Lower bounds for the distribution function of a k-dimensionaln-extensible exchangeable process

A lower bound for the distribution function of a k-dimensional, n-extensible exchangeable process is provided when the marginals are uniform on the unit segment. The result is obtained by means of standard linear programming techniques. The lower bound for infinitely extendible exchangeable processes is the distribution of independent random variables.     

On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations

Summary We consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous Boltzmann type equation, as the number of particles tends to infinity. These laws of large numbers were proved for the Maxwellian molecules by H. Tanaka [Tal] and for the hard spheres by A.S. Sznitman [Szl]. In the present paper we investigate the corresponding large deviations: the large deviation upper bound is obtained and, using convex analysis, a non-variational formulation of the rate function is given. Our results hold for Maxwellian molecules with a cutoff potential and for hard spheres.     

Rates of convergence of random walk on distance regular graphs

When run on any non-bipartite q-distance regular graph from a family containing graphs of arbitrarily large diameter d, we show that d steps are necessary and sufficient to drive simple random walk to the uniform distribution in total variation distance, and that a sharp cutoff phenomenon occurs. For most examples, we determine the set on which the variation distance is achieved, and the precise rate at which it decays. The upper bound argument uses spectral methods – combining the usual Cauchy-Schwarz bound on variation distance with a bound on the tail probability of a first-hitting time, derived from its generating function. Received: 2 April 1997 / Revised version: 10 May 1998     

On exceedances of high levels

The distribution of the excess process describing heights of extreme values can be approximated by the distribution of a Poisson cluster process. An estimate of the accuracy of such an approximation has been derived in [4] in terms of the Wasserstein distance. The paper presents a sharper estimate established in terms of the stronger total variation distance. We derive also a new bound to the accuracy of negative Binomial approximation to the distribution of the number of exceedances.     

Sanov results for Glauber spin-glass dynamics

Summary. In this paper we prove a Sanov result, i.e. a Large Deviation Principle (LDP) for the distribution of the empirical measure, for the annealed Glauber dynamics of the Sherrington-Kirkpatrick spin-glass. Without restrictions on time or temperature we prove a full LDP for the asymmetric dynamics and the crucial upper large deviations bound for the symmetric dynamics. In the symmetric model a new order-parameter arises corresponding to the response function in [SoZi83]. In the asymmetric case we show that the corresponding rate function has a unique minimum, given as the solution of a self-consistent equation. The key argument used in the proofs is a general result for mixing of what is known as Large Deviation Systems (LDS) with measures obeying an independent LDP. Received: 18 May 1995 / In revised form: 14 March 1996     

Scaling limits of interacting diffusions with arbitrary initial distributions

Summary It is remarked that for Brownian particles interacting with a smooth repulsive pair potential the nonlinear diffusion equation which S. Varadhan has derived under an entropy bound for initial densities is valied whatever initial distribution they start with.Research partially supported by Japan Society for the Promotion of Science     

Rearrangements of Gaussian fields

The monotone rearrangement of a function is the non-decreasing function with the same distribution. The convex rearrangement of a smooth function is obtained by integrating the monotone rearrangement of its derivative. This operator can be applied to regularizations of a stochastic process to measure quantities of interest in econometrics.A multivariate generalization of these operators is proposed, and the almost sure convergence of rearrangements of regularized Gaussian fields is given. For the fractional Brownian field or the Brownian sheet approximated on a simplicial grid, it appears that the limit object depends on the orientation of the simplices.     

Probability set functions

A probability set function is interpretable as a probability distribution on binary sequences of fixed length. Cumulants of probability set functions enjoy particularly simple properties which make them more manageable than cumulants of general random variables. We derive some identities satisfied by cumulants of probability set functions which we believe to be new. Probability set functions may be expanded in terms of their cumulants. We derive an expansion which allows the construction of examples of probability set functions whose cumulants are arbitrary, restricted only by their absolute values. It is known that this phenomenon cannot occur for continuous probability distributions. Some particular examples of probability set functions are considered, and their cumulants are computed, leading to a conjecture on the upper bound of the values of cumulants. Moments of probability set functions determined by arithmetical conditions are computed in a final example.Dedicated to our friend, W.A. Beyer. Financial support for this work was derived from the U.S.D.O.E. Human Genome Project, through the Center for Human Genome Studies at Los Alamos National Laboratory, and also through the Center for Nonlinear Studies, Los Alamos National Laboratory, LANL report LAUR-97-323.     

