Blind Source Separation for Operator Self-similar Processes

Self-similarity and operator self-similarity. An R-valued signal {Y (t)} is said to be operator self-similar (o.s.s) when it satisfies the scaling relation {Y (at)}t∈R fdd = {aY (t)}t∈R, a > 0, where fdd denotes the finite dimensional distributions, H is called the Hurst matrix, and a is given by the matrix exponential ∑∞ k=0 log(a) H k! . O.s.s processes naturally generalize the univariate self-similar (s.s) processes, which have been used to model a wide range of phenomena in hydrodynamic turbulence [1], geophysics [2] and Internet traffic [3]. Gaussian case. The celebrated fractional Brownian motion (fBm) is the only Gaussian, self-similar process with stationary increments [4]. The natural multivariate generalization of fBm, called operator fBm (ofBm), was studied in [5, 6], and several papers are now devoted to inferential methods for ofBm (e.g., [7, 8]). Non-Gaussian case. Hermite processes are typically non Gaussian, self-similar, stationary increment processes. They appear as a consequence of non-central limit theorems. The Rosenblatt process (or fractional Rosenblatt motion, fRm) corresponds to the Hermite process of rank 2 [9]. Statistical inference for Hermite-type processes is particularly challenging: it was shown that wavelet-based estimators may display nonstandard convergence rates and asymptotic distributions in [10]. The modeling and statistical inference of multivariate Hermite-type fractional signals, while of great importance in applications, is an essentially unexplored research topic. Outline. This thesis is dedicated to the problem of demixing (multivariate) fractional signals, i.e., extracting a source of independent s.s signals from a set of mixed signals. This blind source problem has been little studied under the framework of fractional stochastic processes. The work considers mainly two types of signals: Gaussian and non-Gaussian (Rosenblatt-type). The basic philosophy is to apply the same wavelet-based demixing procedure to these two types of signals (Section 2). Large size Monte Carlo simulations are used to illustrate that the finite-sample performance of the demixing method is very satisfactory (Section 4). Moreover, the asymptotic properties of the estimators will be studied for Gaussian and non-Gaussian cases separately (Section 3). 2. METHODOLOGY