The paper considers approximations of time-continuous epsilon-correlated random processes by interpolation of time-discrete Moving-Average processes. These approximations are helpful for Monte-Carlo simulations of the response of systems containing random parameters described by epsilon-correlated processes. The paper focuses on the approximation of stationary epsilon-correlated processes with a prescribed correlation function. Numerical results are presented.

This study investigates if momentum effects can be found on the Swedish stock market by testing a cross-sectional momentum strategy on historical data. To explain the results mathematically, a second approach, involving time series regression for predicting future returns is introduced and thereby extends the cross-sectional theory. The result of the study shows that momentum effects through the cross-sectional strategy exist on the Swedish stock market. Although positive return is found, the time series regression do not give any significance for predicting future returns. Hence, there is a contradiction between the two approaches. / Denna studie undersöker om momentumeffekter föreligger på den svenska aktiemarknaden med hjälp av två olika tillvägagångssätt. Först testas momentumstrategin på historisk data och därefter genomförs tidseriesregression för att undersöka om resultaten har statistisk signifikans för att prediktera framtida avkastning. Resultatet visar att momentumeffekter existerar på den svenska aktiemarknaden. Trots att positiv avkastning erhålls ger tidserieregressionen ingen indikation på att prediktering av framtida avkastning är möjlig. Följaktligen finns det en motsägelse mellan de två tillvägagångssätten.

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The paper considers approximations of time-continuous epsilon-correlated random processes by interpolation of time-discrete Moving-Average processes. These approximations are helpful for Monte-Carlo simulations of the response of systems containing random parameters described by epsilon-correlated processes. The paper focuses on the approximation of stationary epsilon-correlated processes with a prescribed correlation function. Numerical results are presented.

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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

Nesta tese consideramos um modelo de regressão não paramétrica, quando a variável explicativa e um processo estritamente estacionário e alpha-mixing. São estudadas as condições sobre o processo Xt e sua estrutura de dependência, assim como do domínio da função f a ser estimada. Também são feitas as adaptações necessárias aos procedimentos para obter as taxas de convergência do risco para a norma Lp, no caso de ondaletas deformadas. Em relação às ondaletas adaptativas de Haar, obtêm-se as taxas de convergência do risco do estimador proposto. Mediante estudos de simulação, e avaliado o desempenho dos procedimentos propostos quando aplicados a amostras finitas sob diferentes níveis de perturbação do sinal e diferentes tamanhos da amostra. Também são feitas aplicações a dados reais. / In this thesis we consider a nonparametric regression model, when the exploratory variables are alpha-mixing stationary processes. We obtain convergence rates for risk for Lp norm, via warped wavelets, under suitable regularity conditions. For estimation using design adapted Haar wavelets we obtain convergence rates for the risk of the proposed estimator. The performance of the estimators are assessed via simulation studies with dierent sample sizes and dierent signal-to-noise ratios. Applications to real data are also given.

Nesta tese consideramos um modelo de regressão não paramétrica, quando a variável explicativa e um processo estritamente estacionário e alpha-mixing. São estudadas as condições sobre o processo Xt e sua estrutura de dependência, assim como do domínio da função f a ser estimada. Também são feitas as adaptações necessárias aos procedimentos para obter as taxas de convergência do risco para a norma Lp, no caso de ondaletas deformadas. Em relação às ondaletas adaptativas de Haar, obtêm-se as taxas de convergência do risco do estimador proposto. Mediante estudos de simulação, e avaliado o desempenho dos procedimentos propostos quando aplicados a amostras finitas sob diferentes níveis de perturbação do sinal e diferentes tamanhos da amostra. Também são feitas aplicações a dados reais. / In this thesis we consider a nonparametric regression model, when the exploratory variables are alpha-mixing stationary processes. We obtain convergence rates for risk for Lp norm, via warped wavelets, under suitable regularity conditions. For estimation using design adapted Haar wavelets we obtain convergence rates for the risk of the proposed estimator. The performance of the estimators are assessed via simulation studies with dierent sample sizes and dierent signal-to-noise ratios. Applications to real data are also given.

