On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion

Abstract

The generalized multifractional Brownian motion (GMBM) is a continuous Gaussian process that extends the classical fractional Brownian motion (FBM) and multifractional Brownian motion (MBM) (SIAM Rev. 10 (1968) 422; INRIA Res. Rept. 2645 (1995); Rev. Mat. Iberoamericana 13 (1997) 19; Fractals: Theory and Applications in Engineering, Springer, Berlin, 1999, pp. 17–32; Statist. Inference Stochastic Process. 3 (2000) 7). As is the case for the MBM, the Hölder regularity of the GMBM varies from point to point. However, and this is the main interest of the GMBM, contrary to the MBM, these variations may be very erratic: As shown in (J. Fourier Anal. Appl. 8 (2002) 581), the pointwise Hölder function { α X ( t)} t of the GMBM may be any lim inf of continuous functions with values in a compact of (0,1). This feature makes the GMBM a good candidate to model complex data such as textured images or multifractal processes. For the GMBM to be useful in applications, it is necessary that its Hölder exponents may be estimated from discrete data. This work deals with the problem of identifying the pointwise Hölder function H of the GMBM: While it does not seem easy to do so when H is an arbitrary lim inf of continuous functions, we obtain below the following a priori unexpected result: As soon as the pointwise Hölder function of GMBM belongs to the first class of Baire ( i.e. when { α X ( t)} t is a limit of continuous functions) it may be estimated almost surely at any point t. We also derive a Central Limit Theorem for our estimator. Thus, even very irregular variations of the Hölder regularity of the GMBM may be detected and estimated in practice. This has important consequences in applications of the GMBM to signal and image processing. It may also lead to new methods for the practical computation of multifractal spectra. We illustrate our results on both simulated and real data.