Characterization of Gaussian self-similar stochastic processes using wavelet-based informational tools L. Zunino * Centro de Investigaciones Ópticas, casilla de correo 124 Correo Central, 1900 La Plata, Argentina and Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata (UNLP), 1900 La Plata, Argentina D. G. Pérez † Instiuto de Física, Pontificia Universidad Católica de Valparaíso (PUCV), 23-40025 Valparaíso, Chile M. T. Martín ‡ and A. Plastino § Instituto de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata (UNLP), casilla de correo 727, 1900 La Plata, Argentina and Argentina’s National Research Council (CONICET), Argentina M. Garavaglia  Centro de Investigaciones Ópticas, casilla de correo 124 Correo Central, 1900 La Plata, Argentina and Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata (UNLP), 1900 La Plata, Argentina O. A. Rosso ¶ Chaos and Biology Group, Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires (UBA), Pabellón II, Ciudad Universitaria, 1428 Ciudad de Buenos Aires, Argentina Received 16 August 2006; published 20 February 2007 Efficient tools to characterize stochastic processes are discussed. Quantifiers originally proposed within the framework of information theory, like entropy and statistical complexity, are translated into wavelet language, which renders the above quantifiers into tools that exhibit the important “localization” advantages provided by wavelet theory. Two important and popular stochastic processes, fractional Brownian motion and fractional Gaussian noise, are studied using these wavelet-based informational tools. Exact analytical expressions are obtained for the wavelet probability distribution. Finally, numerical simulations are used to validate our ana- lytical results. DOI: 10.1103/PhysRevE.75.021115 PACS numbers: 02.50.Ey I. INTRODUCTION The aim of this paper is to explore the ability of some formerly introduced wavelet-based informational quantifiers to characterize stochastic processes. In that sense, we analyze two well-known stochastic processes, namely, 1 fractional Brownian motion FBM and 2 fractional Gaussian noise FGN1,2. We are mainly interested in Gaussian and self- similar stochastic processes. Gaussian processes are impor- tant because they yield the basic model for the analysis of natural phenomena. A process is called Gaussian if all its finite dimensional distributions are Gaussian. Furthermore, considering the ubiquity of Gaussian distributions in prob- ability theory, it is natural to study Gaussian processes. The central limit theorem constitutes a cornerstone of our under- standing of the probabilistic nature of the observable world. When there are reasons to suspect the presence of a large number of small perturbations acting both additively and in- dependently, it is reasonable to assume that the concomitant observations will be Gaussian-distributed 3. That is, if the tails associated to the probability distributions decay fast enough. On the other hand, self-similar stochastic processes are invariant in distribution under suitable scaling of time and space. Formally, a stochastic process Xt is self- similar with index H if, for any c  0, Xt= d c H Xc -1 t , 1 where = d is equality in distribution. The self-similarity ap- pears in a natural way from limit theorems for sums of ran- dom variables 4 –6. Following the arguments mentioned by Beran 7, within the framework of stochastic processes, the role performed by self-similar ones is equivalent to the role of stable distributions among distributions. *Also at Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata UNLP. Electronic address: lucianoz@ciop.unlp.edu.ar † Electronic address: dario.perez@ucv.cl ‡ Electronic address: mtmartin@fisica.unlp.edu.ar § Electronic address: plastino@fisica.unlp.edu.ar  Electronic address: garavagliam@ciop.unlp.edu.ar ¶ Electronic address: oarosso@fibertel.com.ar PHYSICAL REVIEW E 75, 021115 2007 1539-3755/2007/752/02111510 ©2007 The American Physical Society 021115-1