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We introduce an 𝓛₂-type test for testing mutual independence and banded dependence structure for high dimensional data. The test is constructed on the basis of the pairwise distance covariance and it accounts for the non-linear and non-monotone dependences among the data, which cannot be fully captured by the existing tests based on either Pearson correlation or rank correlation. Our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal. It exhibits excellent finite sample performance in our simulation studies even when the sample size is small albeit the dimension is high and is shown to identify non-linear dependence in empirical data analysis successfully. On the theory side, asymptotic normality of our test statistic is shown under quite mild moment assumptions and with little restriction on the growth rate of the dimension as a function of sample size. As a demonstration of good power properties for our distance-covariance-based test, we further show that an infeasible version of our test statistic has the rate optimality in the class of Gaussian distributions with equal correlation.

We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate U-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (J. Multivariate Anal. 34 (1990) 275–289), Nourdin, Peccati and Reinert (Ann. Probab. 38 (2010) 1947–1985) and Döbler and Peccati (Electron. J. Probab. 22 (2017) no. 2), (ii) a new Gamma approximation bound on the Poisson space, refining previous estimates by Peccati and Thäle (ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 525–560) and (iii) new Gamma bounds on a Gaussian space, strengthening estimates by Nourdin and Peccati (Probab. Theory Related Fields 145 (2009) 75–118). As a by-product of our analysis, we also deduce a new inequality for Gamma approximations via exchangeable pairs, that is of independent interest.

We study the distribution of a general class of asymptotically linear statistics which are symmetric functions of N independent observations. The distribution functions of these statistics are approximated by an Edgeworth expansion with a remainder of order o(N-1). The Edgeworth expansion is based on Hoeffding’s decomposition which provides a stochastic expansion into a linear part, a quadratic part as well as smaller higher order parts. The validity of this Edgeworth expansion is proved under Cramér’s condition on the linear part, moment assumptions for all parts of the statistic and an optimal dimensionality requirement for the non linear part.

In this paper, we investigate the limiting spectral distribution of a high-dimensional Kendall’s rank correlation matrix. The underlying population is allowed to have a general dependence structure. The result no longer follows the generalized Marc̆enko-Pastur law, which is brand new. It is the first result on rank correlation matrices with dependence. As applications, we study Kendall’s rank correlation matrix for multivariate normal distributions with a general covariance matrix. From these results, we further gain insights into Kendall’s rank correlation matrix and its connections with the sample covariance/correlation matrix.

We derive a central limit theorem for the number of vertices of convex polytopes induced by stationary Poisson hyperplane processes in${\\Bbb R}^{d}$. This result generalizes an earlier one proved by Paroux [Adv. in Appl. Probab. 30 (1998) 640-656] for intersection points of motion-invariant Poisson line processes in${\\Bbb R}^{2}$. Our proof is based on Hoeffding's decomposition of U-statistics which seems to be more efficient and adequate to tackle the higher-dimensional case than the \"method of moments\" used in [Adv. in Appl. Probab. 30 (1998) 640-656] to treat the case d = 2. Moreover, we extend our central limit theorem in several directions. First we consider k-flat processes induced by Poisson hyperplane processes in${\\Bbb R}^{d}$for 0 ≤ k ≤ d - 1. Second we derive (asymptotic) confidence intervals for the intensities of these k-flat processes and, third, we prove multivariate central limit theorems for the d-dimensional joint vectors of numbers of k-flats and their k-volumes, respectively, in an increasing spherical region.

We prove a multivariate functional version of de Jong’s CLT (J Multivar Anal 34(2):275–289, 1990) yielding that, given a sequence of vectors of Hoeffding-degenerate U-statistics, the corresponding empirical processes on [0, 1] weakly converge in the Skorohod space as soon as their fourth cumulants in t=1 vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified. As an application, we lift to the functional level the ‘universality of Wiener chaos’ phenomenon first observed in Nourdin et al. (Ann Probab 38(5):1947–1985, 2010).

Wasssily Hoeffding's terminal illness and untimely death in 1991 put an end to efforts that were made to interview him for Statistical Science. An account of his scientific work is given in Fisher and Sen [The Collected Works of Wassily Hoeffding (1994) Springer], but the present authors felt that the statistical community should also be told about the life of this remarkable man. He contributed much to statistical science, but will also live on in the memory of those who knew him as a kind and modest teacher and friend, whose courage and learning were matched by a wonderful sense of humor.

We derive a new representation for U-and V-statistics. Using this representation, the asymptotic distribution of U-and V-statistics can be derived by a direct application of the Continuous Mapping theorem. That novel approach not only encompasses most of the results on the asymptotic distribution known in literature, but also allows for the first time a unifying treatment of non-degenerate and degenerate U-and V-statistics. Moreover, it yields a new and powerful tool to derive the asymptotic distribution of very general U-and V-statistics based on long-memory sequences. This will be exemplified by several astonishing examples. In particular, we shall present examples where weak convergence of U-or V-statistics occurs at the rate $a_n^3$ and $a_n^4$, respectively, when an is the rate of weak convergence of the empirical process. We also introduce the notion of asymptotic (non-) degeneracy which often appears in the presence of long-memory sequences.

For a measure preserving transformation T of a probability space ( X , F , μ ) and some d ≥ 1 we investigate almost sure and distributional convergence of random variables of the form x → 1 C n ∑ 0 ≤ i 1 , … , i d < n f ( T i 1 x , … , T i d x ) , n = 1 , 2 , … , where C 1 , C 2 , … are normalizing constants and the kernel f belongs to an appropriate subspace in some L p ( X d , F ⊗ d , μ d ) . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with T and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and d = 2 , we also show that the convergence holds in distribution towards a quadratic form ∑ m = 1 ∞ λ m η m 2 in independent standard Gaussian variables η 1 , η 2 , … .

In this paper the validity of a one-term Edgeworth expansion for Studentized symmetric statistics is proved. We propose jackknife estimates for the unknown constants appearing in the expansion and prove their consistency. As a result we obtain the second-order correctness of the empirical Edgeworth expansion for a very general class of statistics, including U-statistics, L-statistics and smooth functions of the sample mean. We illustrate the application of the bootstrap in the case of a U-statistic of degree two.