Future cognitive radio networks are expected to come as a disruptive technological advance in the currently saturated field of wireless communications. The idea behind cognitive radios is to think of the wireless channels as a pool of communication resources, which can be accessed on-demand by a primary licensed network or opportunistically preempted (or overlaid) by a secondary network with lower access priority. From a physical layer point of view, the primary network is ideally oblivious of the existence of a co-localized secondary networks. The latter are therefore required to autonomously explore the air in search for resource left-overs, and then to optimally exploit the available resource. The exploration and exploitation procedures, which involve multiple interacting agents, are requested to be highly reliable, fast and efficient. The objective of the thesis is to model, analyse and propose computationally efficient and close-to-optimal solutions to the above operations.Regarding the exploration phase, we first resort to the maximum entropy principle to derive communication models with many unknowns, from which we derive the optimal multi-source multi-sensor Neyman-Pearson signal sensing procedure. The latter allows for a secondary network to detect the presence of spectral left-overs. The computational complexity of the optimal approach however calls for simpler techniques, which are recollected and discussed. We then proceed to the extension of the signal sensing approach to the more advanced blind user localization, which provides further valuable information to overlay occupied spectral resources.The second part of the thesis is dedicaded to the exploitation phase, that is, the optimal sharing of available resources. To this end, we derive an (asymptotically accurate) approximated expression for the uplink ergodic sum rate of a multi-antenna multiple-access channel and propose solutions for cognitive radios to adapt rapidly to the evolution of the primary network at a minimum feedback cost for the secondary networks.

Cette thèse, constituée de trois papiers de recherche, est organisée autour de la construction d’indicateurs de crises financières dont les signaux sont ensuite utilisés pour l’élaboration de stratégies de trading algorithmique. Le premier papier traite de l’établissement d’un cadre de travail permettant la construction des indicateurs de crises financière. Le pouvoir de prédiction de nos indicateurs est ensuite démontré en utilisant l’un d’eux pour construire une stratégie de type protective-put active qui est capable de faire mieux en termes de performances qu’une stratégie passive ou, la plupart du temps, que de multiples réalisations d’une stratégie aléatoire. Le second papier va plus loin dans l’application de nos indicateurs de crises à la création de stratégies de trading algorithmique en utilisant le signal combiné d’un grand nombre de nos indicateurs pour gouverner la composition d’un portefeuille constitué d’un mélange de cash et de titres d’un ETF répliquant un indice equity comme le SP500. Enfin, dans le troisième papier, nous construisons des indicateurs de crises financières en utilisant une approche complètement différente. En étudiant l’évolution dynamique de la distribution des spreads des composantes d’un indice CDS tel que l’ITRAXXX Europe 125, une bande de Bollinger est construite autour de la fonction de répartition de la distribution empirique des spreads, exprimée sur une base de deux distributions log-normales choisies à l’avance. Le passage par la fonction de répartition empirique de la frontière haute ou de la frontière basse de cette bande de Bollinger est interprétée en termes de risque et permet de produire un signal de trading. / This thesis is constituted of three research papers and is articulated around the construction of financial crisis indicators, which produce signals, which are then applied to devise successful systematic trading strategies. The first paper deals with the establishment of a framework for the construction of our financial crisis indicators. Their predictive power is then demonstrated by using one of them to build an active protective-put strategy, which is able to beat in terms of performance a passive strategy as well as, most of the time, multiple paths of a random strategy. The second paper goes further in the application of our financial crisis indicators to the elaboration of systematic treading strategies by using the aggregated signal produce by many of our indicators to govern a portfolio constituted of a mix of cash and ETF shares, replicating an equity index like the SP500. Finally, in the third paper, we build financial crisis indicators by using a completely different approach. By studying the dynamics of the evolution of the distribution of the spreads of the components of a CDS index like the ITRAXX Europe 125, a Bollinger band is built around the empirical cumulative distribution function of the distribution of the spreads, fitted on a basis constituted of two lognormal distributions, which have been chosen beforehand. The crossing by the empirical cumulative distribution function of either the upper or lower boundary of this Bollinger band is then interpreted in terms of risk and enables us to construct a trading signal.

