Weak invariance principle in Besov spaces for stationary martingale differences

The classical Donsker weak invariance principle is extended to a Besov spaces framework. Polygonal line processes build from partial sums of stationary martingale differences as well independent and identically distributed random variables are considered. The results obtained are shown to be optimal

On the rate of convergence in the martingale CLT

Bounds are found on the accuracy of the Gaussian approximation of discreet time martingales with values in a multidimensional space. The bounds are with respect to the Kantorovich and the Prohorov metrics.Central limit theorem Martingale Banach space Convergence rates

A central limit theorem for self-normalized sums of a linear process

Let be a linear process, where and [var epsilon]t, t[set membership, variant]Z, are i.i.d. r.v.'s in the domain of attraction of a normal law with zero mean and possibly infinite variance. We prove a central limit theorem for self-normalized sums where is a sum of squares of block-sums of size m, as m and the number of blocks N=n/m tend to infinity.Linear process Normal law Self-normalization

Invariance principles for adaptive self-normalized partial sums processes

Let [zeta]nse be the adaptive polygonal process of self-normalized partial sums Sk=[summation operator]1[less-than-or-equals, slant]i[less-than-or-equals, slant]kXi of i.i.d. random variables defined by linear interpolation between the points (Vk2/Vn2,Sk/Vn), k[less-than-or-equals, slant]n, where Vk2=[summation operator]i[less-than-or-equals, slant]k Xi2. We investigate the weak Hölder convergence of [zeta]nse to the Brownian motion W. We prove particularly that when X1 is symmetric, [zeta]nse converges to W in each Hölder space supporting W if and only if X1 belongs to the domain of attraction of the normal distribution. This contrasts strongly with Lamperti's FCLT where a moment of X1 of order p>2 is requested for some Hölder weak convergence of the classical partial sums process. We also present some partial extension to the nonsymmetric case.Functional central limit theorem Domain of attraction Holder space Randomization

Principe d'invariance pour processus de sommation multiparamétrique et applications

A thèse a pour objet de prouver le principe d'invariance dans des espaces de Hölder pour le processus de sommation multiparamétrique et d'utiliser ce résultat en détection de rupture dans des données de panel. On caractérise d'abord la convergence en loi dans un espace de Hölder, du processus de sommation multiparamétrique dans le cas d'un champ aléatoire i.i.d. d'éléments aléatoires centrés et de carré intégrable d'un espace de Hilbert séparable, par la finitude d'un certain moment faible dont l'ordre croît avec l'exposant de Hölder, depuis deux lorsque l'exposant est nul, jusqu'à l'infini lorsque l'exposant est un demi. Ensuite on considère les tableaux triangulaires centrés à valeurs réelles. On propose une construction adaptative du processus de sommation qui coïncide avec la construction classique pour le cas d'un seul paramètre. On prouve le théorème limite central fonctionnel hölderien pour ce processus. Le processus limite est gaussien sous certaines conditions de régularité pour les variances du tableau triangulaire, le drap de Wiener n'étant qu'un cas particulier. Enfin on fournit des applications de ces résultats théoriques en construisant des statistiques de détection de rupture épidémique dans un ensemble de données multi-indexées. On construit un test de détection d'un changement d'espérance dans un rectangle épidémique, trouve sa loi limite et donne des conditions pour sa consistance. On adapte notre statistique pour la détection de rupture du coefficient dans les modèles classiques de régression pourpanel.The thesis is devoted to proving invariance principle in Hëlder spaces for the multi-parameter summation process and then using this resull to construct the tests for detecting' structural breaks in panel data. First we characterize the weak convergence in Hëlder space of multi-parameter summation process in the case of Li.d. random field of square integrable centered random elements in the separable Hilbert space by the finiteness of the certain weak moment, whose order increases with the Hblder exponent, turning to two, when exponent is zero and ta infinity when exponent is one hait. Next we consider real valued centered triangular arrays. We propose adaptive construction of the summation process which coincides with classical construction for the one parameter case. We prove the functional central Iimit theorem for this process in Hëlder space. The limiting process is Gaussian under certain regularity condition for variances of the triangular array, Wlener sheet being the special case. Finally we provide sorne application of the theoretical results by constructing statistics for detecting the epidemic change in a given data with multi-dimensional indexes. We construct a test for detecting the change of the mean in a epidemic rectangle, find its asymptotic distribution and give the conditions for the consistency. We adapt our proposed statistic for detecting the change of the coefficient in the classical panel regression models.LILLE1-Bib. Electronique (590099901) / SudocSudocFranceF

Invariance principle for multiparameter summation processes and applications

LILLE1-BU (590092102) / SudocSudocFranceF

A Hölderian functional central limit theorem for a multi-indexed summation process

Let be an i.i.d. random field of square integrable centered random elements in the separable Hilbert space and , , be the summation processes based on the collection of sets [0,t1]x...x[0,td], 0 =2, we characterize the weak convergence of in the Hölder space by the finiteness of the weak p moment of for p=(1/2-[alpha])-1. This contrasts with the Hölderian FCLT for d=1 and [A. Rackauskas, Ch. Suquet, Necessary and sufficient condition for the Lamperti invariance principle, Theory Probab. Math. Statist. 68 (2003) 115-124] where the necessary and sufficient condition is P(X1>t)=o(t-p).Brownian sheet Hilbert space valued Brownian sheet Hilbert space Functional central limit theorem Holder space Invariance principle Summation process