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In the recent years, Independent Component Analysis (ICA) has become a fundamental tool in adaptive signal and data processing, especially in the field of Blind Source Separation (BSS). Even though there exist some methods for which an algebraic solution to the ICA problem may be found, other iterative methods are very popular. Among them is the class of information-theoretic approaches, laying on entropies. The associated objective functions are maximized based on optimization schemes, and on gradient-ascent techniques in particular. Two major issues in this field are the following: 1) Does the global maximum point of these entropic objectives correspond to a satisfactory solution of BSS ? and 2) as gradient techniques are used, optimization algorithms look in fact for local maximum points, so what about the meaning of these local optima from the BSS problem point of view? Even though there are some partial answers to these questions in the literature, most of them are based on simulation and conjectures; formal developments are often lacking. This thesis aims at filling this lack and providing intuitive justifications, too. We focus the analysis on Rényi's entropy-based contrast functions. Our results show that, generally speaking, Rényi's entropy is not a suitable contrast function for BSS, even though we recover the well-known results saying that Shannon's entropy-based objectives are contrast functions. We also show that the range-based contrast functions can be built under some conditions on the sources. The BSS problem is stated in the first chapter, and viewed under the information (theory) angle. The two next chapters address specifically the above questions. Finally, the last chapter deals with range-based ICA, the only ``entropy-based contrast' which, based on the enclosed results, is also a <i>discriminant</i> contrast function, in the sense that it is theoretically free of spurious local optima. Geometrical interpretations and surprising examples are given. The interest of this approach is confirmed by testing the algorithm on the MLSP 2006 data analysis competition benchmark; the proposed method outperforms the previously obtained results on large-scale and noisy mixture samples obtained through ill-conditioned mixing matrices.

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Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.