L'une des stratégies utilisée pour augmenter l'efficacité spectrale (ES) des réseaux cellulaires est de réutiliser la bande de fréquences sur des zones relativement petites. Le problème majeur dans ce cas est un plus grand niveau d'interférence, diminuant l'efficacité énergétique (EE). En plus d'une plus grande largeur de bande, la densification des réseaux (cellules de petite taille ou multi-utilisateur à entrées multiples et sortie unique, MU-EMSO), peut augmenter l'efficacité spectrale par unité de surface (ESuS). La consommation totale d'énergie des réseaux sans fil augmente en raison de la grande quantité de puissance de circuit consommée par les structures de réseau denses, réduisant l'EE. Dans cette thèse, la région EE-SE est caractérisé dans un réseau cellulaire hexagonal en considérant plusieurs facteurs de réutilisation de fréquences (FRF), ainsi que l'effet de masquage. La région EE-ESuS est étudiée avec des processus de Poisson ponctuels (PPP) pour modéliser un réseau MU-EMSO avec un précodeur à rapport signal sur fuite plus bruit (RSFB). Différentes densités de station de base (SB) et nombre d'antennes aux SB avec une consommation d'énergie statique sont considérées.Nous caractérisons d'abord la région EE-SE dans le réseau cellulaire hexagonal pour différentes FRF, avec et sans masquage. Avec le masquage en plus de la perte de propagation, la mesure de coupure ε-EE-ES est proposée pour évaluer les performances. Les courbes EE-ES présentent une grande partie linéaire, due à la consommation de puissance statique, suivie d'une forte diminution de l'EE, puisque le réseau est homogène et limité par les interférences. Les résultats montrent qu'un FRF de 1 pour les régions proches de la SB et des FRF plus élevés dans la région plus proche du bord de la cellule améliorent le point optimal du EE-ES. De plus, un meilleur compromis EE-ES peut être obtenu avec une valeur plus élevée de coupure. En outre, un FRF de 1 est le meilleur choix pour une valeur élevée de coupure en raison d'une réduction du rapport signal sur interférence plus bruit (RSIB).Les précodeurs sont utilisés en liaison descendante des réseaux cellulaires MU-EMSO à accès multiple par division spatiale (AMDS) pour améliorer le RSIB. La géométrie stochastique a été utilisée intensivement pour analyser de tels systèmes complexes. Nous obtenons une expression analytique de l'ESuS en régime asymptotique, c.-à-d. nombre d'antennes et d'utilisateurs infinis, en utilisant des résultats de matrices aléatoires et de géométrie stochastique. Les SBs et les utilisateurs sont modélisés par deux PPP indépendants et le précodage RSFB est utilisé. L'EE est dérivée d'un modèle de consommation de puissance linéaire. Les simulations de Monte Carlo montrent que les expressions analytiques sont précises même pour un nombre faible d'antennes et d'utilisateurs. De plus, les courbes d'EE-ESuS ont une grande partie linéaire avant une forte décroissante de l'EE, comme pour les réseaux hexagonaux. Les résultats montrent également que le précodeur RSFB offre de meilleurs performances que le précodeur forçage à zéro (FZ), qui est typiquement utilisé dans la literature. Les résultats numériques pour le précodeur RSFB montrent que déployer plus de SBs ou d'antennes aux BSs augmente l'ESuS, mais que le gain dépend du rapport des densités SB-utilisateurs et du nombre d'antennes lorsque la densité de l'utilisateur est fixe. L'EE augmente seulement lorsque l'augmentation de l'ESuS est plus importante que l'augmentation de la consommation d'énergie par unité de surface. D'autre part, lorsque la densité d'utilisateur augmente, l'ESuS dans la région limitée par les interférences peut être améliorée en déployant davantage de SB sans sacrifier l'EE et le débit ergodique des utilisateurs. / One of the used strategies to increase the spectral efficiency (SE) of cellular network is to reuse the frequency bandwidth over relatively small areas. The major issue in this case is higher interference, decreasing the energy efficiency (EE). In addition to the higher bandwidth, densification of the networks (e.g. small cells or multi-user multiple input single output, MU-MISO) potentially increases