In time series analysis, most of the models are based on the assumption of covariance stationarity. However, many time series in the applied sciences show a time-varying second-order structure. That is, variance and covariance, or equivalently the spectral structure, are likely to change over time. Examples may be found in a growing number of fields, such as biomedical time series analysis, geophysics, telecommunications, or financial data analysis, to name but a few. In this thesis, we are concerned with the modelling of such nonstationary time series, and with the subsequent questions of how to estimate their second-order structure and how to forecast these processes. We focus on univariate, discrete-time processes with zero-mean arising, for example, when the global trend has been removed from the data. The first chapter presents a simple model for nonstationarity, where only the variance is time-varying. This model follows the approach of "local stationarity" introduced by [1]. We show that our model satisfactorily explains the nonstationary behaviour of several economic data sets, among which are the U.S. stock returns and exchange rates. This chapter is based on [5]. In the second chapter, we study more complex models, where not only the variance is evolutionary. A typical example of these models is given by time-varying ARMA(p,q) processes, which are ARMA(p,q) with time-varying coefficients. Our aim is to fit such semiparametric models to some nonstationary data. Our data-driven estimator is constructed from a minimisation of a penalised contrast function, where the contrast function is an approximation to the Gaussian likelihood of the model. The theoretical performance of the estimator is analysed via non asymptotic risk bounds for the quadratic risk. In our results, we do not assume that the observed data follow the semiparamatric structure, that is our results hold in the misspecified case. The third chapter introduces a fully nonparametric model for local nonstationarity. This model is a wavelet-based model of local stationarity which enlarges the class of models defined by Nason et al. [3]. A notion of time-varying "wavelet spectrum' is uniquely defined as a wavelet-type transform of the autocovariance function with respect to so-called "autocorrelation wavelets'. This leads to a natural representation of the autocovariance which is localised on scales. One particularly interesting subcase arises when this representation is sparse, meaning that the nonstationary autocovariance may be decomposed in the autocorrelation wavelet basis using few coefficients. We present a new test of sparsity for the wavelet spectrum in Chapter 4. It is based on a non-asymptotic result on the deviations of a functional of a periodogram. In this chapter, we also present another application of this result given by the pointwise adaptive estimation of the wavelet spectrum. Chapters 3 and 4 are based on [6] Computational aspects of the test of sparsity and of the pointwise adaptive estimator are considered in Chapter 5. We give a description of a full algorithm, and an application in biostatistics. In this chapter, we also derive a new test of covariance stationarity, applied to another case study in biostatistics. This chapter is based on [7]. Finally, Chapter 6 address the problem how to forecast the general nonstationary process introduced in Chapter 3. We present a new predictor and derive the prediction equations as a generalisation of the Yule-Walker equations. We propose an automatic computational procedure for choosing the parameters of the forecasting algorithm. Then we apply the prediction algorithm to a meteorological data set. This chapter is based on [2,4]. References [1] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist., 25, 1-37, 1997. [2] Fryzlewicz, P., Van Bellegem, S. and von Sachs, R. (2003). Forecasting non-stationary time series by wavelet process modelling. Annals of the Institute of Statistical Mathematics. 55, 737-764. [3] Nason, G.P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of evolutionary wavelet spectra. Journal of the Royal Statistical Society Series B. 62, 271-292. [4] Van Bellegem, S., Fryzlewicz, P. and von Sachs, R. (2003). A wavelet-based model for forecasting non-stationary processes. In J-P. Gazeau, R. Kerner, J-P. Antoine, S. Metens and J-Y. Thibon (Eds.). GROUP 24: Physical and Mathematical Aspects of Symmetries. Bristol: IOP Publishing (in press). [5] Van Bellegem, S. and von Sachs, R. (2003). Forecasting economic time series with unconditional time-varying variance. International Journal of Forecasting (in press). [6] Van Bellegem, S. and von Sachs, R. (2003). Locally adaptive estimation of sparse, evolutionary wavelet spectra (submitted). [7] Van Bellegem, S. and von Sachs, R. (2003). On adaptive estimation for locally stationary wavelet processes and its applications (submitted).