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.     

Extremal properties of some geometric processes

In this paper some isoperimetric inequalities for stationary random tessellations are discussed. At first, classical results on deterministic tessellations in the Euclidean plane are extended to the case of random tessellations. An isoperimetric inequality for the random Poisson polygon is derived as a consequence of a theorem of Davidson concerning an extremal property of tessellations generated by random lines inR ². We mention extremal properties of stationary hyperplane tessellations inR ^d related to Davidson's result in cased=2. Finally, similar problems for random arrangements ofr-flats inR ^d are considered (r).This work was done while the author was visiting the University of Strathclyde in Glasgow.     

Change point estimators by local polynomial fits under a dependence assumption

We study a random design regression model generated by dependent observations, when the regression function itself (or its ν-th derivative) may have a change or discontinuity point. A method based on the local polynomial fits with one-sided kernels to estimate the location and the jump size of the change point is applied in this paper. When the jump location is known, a central limit theorem for the estimator of the jump size is established; when the jump location is unknown, we first obtain a functional limit theorem for a local dilated-rescaled version estimator of the jump size and then give the asymptotic distributions for the estimators of the location and the jump size of the change point. The asymptotic results obtained in this paper can be viewed as extensions of corresponding results for independent observations. Furthermore, a simulated example is given to show that our theory and method perform well in practice.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Reconstruction of convex bodies from surface tensors

We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. When only measurements subject to noise of surface tensors are available for reconstruction, we recommend to use certain values of the surface tensors, namely harmonic intrinsic volumes instead of the surface tensors evaluated at the standard basis. The second algorithm we present is based on harmonic intrinsic volumes and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise variables.     

Maximal Inequalities for Fractional Lévy and Related Processes

In this article we study processes that are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding “convoluted martingale” is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.     

Central limit theorem for integrated square error of multivariate nonparametric density estimators

Martingale theory is used to obtain a central limit theorem for degenerate U-statistics with variable kernels, which is applied to derive central limit theorems for the integrated square error of multivariate nonparametric density estimators. Previous approaches to this problem have employed Komlós-Major-Tusnády type approximations to the empiric distribution function, and have required the following two restrictive assumptions which are not necessary using the present approach: (i) the data are in one or two dimensions, and (ii) the estimator is constructed suboptimally.     

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.     

Properties of nonparametric estimators of autocovariance for stationary random fields

Summary We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovatiance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.     

Inequalities for absolutely regular sequences: application to density estimation

This paper investigates the problem of density estimation for absolutely regular observations. In a first part, we state two important results: a new variance inequality and a Rosenthal type inequality. This allows us to study the ? _p -integrated risk, p≧ 2, of a large class of density estimators including kernel or projection estimators. Under the summability condition on the mixing coefficients ∑ _{k≧ 0} (k+1) ^{p− 2} β _k <∞, the rates obtained are those known to be optimal in the independent setting. Received: 17 October 1995 / In revised form: 26 October 1996     

Local invariance principles and their application to density estimation

Summary Letx ₁,...,x _n be independent random variables with uniform distribution over [0, 1] ^d , andX( ⁿ ) be the centered and normalized empirical process associated tox ₁,...,x _n . Given a Vapnik-Chervonenkis classL of bounded functions from [0, 1] ^d intoR of bounded variation, we apply the one-dimensional dyadic scheme of Komlós, Major and Tusnády to get the best possible rate in Dudley's uniform central limit theorem for the empirical process {E ⁽ⁿ⁾(h):hL}. WhenL fulfills some extra condition, we prove there exists some sequenceB _n of Brownian bridges indexed byL such that whereK (L) denotes the maximal variation of the elements ofL. This result is then applied to maximal deviations distributions for kernel density estimators under minimal assumptions on the sequence of bandwith parameters. We also derive some results concerning strong approximations for empirical processes indexed by classes of sets with uniformly small perimeter. For example, it follows from Beck's paper that the above result is optimal, up to a possible factor , whenL is the class of Euclidean balls with radius less thanr.     

Volatility inference in the presence of both endogenous time and microstructure noise

In this article we consider the volatility inference in the presence of both market microstructure noise and endogenous time. Estimators of the integrated volatility in such a setting are proposed, and their asymptotic properties are studied. Our proposed estimator is compared with the existing popular volatility estimators via numerical studies. The results show that our estimator can have substantially better performance when time endogeneity exists.     

Local Stereology of Tensors of Convex Bodies

In this paper, we present local stereological estimators of Minkowski tensors defined on convex bodies in ? ^d . Special cases cover a number of well-known local stereological estimators of volume and surface area in ?³, but the general set-up also provides new local stereological estimators of various types of centres of gravity and tensors of rank two. Rank two tensors can be represented as ellipsoids and contain information about shape and orientation. The performance of some of the estimators of centres of gravity and volume tensors of rank two is investigated by simulation.     

A CLT concerning critical points of random functions on a Euclidean space

We prove a central limit theorem concerning the number of critical points in large cubes of an isotropic Gaussian random function on a Euclidean space.     

A bivariate Lévy process with negative binomial and gamma marginals

The joint distribution of X and N, where N has a geometric distribution and X is the sum of N IID exponential variables (independent of N), is infinitely divisible. This leads to a bivariate Lévy process {(X(t),N(t)),t≥0}, whose coordinates are correlated negative binomial and gamma processes. We derive basic properties of this process, including its covariance structure, representations, and stochastic self-similarity. We examine the joint distribution of (X(t),N(t)) at a fixed time t, along with the marginal and conditional distributions, joint integral transforms, moments, infinite divisibility, and stability with respect to random summation. We also discuss maximum likelihood estimation and simulation for this model.     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.     

The almost equivalence of pairwise and mutual independence and the duality with exchangeability

For a large collection of random variables in an ideal setting, pairwise independence is shown to be almost equivalent to mutual independence. An asymptotic interpretation of this fact shows the equivalence of asymptotic pairwise independence and asymptotic mutual independence for a triangular array (or a sequence) of random variables. Similar equivalence is also presented for uncorrelatedness and orthogonality as well as for the constancy of joint moment functions and exchangeability. General unification of multiplicative properties for random variables are obtained. The duality between independence and exchangeability is established through the random variables and sample functions in a process. Implications in other areas are also discussed, which include a justification for the use of mutually independent random variables derived from sequential draws where the underlying population only satisfies a version of weak dependence. Macroscopic stability of some mass phenomena in economics is also characterized via almost mutual independence. It is also pointed out that the unit interval can be used to index random variables in the ideal setting, provided that it is endowed together with some sample space a suitable larger measure structure. Received: 16 April 1997 / Revised version: 18 May 1998     

On the second derivative of the spherical contact distribution function of smooth grain models

We consider a stationary grain model Ξ in ℝ ^d with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula for the(d− 1)-st quermass density. Received: 22 November 1999 / Revised version: 2 November 2000 /?Published online: 14 June 2001     

Reconstruction of n-dimensional convex bodies from surface tensors

In this paper, we derive uniqueness and stability results for surface tensors. Further, we develop two algorithms that reconstruct shape of n-dimensional convex bodies. One algorithm requires knowledge of a finite number of surface tensors, whereas the other algorithm is based on noisy measurements of a finite number of harmonic intrinsic volumes. The derived stability results ensure consistency of the two algorithms. Examples that illustrate the feasibility of the algorithms are presented.     

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.