the area spectral efficiency (ASE). The total energy consumption of the wireless networks increases due to the large amount of circuit power consumed by the dense network structures, leading to the decrease of EE. In this thesis, the EE-SE achievable region is characterized in a hexagonal cellular network considering several frequency reuse factors (FRF), as well as shadowing. The EE-ASE region is also studied using Poisson point processes (PPP) to model the MU-MISO network with signal-to-leakage-and-noise ratio (SLNR) precoder. Different base station (BS) densities and different number of BS antennas with static power consumption are considered.The EE-SE region in a hexagonal cellular network for different FRF, both with and without shadowing is first characterized. When shadowing is considered in addition to the path loss, the ε-SE-EE tradeoff is proposed as an outage measure for performance evaluation. The EE-SE curves have a large linear part, due to the static power consumption, followed by a sharp decreasing EE, since the network is homogeneous and interference-limited. The results show that FRF of 1 for regions close to BS and higher FRF for regions closer to the cell edge improve the EE-SE optimal point. Moreover, better EE-SE tradeoff can be achieved with higher outage values. Besides, FRF of 1 is the best choice for very high outage value due to the significant signal-to-interference-plus-noise ratio (SINR) decrease.In downlink, precoders are used in space division multiple access (SDMA) MU-MISO cellular networks to improve the SINR. Stochastic geometry has been intensively used to analyse such a complex system. A closed-form expression for ASE in asymptotic regime, i.e. number of antennas and number of users grow to infinity, has been derived using random matrix theory and stochastic geometry. BSs and users are modeled by two independent PPP and SLNR precoder is used at BS. EE is then derived from a linear power consumption model. Monte Carlo simulations show that the analytical expressions are tight even for moderate number of antennas and users. Moreover, the EE-ASE curves have a large linear part before a sharply decreasing EE, as observed for hexagonal network. The results also show that SLNR outperforms the zero-foring (ZF) precoder, which is typically used in literature. Numerical results for SLNR show that deploying more BS or a large number of BS antennas increase ASE, but the gain depends on the BS-user density ratio and on the number of antennas when user density is fixed. EE increases only when the increase in ASE dominates the increase of the power consumption per unit area. On the other hand, when the user density increases, ASE in interference-limited region can be improved by deploying more BS without sacrificing EE and the ergodic rate of the users.

Três ensembles, respectivamente relacionados com as distribuições Gaussiana, Lognormal e de Levy, são abordados neste trabalho primordialmente do ponto de vista da termodinâmica de seus espectros. Novas expressões para as grandezas termodinâmicas sao encontradas para os ensembles de Stieltjes e de Bertuola-Pato, e a conexão destes com os ensembles Gaussianos e estabelecida. Esta tese também se compromete com a continuação do desenvolvimento e aprimorarão do ensemble generalizado de Bertuola-Pato, estendendo alguns resultados para os ensembles simplifico e unitário generalizados, alem do ortogonal generalizado já introduzido anteriormente por A. C. Bertuola e M. P. Pato. / Three ensembles, related to the Gaussian, the Lognormal and the L´evy distributions respectively, have been studied in this work and were investigated most of all in what concerns their spectral thermodynamics. New expressions for the thermodynamics quantities were found for the Stieltjes and the Bertuola-Pato ensembles, and the connection with the gaussian ensembles is established. This work concerned with the development continuity and with the improvement of Bertuola-Pato generalized ensemble, extending some of the results to the simplectic and unitary generalized ensembles, besides the orthogonal generalized ensemble introduced before by A. C. Bertuola and M. P. Pato.

