Identifying the multifractional function of a Gaussian process.

Gaussian processes that are multifractional are studied in this paper. By multifractional processes we mean that they behave locally like a fractional Brownian motion, but the fractional index is no more a constant: it is a function. We introduce estimators of this multifractional function based on discrete observations of one sample path of the process and we study their asymptotical behaviour as the mesh decreases to zero. Key-words: Gaussian processes, identification, multifractional function. AMS classification (1991): 60G15,60G17,62G05. Universit'e Blaise Pascal, U.R.A. 1501 C.N.R.S. Les C'ezeaux, 63170 Aubi`eres, France. y D'epartement de Math'ematiques Universit'e de Versailles-St Quentin en Yvelines 45, avenue des EtatsUnis, 78035 Versailles France and CERMICS ENPC. z Laboratoire de Biom'etrie I.N.R.A., Domaine de Vilvert 78350 Jouy-en-Josas, France. 1 Introduction The fractional Brownian motion introduced in Mandelbrot and Van Ness (1968) provides a very powerful mod..