Peter Eichelsbacher

Large deviations of U -empirical measures in strong topologies and applications

Abstract We prove large deviation principles (LDP) for m-fold products of empirical measures and for U-empirical measures, where the underlying i.i.d. random variables take values in a measurable (not necessarily Polish) space (S, S ) . The results can be formulated on suitable subsets of all probability measures on (S m , S ⊗m ) . We endow the spaces with topologies, which are stronger than the usual τ-topology and which make integration with respect to certain unbounded, Banach-space valued functions a continuous operation. A special feature is the non-convexity of the rate function for m⩾2. Improved versions of LDPs for Banach-space valued U- and V-statistics are obtained as a particular application. Some further applications concerning the Gibbs conditioning principle and a process level LDP are mentioned.

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 Statulevičius. 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.

Moderate deviations for stabilizing functionals in geometric probability

The purpose of the present paper is to establish explicit bounds on moderate deviation probabilities for a rather general class of bounded geometric functionals enjoying the exponential stabilization property. Compared to our previous work [1], the price to pay for the considerably larger generality of our estimates is a narrower scale range in which our moderate deviation results hold; we argue that a range limitation seems inevitable under our general assumptions though. Our proof techniques rely on cumulant expansions and cluster measures and yield completely explicit bounds on deviation probabilities. The examples of geometric functionals we treat include bounded statistics of random packing models and random graphs arising in computational geometry, such as Euclidean nearest neighbor graphs, Voronoi graphs and sphere of influence graphs.

Moderate Deviations for Some Point Measures in Geometric Probability

Functionals in geometric probability are often expressed as sums of bounded functions exhibiting expo- nential stabilization. Methods based on cumulant techniques and exponential modifications of measures show that such functionals satisfy moderate deviation principles. This leads to moderate deviation principles and laws of the iterated logarithm for random packing models as well as for statistics associated with germ-grain models and k nearest neighbor graphs.

LARGE DEVIATIONS FOR DISORDERED BOSONS AND MULTIPLE ORTHOGONAL POLYNOMIAL ENSEMBLES

We prove a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles that includes biorthogonal Laguerre, Jacobi, and Hermite ensembles, the matrix model of Lueck, Sommers, and Zirnbauer for disordered bosons, the Stieltjes-Wigert matrix model of Chern-Simons theory, and Angelesco ensembles.

EXPONENTIAL APPROXIMATIONS IN COMPLETELY REGULAR TOPOLOGICAL SPACES AND EXTENSIONS OF SANOV'S THEOREM

This paper is devoted to the well known transformations that preserve a large deviation principle (LDP), namely, the contraction principle with approximately continuous maps and the concepts of exponential equivalence and exponential approximations. We generalize these transformations to completely regular topological state spaces, give some examples and, as an illustration, reprove a generalization of Sanovs theorem, due to de Acosta (J. Appl. Probab. 31 A (1994) 41-47). Using partition-dependent couplings, we then extend this version of Sanovs theorem to triangular arrays and prove a full LDP for the empirical measures of exchangeable sequences with a general measurable state space.

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. Sanovs 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.

Stein’s method for discrete Gibbs measures

Steins method provides a way of bounding the distance of a probability distribution to a target distribution

Compound Poisson Approximation for Dissociated Random Variables via Stein's Method

\mu

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

. Here we develop Steins method for the class of discrete Gibbs measures with a density