In this thesis, we have explored the commonalities and connections between different classes of quantum systems that do not thermalize. Specifically, we have (1) shown that localized systems possess conservation laws like integrable systems, which can be constructed in a systematic way and used to detect localization-delocalization transitions , (2) studied the phenomenon of many-body localization in a model with a single particle mobility edge, (3) shown that interesting finite-size scaling emerges, with universal exponents, when athermal quantum systems are forced to thermalize through the application of perturbations and (4) shown that these scaling laws also arise when a perturbation causes a crossover between quantum systems described by different random matrix ensembles. We conclude with a brief summary of each chapter. In Chapter 2, we have investigated the effects of finite size on the crossover between quantum integrable systems and non-integrable systems. Using exact diagonalization of finite-sized systems, we have studied this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L → ∞, non-integrability sets in for an arbitrarily small integrabilitybreaking perturbation. The crossover value of the perturbation scales as a power law ∼ L−3 when the integrable system is gapless and the scaling appears to be robust to microscopic details and the precise form of the perturbation. In Chapter 3, we have studied the crossover among different random matrix ensembles CHAPTER 6. CONCLUSION 127 [Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE)] realized in different microscopic models. We have found that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We have also found that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. Finally,we have conjectured that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system. In Chapter 4, we have outlined a procedure to construct conservation laws for Anderson localized systems. These conservation laws are found as power series in the hopping parameters. We have also obtained the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended depending on the strength of a coupling constant. We have formulated a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure for the Aubry-Andre model, we show that integrals of motion given by our construction are well-defined in the localized phase but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction. In Chapter 5, we have studied many body localization and investigated its nature in the presence of a single particle mobility edge. Employing the technique of exact diagonalization for finite-sized systems, we have calculated the level spacing distribution, time evolution of entanglement entropy, optical conductivity and return probability to characterize the nature of localization. The localization that develops in the presence of interactions in these systems appears to be different from regular Many-Body Localization (MBL) in that the growth of entanglement entropy with time is linear (like in CHAPTER 6. CONCLUSION 128 a thermal phase) instead of logarithmic but saturates to a value much smaller than the thermal value (like for MBL). All other diagnostics seem consistent with regular MBL

We reinvestigate the validity of mapping the problem of two onsite interacting particles in a random potential onto an effective random matrix model. To this end we first study numerically how the non-interacting basis is coupled by the interaction. Our results indicate that the typical coupling matrix element decreases significantly faster with increasing single-particle localization length than is assumed in the random matrix model. We further show that even for models where the dependency of the coupling matrix element on the single-particle localization length is correctly described by the corresponding random matrix model its predictions for the localization length can be qualitatively incorrect. These results indicate that the mapping of an interacting random system onto an effective random matrix model is potentially dangerous. We also discuss how Imry's block-scaling picture for two interacting particles is influenced by the above arguments.

At a first glance, artificial neural networks, with engineered learning algorithms and carefully chosen nonlinearities, are nothing like the complicated self-organized spiking neural networks studied by theoretical neuroscientists. Yet, both adapt to their inputs, keep information from the past in their state space and are able of learning, implying that some information processing principles should be common to both. In this thesis we study those principles by incorporating notions of systems theory, statistical physics and graph theory into artificial neural networks and theoretical neuroscience models. % TO DO: What is different in this thesis? -> classical signal processing with complex systems on top The starting point for this thesis is \ac{RC}, a learning paradigm used both in machine learning\cite{jaeger2004harnessing} and in theoretical neuroscience\cite{maass2002real}. A neural network in \ac{RC} consists of two parts, a reservoir – a directed and weighted network of neurons that projects the input time series onto a high dimensional space – and a readout which is trained to read the state of the neurons in the reservoir and combine them linearly to give the desired output. In classical \ac{RC}, the reservoir is randomly initialized and left untrained, which alleviates the training costs in comparison to other recurrent neural networks. However, this lack of training implies that reservoirs are not adapted to specific tasks and thus their performance is often lower than that of other neural networks. Our contribution has been to show how knowledge about a task can be integrated into the reservoir architecture, so that reservoirs can be tailored to