Rank-dependent moderate deviations of U-empirical measures in strong topologies

 We prove a rank-dependent moderate deviation principle for U-empirical measures, where the underlying i.i.d. random variables take values in a measurable (not necessarily Polish) space (S,𝒮). The result can be formulated on a suitable subset of all signed measures on (S ^m ,𝒮^{⊗ m} ). We endow this space with a topology, which is stronger than the usual τ-topology. A moderate deviation principle for Banach-space valued U-statistics is obtained as a particular application. The advantage of our result is that we obtain in the degenerate case moderate deviations in non-Gaussian situations with non-convex rate functions. Received: 22 February 2000 / Revised version: 15 November 2002 / Published online: 28 March 2003 Research partially supported by the Swiss National Foundation, Contract No. 21-298333.90. Mathematics Subject Classification (2000): Primary 60F10; Secondary 62G20, 28A35 Key words or phrases: Rank-dependent moderate deviations – Empirical measures – Strong topology – U-statistics     

Rates of Convergence in the Blume–Emery–Griffiths Model

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature $\beta $ and the interaction strength $K$ . The rates of convergence results are obtained as $(\beta ,K)$ converges along appropriate sequences $(\beta _n,K_n)$ to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein’s method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry–Esseen quality, on approximation error.     

Large Deviations for Products of Empirical Probability Measures in the τ-Topology

We prove a large deviation principle (LDP) for products of empirical measures, where the state space S of the underlying sequence of i.i.d. random variables is Polish and the set of probability measures on S respectively S×S is endowed with the -topology. An improved form of a LDP for U-statistics and some conclusions from that are obtained as a particular application.     

A large deviation principle form-variate von Mises-statistics and U-statistics

A large deviation principle form-variate von Mises-statistics and U-statistics with a kernel function satisfying natural moment conditions is proved. Sanov's large deviation result for the empirical distribution function and two fundamental conservation principles in large deviation theory are the main tools. The rate functions are “drawback”-entropy functionals.     

Moderate Deviations via Cumulants

The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations probabilities due to Rudzkis, Saulis, and Statulevi?ius. The examples of random objects we treat include dependency graphs, subgraph-counting statistics in Erdös–Rényi random graphs and U-statistics. Moreover, we prove moderate deviation principles for certain statistics appearing in random matrix theory, namely characteristic polynomials of random unitary matrices and the number of particles in a growing box of random determinantal point processes such as the number of eigenvalues in the GUE or the number of points in Airy, Bessel, and sine random point fields.