Existence et régularité höldérienne des fonctions de bosses

Abstract

Abstract. We discuss the almost sure existence of random functions that can be written as sums of elementary pulses. We then estimate their uniform Hölder regularity by applying some results on coverings by random intervals.

A note on the Hausdorff dimension of general sums of pulses graphs

“…In this paper, we investigate the Hausdorff dimension of their graphs which provides a measure of the irregularity of the process and gives a positive answer to the question of Tricot. In particular our result is an improvement of the result of [1] who gives only an upper bound of the upper box dimension of the graph of F .…”

“…The existence and regularity of bumps sums functions defined by (1) have been studied in [1]. In particular Abid proved the following results.…”

“…In the same particular case and in higher dimension, the analysis of the fractal sums of pulses has been treated in [3] and [2]. The existence and regularity of functions defined by (1) have been studied in [1]. Notice that this kind of functions are important for the purpose of modeling strange phenomena which are known to exhibit multifractal behaviors.…”

“…Such behaviors occur for instance in geophysics [5] when considering the spatial-temporal position and the intensity of seismic events, in telecommunications where the TCP Internet traffic is known to be multifractal [8], and also when studying financial time series [10]. This work was motivated by a question given in [13] about the Hausdorff dimension of the graph of functions defined by (1). In this paper, we investigate the Hausdorff dimension of their graphs which provides a measure of the irregularity of the process and gives a positive answer to the question of Tricot.…”

Multifractal analysis of sums of random pulses

“…When , almost surely . In [ 10 ], Ben Abid gave alternative conditions for the convergence of such processes G , also determining the uniform regularity of G , i.e. to which global Hölder space G may belong to, almost surely.…”