specific problems without training. We do this design by identifying two features that are useful for machine learning: the memory of the reservoir and its power spectra. First we show that the correlations between neurons limit the capacity of the reservoir to retain traces of previous inputs, and demonstrate that those correlations are controlled by moduli of the eigenvalues of the adjacency matrix of the reservoir. Second, we prove that when the reservoir resonates at the frequencies that are present on the desired output signal, the performance of the readout increases. Knowing the features of the reservoir dynamics that we need, the next question is how to impose them. The simplest way to design a network with that resonates at a certain frequency is by adding cycles, which act as feedback loops, but this also induces correlations and hence memory modifications. To disentangle the frequencies and the memory design, we studied how the addition of cycles modifies the eigenvalues in the adjacency matrix of the network. Surprisingly, the shape of the eigenvalues is quite beautiful \cite{aceituno2019universal} and can be characterized using random matrix theory tools. Combining this knowledge with our result relating eigenvalues and correlations, we designed an heuristic that tailors reservoirs to specific tasks and showed that it improves upon state of the art \ac{RC} in three different machine learning tasks. Although this idea works in the machine learning version of \ac{RC}, there is one fundamental problem when we try to translate to the world of theoretical neuroscience: the proposed frequency adaptation requires prior knowledge of the task, which might not be plausible in a biological neural network. Therefore the following questions are whether those resonances can emerge by unsupervised learning, and which kind of learning rules would be required. Remarkably, these resonances can be induced by the well-known Spike Time-Dependent Plasticity (STDP) combined with homeostatic mechanisms. We show this by deriving two self-consistent equations: one where the activity of every neuron can be calculated from its synaptic weights and its external inputs and a second one where the synaptic weights can be obtained from the neural activity. By considering spatio-temporal symmetries in our inputs we obtained two families of solutions to those equations where a periodic input is enhanced by the neural network after STDP. This approach shows that periodic and quasiperiodic inputs can induce resonances that agree with the aforementioned \ac{RC} theory. Those results, although rigorous, are expressed on a language of statistical physics and cannot be easily tested or verified in real, scarce data. To make them more accessible to the neuroscience community we showed that latency reduction, a well-known effect of STDP\cite{song2000competitive} which has been experimentally observed \cite{mehta2000experience}, generates neural codes that agree with the self-consistency equations and their solutions. In particular, this analysis shows that metabolic efficiency, synchronization and predictions can emerge from that same phenomena of latency reduction, thus closing the loop with our original machine learning problem. To summarize, this thesis exposes principles of learning recurrent neural networks that are consistent with adaptation in the nervous system and also improve current machine learning methods. This is done by leveraging features of the dynamics of recurrent neural networks such as resonances and correlations in machine learning problems, then imposing the required dynamics into reservoir computing through control theory notions such as feedback loops and spectral analysis. Then we assessed the plausibility of such adaptation in biological networks, deriving solutions from self-organizing processes that are biologically plausible and align with the machine learning prescriptions. Finally, we relate those processes to learning rules in biological neurons, showing how small local adaptations of the spike times can lead to neural codes that are efficient and can be interpreted in machine learning terms.

In this thesis, we review some applications of random matrix theory in machine learning and theoretical deep learning. More specifically, we review data modelling in the regime of numerous and large dimensional data, a method for estimating covariance matrix distances in the aforementioned regime, as well as an asymptotic analysis of a simple neural network model in the limit where the number of neurons is large and the data is both numerous and large dimensional. We also review some recent research where random matrix models and methods have been applied to Hessian matrices of neural networks with interesting results. As becomes apparent, random matrix theory is a useful tool for various machine learning applications and it is a fruitful field of mathematics toexplore, in particular, in the context of theoretical deep learning. / I denna uppsatsen undersöker vi några tillämpningar av slumpmatristeori inom maskininlärning och teoretisk djupinlärning. Mer specifikt undersöker vi datamodellering i domänet där både datamängden och dimensionen på datan är stor, en metod för att uppskatta avstånd mellan kovariansmatriser i det tidigare nämnda domänet, samt en asymptotisk analys av en enkel neuronnätsmodell i gränsen där antalet neuroner är stort och både datamängden och dimensionen pådatan är stor. Vi undersöker också en del aktuell forskning där slumpmatrismodeller och metoder från slumpmatristeorin har tillämpats på Hessianska matriserför artificiella neuronnätverk med intressanta resultat. Det visar sig att slumpmatristeori är ett användbart verktyg för olika maskininlärningstillämpningaroch är ett område av matematik som är särskilt givande att utforska inom kontexten för teoretisk djupinlärning.