Semiparametric statistical inference in global random search

The multivariate multiextremal optimization problem is considered. Various statistical procedures based on the use of the asymptotic theory of extreme order statistics are thoroughly described. These procedures are used to infer about the maximal value of a function by its values at random points. A class of global random search methods underlying the procedures is considered. These methods generalize the well-known branch and bound methods. The article is mainly of a survey nature. It also contains new results.     

On the Gerber-Shiu function for a risk model with multi-layer dividend strategy

In this paper, we present a new approach to the study of the Gerber-Shiu discounted function for the risk model with multi-layer dividend strategy. The formulae for the Gerber-Shiu discounted function and ruin probability were obtained and the special case where the claim size distribution is a combination of exponentials is considered in detail.     

Large deviations w.r.t. quasi-every starting point for symmetric right processes on general state spaces

Summary In the work of Donsker and Varadhan, Fukushima and Takeda and that of Deuschel and Stroock it has been shown, that the lower bound for the large deviations of the empirical distribution of an ergodic symmetric Markov process is given in terms of its Dirichlet form. We give a short proof generalizing this principle to general state spaces that include, in particular, infinite dimensional and non0metrizable examples. Our result holds w.r.t. quasi-every starting point of the Markov process. Moreover we show the corresponding weak upper bound w.r.t. quasi-every starting point.This research was supported by the Graduiertenkolleg Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik, Bonn     

On the distribution function with respect to conditional expectation on Riesz spaces

The notion of distribution function with respect to a conditional expectation is defined and studied in the framework of Riesz spaces.     

L 2-lower bounds for a special class of random walks

Summary We investigate theL ₂-speed of convergence to stationarity for a certain class of random walks on a compact connected Lie group. We give a lower bound on the number of stepsk necessary such that thek-fold convolution power of the original step distribution has anL ₂-density. Our method uses work by Heckman on the asymptotics of multiplicities along a ray of representations. Several examples are presented.This paper is based on parts of the author's doctoral dissertation written at The Johns Hopkins University     

An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t), starting from a random position η. Let S(t) be an assigned boundary, such that P(η≥S(0))=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the FPT of X(t) below S(t) has distribution F. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t) is Brownian motion and S(t) is a straight line across the origin.     

Zeno's walk: A random walk with refinements

Summary. A self-modifying random walk on is derived from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed. This process converges almost surely to a random variable which is totally singular with respect to Lebesgue measure, and which is supported on a subset of having Hausdorff dimension less than , which we calculate by a theorem of Billingsley. By generating function techniques we then calculate the exponential rate of convergence of the process to its limit point, which may be taken as a bound for the convergence of the measure in the Wasserstein metric. We describe how the process may viewed as a random walk on the space of monotone piecewise linear functions, where moves are taken by successive compositions with a randomly chosen such function. Received: 20 November 1995 / In revised form: 14 May 1996     

On a modification of the classical risk process

In this paper, we present the classical risk process with two-step premium function. This means that the gross risk premium rate changes if the insurer’s surplus reaches a certain threshold level. The formula for the infinite-time ruin probability is obtained. The asymptotic behaviour of the ruin probability in the case where the claim size distribution has a light tail is considered as well.     

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S∈(0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.     

Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

Stationary distribution function estimation for ergodic diffusion process

The problem of nonparametric stationary distribution function estimation by the observation of an ergodic diffusion process is considered. The local asymptotic minimax lower bound on the risk of all the estimators is found, and it is proved that the empirical distribution function is asymptotically efficient in the sense of this bound.