global Hölder Index

“…Proposition 10. Let α > 0; if f is uniformly Hölderian and if for any C > 0, there exists a strictly increasing sequence of integers (j n ) n∈N such that (7) holds, let h ∈ R d and define J = sup{j n : |h| < 2 −jn }. We have, for h sufficiently small,…”

“…Appendix A.1. A uniform irregular function satisfying Property (7) Let α ∈ (0, 1), ℓ 0 ∈ N and define the two following sequences of integers (j n ) n∈N and (j n,α ) n∈N as…”

“…Proposition 15. For any (α, ε, β) ∈ (0, 1) 2 × (1, +∞), the functions f α,β,ε defined by the relation (A.5) are uniformly Hölderian, satisfy (7) and belong to UI α 1−ε (R).…”

“…Replacing the sequence j n by ℓ n = j n+n 0 +1 , property(7) is equivalent to the existence, for any C > 0, of a strictly increasing sequence of integers (j n ) n∈N such that for any n ∈ N,sup ℓ≥jn sup i,k |c (i) ℓ,k | ≤ C2 −jnα ,…”