Les grandes matrices de covariance constituent certainement l’un des modèles les plus utiles pour les applications en statistiques en grande dimension, en communication numérique, en biologie mathématique, en finance, etc. Les travaux de Marcenko et Pastur (1967) ont permis de décrire le comportement asymptotique de la mesure spectrale de telles matrices formées à partir de N copies indépendantes de n observations d’une suite de variables aléatoires iid et sa convergence vers une distribution de probabilité déterministe lorsque N et n convergent vers l’infini à la même vitesse. Plus récemment, Merlevède et Peligrad (2016) ont démontré que dans le cas de grandes matrices de covariance issues de copies indépendantes d’observations d’un processus strictement stationnaire centré, de carré intégrable et satisfaisant des conditions faibles de régularité, presque sûrement, la distribution spectrale empirique convergeait étroitement vers une distribution non aléatoire ne dépendant que de la densité spectrale du processus sous-jacent. En particulier, si la densité spectrale est continue et bornée (ce qui est le cas des processus linéaires dont les coefficients sont absolument sommables), alors la distribution spectrale limite a un support compact. Par contre si le processus stationnaire exhibe de la longue mémoire (en particulier si les covariances ne sont pas absolument sommables), le support de la loi limite n'est plus compact et des études plus fines du comportement des valeurs propres sont alors nécessaires. Ainsi, cette thèse porte essentiellement sur l’étude des asymptotiques et des fluctuations des plus grandes valeurs propres de grandes matrices de covariance associées à des processus stationnaires à longue mémoire. Dans le cas où le processus stationnaire sous-jacent est Gaussien, l’étude peut être simplifiée via un modèle linéaire dont la matrice de covariance de population sous-jacente est une matrice de Toeplitz hermitienne. On montrera ainsi que dans le cas de processus stationnaires gaussiens à longue mémoire, les fluctuations des plus grandes valeurs propres de la grande matrice de covariance empirique convenablement renormalisées sont gaussiennes. Ce comportement indique une différence significative par rapport aux grandes matrices de covariance empirique issues de processus à courte mémoire, pour lesquelles les fluctuations de la plus grande valeur propre convenablement renormalisée suivent asymptotiquement la loi de Tracy-Widom. Pour démontrer notre résultat de fluctuations gaussiennes, en plus des techniques usuelles de matrices aléatoires, une étude fine du comportement des valeurs propres et vecteurs propres de la matrice de Toeplitz sous-jacente est nécessaire. On montre en particulier que dans le cas de la longue mémoire, les m plus grandes valeurs propres de la matrice de Toeplitz convergent vers l’infini et satisfont une propriété de type « trou spectral multiple ». Par ailleurs, on démontre une propriété de délocalisation de leurs vecteurs propres associés. Dans cette thèse, on s’intéresse également à l’universalité de nos résultats dans le cas du modèle simplifié ainsi qu’au cas de grandes matrices de covariance lorsque les matrices de Toeplitz sont remplacées par des matrices diagonales par blocs / Large covariance matrices play a fundamental role in the multivariate analysis and high-dimensional statistics. Since the pioneer’s works of Marcenko and Pastur (1967), the asymptotic behavior of the spectral measure of such matrices associated with N independent copies of n observations of a sequence of iid random variables is known: almost surely, it converges in distribution to a deterministic law when N and n tend to infinity at the same rate. More recently, Merlevède and Peligrad (2016) have proved that in the case of large covariance matrices associated with independent copies of observations of a strictly stationary centered process which is square integrable and satisfies some weak regularity assumptions, almost surely, the empirical spectral distribution converges weakly to a nonrandom distribution depending only on the spectral density of the underlying process. In particular, if the spectral density is continuous and bounded (which is the case for linear processes with absolutely summable coefficients), the limiting spectral distribution has a compact support. However, if the underlying stationary process exhibits long memory, the support of the limiting distribution is not compact anymore and studying the limiting behavior of the eigenvalues and eigenvectors of the associated large covariance matrices can give more