arXiv subjects
Publications and source records attributed to Davide Giraudo.
We investigate the convergence of partial sum processes based on a strictly stationary $\beta$-mixing sequence of random variables. The convergence in the space of continuous function as well as in H{\"o}lder spaces is considered. The conditions are close to optimality.
In this paper, we establish an exponential inequality for U-statistics of i.i.d. data, varying kernel and taking values in a separable Hilbert space. The bound are expressed as a sum of an exponential term plus an other one involving the tail of a sum of squared norms. We start by the degenerate case. Then we provide applications to U-statistics of not necessarily degenerate fixed kernel, weighted U-statistics and incomplete U-statistics.
The now classical convergence in distribution theorem for well normalized sums ofstationary martingale increments has been extended to multi-indexed martingaleincrements (see Volný (2019) and references in there). In the presentarticle we make progress in the identification of the limit law.In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is stillthe case for multi-indexed martingale increments when one of the processes defined by one coordinate of the{\it multidimensional time} is ergodic. In the general case, the limit may be non normal.The dynamical properties of the $\mathbb{Z}^d$-measure preserving action associatedto the stationary random field allows us to give a necessary and sufficient conditionfor the existence of a non-normal limit law, in terms of entropy of some random processes.The identification of a {\it natural} factor on which the $\mathbb{Z}^d$-action is {\it of product type
We show a deviation inequality for U-statistics of independent data taking values in a separable Banach space which satisfies some smoothness assumptions. We then provide applications to rates in the law of large numbers for U-statistics, a H{ö}lderian functional central limit theorem and a moment inequality for incomplete $U$-statistics.
In this note, we give sufficient conditions for the almost sure and the convergence in $\mathbb{L}^p$ of a $U$-statistic of order $m$ built on a strictly stationary but not necessarily ergodic sequence.
In this paper, we investigate the law of large numbers for strictly stationary random fields, that is, we provide sufficient conditions on the moments and the dependence of the random field in order to guarantee the almost sure convergence to $0$ and the convergence in $\mathbb L^p$ of partials sums over squares or rectangles of $\mathbb Z^d$. Approximation by multi-indexed martingales as well as by $m$-dependent random fields are investigated. Applications to functions of $d$-independent Bernoulli shifts and to functionals of i.i.d.\ random fields are also provided.
In this paper, we investigate the functional central limit theorem and the Marcinkiewicz strong law of large numbers for U-statistics having absolutely regular data and taking value in a separable Hilbert space. The novelty of our approach consists in using coupling in order to formulate a deviation inequality for original $U$-statistic, where the upper bound involves the mixing coefficient and the tail of several U-statistics of i.i.d. data. The presented results improve the known results in several directions: the case of metric space valued data is considered as well as Hilbert space valued, and the mixing rates are less restrictive in a wide range of parameters.
In this paper, we study the asymptotic distribution of some U-statistics whose entries are functions of empirical moments computed from non-overlapping consecutive blocks of an underlying weakly dependent process. The length of these blocks converges to infinity, and thus we consider U-statistics of triangular arrays. We establish asymptotic normality of such U-statistics. The results can be used to construct tests for changes of higher order moments.
In this paper, we establish an exponential inequality for random fields, which is applied in the context of convergence rates in the law of large numbers and Hölderian weak invariance principle.
In this note, we investigate the convergence of a $U$-statistic of order two having stationary ergodic data. We will find sufficient conditions for the almost sure and $L^1$ convergence and present some counter-examples showing that the $U$-statistic itself might fail to converge: centering is needed as well as boundedness of $\sup_{j\geq 2}\mathbb{E}[|h(X_1,X_j)|]$.
We show a deviation inequality inequalities for multi-indexed martingale We then provide applications to kernel regression for random fields and rates in the law of large numbers for orthomartingale difference random fields.
We provide a sufficient condition for the bounded law of the iterated logarithms for strictly stationary random fields expressable as a functional of i.i.d. random fields when the summation is done on rectangles. The study is done via the control of the moments of an appropriated maximal function. Applications to functionals of linear random fields, functions of a Gaussian linear random field and Volterra process are given.
We give sufficient conditions for the bounded law of the iterated logarithms for strictly stationary random fields when the summation is done on rectangle. The study is done by the control of an appropriated maximal function. The case of orthomartingales is treated. Then results on projective conditions are derived.
In this paper, we consider U-statistics whose data is a strictly stationary sequence which can be expressed as a functional of an i.i.d. one. We establish a strong law of large numbers, a bounded law of the iterated logarithms and a central limit theorem under a dependence condition. The main ingredients for the proof are an approximation by U-statistics whose data is a functional of $\ell$ i.i.d. random variables and an analogue of the Hoeffding's decomposition for U-statistics of this type.
We consider the empirical two-sample $U$-statistic with strictly $β$-mixing strictly stationary data and inverstigate its convergence in Skorohod spaces. We then provide an application of such convergence.
We consider a change-point test based on the Hill estimator to test for structural changes in the tail index of Long Memory Stochastic Volatility time series. In order to determine the asymptotic distribution of the corresponding test statistic, we prove a uniform reduction principle for the tail empirical process in a two-parameter Skorohod space. It is shown that such a process displays a dichotomous behavior according to an interplay between the Hurst parameter, i.e., a parameter characterizing the dependence in the data, and the tail index. Our theoretical results are accompanied by simulation studies and the analysis of financial time series with regard to structural changes in the tail index.
We establish an exponential inequality for degenerated $U$-statistics of order $r$ of i.i.d. data. This inequality gives a control of the tail of the maxima absolute values of the $U$-statistic by the sum of two terms: an exponential term and one involving the tail of $h\left(X_1,\dots,X_r\right)$. We also give a version for not necessarily degenerated $U$-statistics having a symmetric kernel and furnish an application to the convergence rates in the Marcinkiewicz law of large numbers. Application to invariance principle in Hölder spaces is also considered.
Let $(X_i, \mathcal{F}_i)_{i\geq1}$ be a martingale difference sequence in a smooth Banach space. Let $S_n=\sum_{i=1}^nX_i, n\geq 1,$ be the partial sums of $(X_i, \mathcal{F}_i)_{i\geq 1}$. We give upper bounds on the quantity $\mathbb{P}\left(\max_{1\leq k\leq n}\lVert S_k\rVert>nx\right)$ in terms of $ n\geq 1$ and $x>0$ in two different situations: when the martingale differences have uniformly bounded exponential moments and when the decay of the tail of the increments is polynomial.