U-statistics are widely used in fields such as economics, machine learning, and statistics. However, while they enjoy desirable statistical properties, have an obvious drawback that the computation becomes impractical data size $n$ increases. Specifically, number of combinations, say $m$, a U-statistic order $d$ has to evaluate is $O(n^d)$. Many efforts been made approximate original using smal...

This paper investigates weak convergence of U -statistics via approximation in probability. The classical condition that the second moment of the kernel of the underlying U -statistic exists is relaxed to having 43 moments only (modulo a logarithmic term). Furthermore, the conditional expectation of the kernel is only assumed to be in the domain of attraction of the normal law (instead of the c...

This paper considers testing a covariance matrix in the high dimensional setting where the dimension p can be comparable or much larger than the sample size n. The problem of testing the hypothesis H0 : = 0 for a given covariance matrix 0 is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius n...

Abstract A uniform in probability approximation is established for Studentized processes of non degenerate U -statistics of order m ≥ 2 in terms of a standard Wiener process. The classical condition that the second moment of kernel of the underlying U-statistic exists is relaxed to having 5 3 moments. Furthermore, the conditional expectation of the kernel is only assumed to be in the domain of ...

A weighted U -statistic based on a random sample X1, . . . ,Xn has the form Un = ∑1≤i,j≤n wi−jK(Xi,Xj ), where K is a fixed symmetric measurable function and the wi are symmetric weights. A large class of statistics can be expressed as weighted U -statistics or variations thereof. This paper establishes the asymptotic normality of Un when the sample observations come from a nonlinear time serie...

A weighted U -statistic based on a random sample X1, . . . , Xn has the form Un = ∑ 1≤i,j≤n wi−jK(Xi, Xj) where K is fixed symmetric measurable function and the wi are symmetric weights. A large class of statistics can be expressed as weighted U -statistics or variations thereof. This paper establishes the asymptotic normality of Un when the sample observations come from a non-linear time serie...

The subject of this talk was the review of our study of three (2 + 1) dimensional Quantum Chromodynamics. In our previous works, we showed the existence of a phase where parity is unbroken and the flavor group U (2n) is broken to a subgroup U (n) × U (n). We derived the low energy effective action for the theory and showed that it has solitonic excitations with Fermi statistic, to be identified...

A general model is proposed for studying ranking problems. We investigate learning methods based on empirical minimization of the natural estimates of the ranking risk. The empirical estimates are of the form of a U -statistic. Inequalities from the theory of U -statistics and U processes are used to obtain performance bounds for the empirical risk minimizers. Convex risk minimization methods a...

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of U and V -type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centered chisquared random variables, a direct proof for the bootstrap counterpart can be difficult for several reasons. We employ simple coupling...