information on the underlying process. This thesis is in this direction and aims at studying the asymptotics and the fluctuations of the largest eigenvalues of large covariance matrices associated with stationary processes exhibiting long memory. In the case where the underlying stationary process is Gaussian, the study can be simplified by a linear model whose underlying population covariance matrix is a Hermitian Toeplitz matrix. In the case of stationary Gaussian processes exhibiting long memory, we then show that the fluctuations of the largest eigenvalues suitably renormalized are Gaussian. This limiting behavior shows a difference compared to the one when large covariance matrices associated with short memory processes are considered. Indeed in this last case, the fluctuations of the largest eigenvalues suitably renormalized follow asymptotically the Tracy-Widom law. To prove our results on Gaussian fluctuations, additionally to usual techniques developed in random matrices analysis, a deep study of the eigenvalues and eigenvectors behavior of the underlying Toeplitz matrix is necessary. In particular, we show that in the case of long memory, the largest eigenvalues of the Toeplitz matrix converge to infinity and satisfy a property of “multiple spectral gaps”. Moreover, we prove a delocalization property of their associated eigenvectors. In this thesis, we are also interested in the universality of our results in the case of the simplified model and also in the case of large covariance matrices when the Toeplitz matrices are replaced by bloc diagonal matrices

La synchronisation temporelle des systèmes MIMO a été abondamment étudiée dans les quinze dernières années, mais la plupart des techniques existantes supposent que le bruit est blanc temporellement et spatialement, ce qui ne permet pas de modéliser la présence d'interférence. Nous considérons donc le cas de bruits blancs temporellement mais pas spatialement, dont la matrice de covariance spatiale est inconnue. En formulant le problème de l'estimation de l'instant de synchronisation comme un test d'hypothèses, nous aboutissons au test du rapport de vraisemblance généralisé (GLRT) qui donne lieu à la comparaison avec un seuil d'une statistique de test eta_GLRT. Cependant, pour des raisons de complexité, l'utilisation de cette statistique n'est pas toujours considérée comme réaliste. La première partie de ce travail a donc été consacrée à mettre en évidence des tests alternatifs moins complexes à mettre en œuvre, tout en ayant des performances similaires. Une analyse comparative exhaustive, prenant en considération le bruit et l'interférence, le type de canal, le nombre d'antennes en émission et en réception, et l'orthogonalité de la séquence de synchronisation est réalisée. Enfin, nous étudions le problème de l'optimisation du nombre d'antennes en émission K pour la synchronisation temporelle, montrant que pour un RSB élevé, les performances augmentent avec K dès que le produit de K avec le nombre d'antennes de réception M n'est pas supérieur à 8.Le deuxième aspect de ce travail est une analyse statistique de eta_GLRT dans le cas où la taille de la séquence d'apprentissage N est du même ordre de grandeur que M, ce qui conduit naturellement à étudier le comportement de eta_GLRT dans le régime asymptotique des grands systèmes M tend vers l'infini, N tend l'infini de telle sorte que M/N tende vers une constante non nulle. Nous considérons le cadre applicatif d'un système muni d'une unique antenne d'émission et d'un canal à trajets multiples, qui est formellement identique à celui d'un système MIMO dont le nombre d'antennes d'émissions correspondrait au nombre de trajets. Lorsque le nombre de trajets L est beaucoup plus faible que N et M, nous établissons que eta_GLRT a un comportement gaussien avec l'espérance asymptotique L log (1 / (1-M/N)) et la variance (L/N)*(M/N)/(1-M/N). Ceci est en contraste avec le régime asymptotique standard quand N tend vers l'infini et M et L fixe où eta_GLRT a un comportement chi2. Sous l'hypothèse H_1, eta_GLRT a aussi un comportement gaussien. Nous considérons également le cas où le nombre de trajets L tend vers l'infini à la même vitesse que M et N. Nous utilisons des résultats connus concernant le comportement des statistiques linéaires des valeurs propres des grandes F matrices, et déduisons que dans le régime où L,M,N tendent vers l'infini à la même vitesse, eta_GLRT a encore un comportement gaussien sous H_0, mais avec une espérance et variance différentes. L'analyse de eta_GLRT sous H_1 lorsque L,M,L convergent vers l'infini nécessite l'établissement d'un