A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order $q$ generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for $q=2$ and $q=4$. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and $U$-statistics on the Poisson space

Four moments theorems on Markov chains

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumptionwemake on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.First author draf

Four moments theorems on Markov chaos

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carr\'e du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.Comment: 24 page

On Higher Order Elicitability and Some Limit Theorems on the Poisson and Wiener Space

This PhD thesis consists of two independent parts. The first one is dedicated to a thorough study of higher order elicitability whereas the second part is concerned with qualitative and quantitative limit theorems for Poisson and Gaussian functionals. It comprises a total number of four articles, three of them already published in peer-reviewed journals (Annals of Statistics, Risk Magazine, and ALEA), the fourth one in a preprint version. The articles are accompanied by detailed additional material, primarily concerning questions of order-sensitivity, order-preservingness and convexity of strictly consistent scoring functions

Quantitative de Jong theorems in any dimension

We develop a new quantitative approach to a multidimensional version of the well-known {\it de Jong's central limit theorem} under optimal conditions, stating that a sequence of Hoeffding degenerate $U$-statistics whose fourth cumulants converge to zero satisfies a CLT, as soon as a Lindeberg-Feller type condition is verified. Our approach allows one to deduce explicit (and presumably optimal) Berry-Esseen bounds in the case of general $U$-statistics of arbitrary order $d\geq1$. One of our main findings is that, for vectors of $U$-statistics satisfying de Jong' s conditions and whose covariances admit a limit, componentwise convergence systematically implies joint convergence to Gaussian: this is the first instance in which such a phenomenon is described outside the frameworks of homogeneous chaoses and of diffusive Markov semigroups.Comment: 40 pages, to appear in: Electronic Journal of Probabilit

Asymptotic Behaviour of Random Vandermonde Matrices with Entries on the Unit Circle

Analytical methods for finding moments of random Vandermonde matrices with entries on the unit circle are developed. Vandermonde Matrices play an important role in signal processing and wireless applications such as direction of arrival estimation, precoding, and sparse sampling theory, just to name a few. Within this framework, we extend classical freeness results on random matrices with independent, identically distributed (i.i.d.) entries and show that Vandermonde structured matrices can be treated in the same vein with different tools. We focus on various types of matrices, such as Vandermonde matrices with and without uniform phase distributions, as well as generalized Vandermonde matrices. In each case, we provide explicit expressions of the moments of the associated Gram matrix, as well as more advanced models involving the Vandermonde matrix. Comparisons with classical i.i.d. random matrix theory are provided, and deconvolution results are discussed. We review some applications of the results to the fields of signal processing and wireless communications.Comment: 28 pages. To appear in IEEE Transactions on Information Theor

Characterizations, Sub and Resampling, and Goodness of Fit

We present a general proposal for testing for goodness of fit, based on resampling and subsampling methods, and illustrate it with graphical and analytical tests for the problems of testing for univariate or multivariate normality. The proposal shows promising, and in some cases dramatic, success in detecting nonnormality. Compared to common competitors, such as a Q-Q plot or a likelihood ratio test against a specified alternative, our proposal seems to be the most useful when the sample size is small, such as 10 or 12, or even very small, such as 6! We also show how our proposal provides tangible information about the nature of the true cdf from which one is sampling. Thus, our proposal also has data analytic value. Although only the normality problem is addressed here, the scope of application of the general proposal should be much broader

Four moments theorems on Markov chaos

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumptionwe make on the Pearson distribution is that it admits four moments. These results are obtained by rst proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diusion generator and invariant measures of diusions, which is of independent interest, and making use of the new concept of chaos grade. For the heavy-tailed Pearson distributions, this seems to be the rst time that sucient conditions in terms of (nitely many) moments are given in order to converge to a distribution that is not characterized by its moments