théorème central limite pour les statistiques linéaires des valeurs propres de matrices F de moyennes non-nulles, une tâche difficile. Motivé par les résultats obtenus dans le cas où L reste fini, nous proposons d'approximer la distribution asymptotique par une distribution gaussienne dont l'espérance et la variance sont la somme de l'espérance et la variance asymptotique sous H_0quand L tend vers l'infini avec l'espérance et la variance asymptotique sous H_1 dans le régime classique N tend vers l'infini et M fixé. Des simulations numériques permettent de comparer les courbes ROC des différents approximant avec des courbes ROC empiriques. Les résultats montrent que nos approximant de grandes dimensions fournissent de meilleurs résultats quand M/N augmente, tout en permettant de capturer la performance réelle pour les petites valeurs de M/N / Time synchronization of MIMO systems have been strongly studied in the last fifteen years, but most of the existing techniques assume a spatially and temporally white noise, which does not allow modeling the presence of interference. We consider thus a temporally white but spatially colored noise, with an unknown covariance matrix. Formulating the estimation problem as a hypothesis testing problem, we obtain a Generalized likelihood ratio test (GLRT), which gives us a synchronization statistics eta_GLRT. However, for complexity reasons, it is not always considered realistic for practical situations. A part of this work has thus been devoted to showing that there exist non-GLRT statistics that are less complex to implement than theet a_GLRT, while having similar performance. Furthermore, we perform a comparative parameter analysis, taking into consideration the noise type, channel type, the number of transmit and receive antennas, and the orthogonality of the synchronization sequence. Lastly, the problem of optimization of the number of transmit antennas K for time synchronization has been investigated. showing, for high SNR, increasing performance with K as long as the product KM is not larger than 8, where M is the number of receive antennas. The second aspect of MIMO synchronization studied in thesis is asymptotic analysis of the same GLRT, but for large M. In this context, the synchronization sequence length N is the same order of magnitude as M, and this leads us naturally to the study of the the behavior of eta_GLRT in the asymptotic regime where M,N go towards infinity such that M/N go towards a non-zero constant. We consider the case of a single transmit antenna in a multi-path channel, which formally is equivalent to the MIMO system where the transmit antennas correspond to the number of paths. We address the case When the number of paths L does not scale with M and N, we establish that eta_GLRT has a Gaussian behavior with asymptotic mean L log (1/ (1 - M/N))and variance (L/N)*(M/N)/(1-M/N).This is in contrast with the standard asymptotic regime N goes to infinity and M fixed where eta_GLRT has a chi^2 behaviour. Under hypothesis H_1, eta_GLRT still has a Gaussian behaviour. The corresponding asymptotic mean and variance are obtained as the sum of the asymptotic mean and variance in the standard regime N goes to infinity and M fixed, and L log(1/(1-/M/N))L log (1 / (1-M/N)) and (L/N)*(M/N)/(1-M/N)respectively, i.e. the asymptotic mean and variance under H_0.We also consider the case where the number of paths L converges towards infinity at the same rate as M and N. Using known results of concerning the behaviour of linear statistics of the eigenvalues of large F-matrices, we deduce that in the regime where L,M,N converge to infinity at the same rate, eta_GLRT still has a Gaussian behaviour under H_0, but with a different mean and variance. The analysis of eta_GLRT under H_1 whenL,M,N converge to infinity needs to establish a central limit theorem for linear statistics of the eigenvalues of large non zero-mean F-matrices, a difficult ask. Motivated by the results obtained in the case where L remains finite, we propose to approximate the asymptotic distribution of eta_GLRT by a Gaussian distribution whose mean and variance are the sum of the asymptotic mean and variance under H_0when L goes to infinity with the asymptotic mean and variance under H_1 in the standard regime N goes to infinity and M fixed. Numerical simulations allow to compare the ROC curves obtained with the different approximations with the empirical ROC curves. The results show that the large-system approximations provide better results when M/N increases, while also allowing to capture the actual performance for small values of M/N