On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals F_n towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of F_n to zero; (ii) the covariance matrix of F_n to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a R^d-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.     

On hypersurfaces in a Riemannian vector bundle with prescribed Gaussian curvature and convexity

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and the unit subbundle of E. We prove that there is no radial graph on with a strictly positive Gaussian curvature and that the Gaussian curvature of a convex radial graph must be identically equal to zero. Moreover, by solving on a nonlinear degenerate equation of Monge-Ampère type, we prove the existence of radial graphs having simultaneously a Gaussian curvature identically equal to zero and a prescribed strictly positive vertical Gaussian curvature.Mathematics Subject Classification (2000):35J60, 53C55, 58G30     

On Gaussian approximation of Hilbert space valued discrete time martingales

Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 33, No. 4, pp. 476–491, October–December, 1993.     

Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures

We define a notion of quasistatic evolution for the elliptic approximation of the Mumford-Shah functional proposed by Ambrosio and Tortorelli. Then we prove that this regular evolution converges to a quasi static growth of brittle fractures in linearly elastic bodies.Received: 1 January 2003, Accepted: 29 January 2004, Published online: 12 May 2004Mathematics Subject Classification (2000): 35R35, 74R10, 35J25     

Mathematical simulation of failure processes of composite materials reinforced with brittle fibers

Conclusions 1. It is concluded on the basis of an analysis of experimental data and also from theoretical investigations with respect to stress redistribution upon the breaking of fibers that the successive breaking of a number of fibers, caused by the overload from the breaking of individual fibers, is one of the principal mechanisms according to which the complete failure of a material reinforced with brittle fibers takes place.2. A discrete model of a composite material has been worked out. A random fiber strength distribution over the surfaces of the cross sections of the composite material is produced on the computer by the application of Monte Carlo methods.3. A program was written for the computer which simulates the testing of composite materials, permitting the investigation of the statistical accumulation of damage in failure processes as well as the avalanchetype processes of the complete failure of a material.4. The effect of the statistical distribution of the strength of the reinforcing fibers, the ratio of properties, and the volume fractions of composites on the failure processes of composite materials is investigated. Deformation diagrams of a D-16 aluminum alloy-boron fiber composite material, constructed on the basis of an anlysis of the simulated process of fiber breaking in a composite, agree well with the experimental relations.5. The opinion is expressed that the development of cybernetic simulation of failure processes will permit giving an answer to a number of actual questions in the study of materials and the mechanics of failure.Baikov Institute of Metallurgy, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 5, pp. 800–808, September–October, 1976.     

On the approximation by convolution of the generalized Poisson measure and the Gaussian distribution. I

Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 32, No. 3, pp. 333–345, July–September, 1992.     

On the approximation by convolution of the generalized Poisson measure and the Gaussian distribution. II

Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 32, No. 4, pp. 454–462, October–December, 1992.     

Gaussian skewness approximation for dynamic rate multi-server queues with abandonment

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally when a manager updates a staffing schedule in response to a forecast of increased customer demand. Mathematically, this type of scaling ultimately gives us the fluid and diffusion limits as found in Mandelbaum et al., Queueing Syst 30:149–201 (1998) for Markovian service networks. The asymptotics used here reduce to the Halfin and Whitt, Oper Res 29:567–588 (1981) scaling for multi-server queues. The diffusion limit suggests a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are easily computed from a two-dimensional dynamical system for the fluid and diffusion limiting processes. Recent work by Ko and Gautam, INFORMS J Comput, to appear (2012) found that a modified version of these differential equations yield better Gaussian estimates of the original queueing system distribution. In this paper, we introduce a new three-dimensional dynamical system that is based on estimating the mean, variance, and third cumulant moment. This improves on the previous approaches by fitting the distribution from a quadratic function of a Gaussian random variable.     

Strong approximation for certain quadric fibrations with compact fibers

By developing the representation theory of quadratic Diophantine equations, we prove strong approximation with Brauer–Manin obstruction for certain quadratic fibrations where each fiber is compact.     

On the Picard bundle

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g?2, and also fix an integer r such that degree(ξ)>r(2g−1). Let Mξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier-Mukai transform, with respect to a Poincaré line bundle on X×J(X), of any F∈Mξ(r) is a stable vector bundle on J(X). This gives an injective map of Mξ(r) in a moduli space associated to J(X). If g=2, then Mξ(r) becomes a Lagrangian subscheme.     

On the approximation of curves with polylines

The exact values of the estimation of the approximation error of parametrically defined curves by inscribed polylines in them-dimensional space R ^m for classes of functions defined by moduli of continuity are presented. The result is a sort of generalization of the results of B.N. Malozemov on the approximation of continuous functions with polylines. Also, the problem of finding the upper bounds of deviations of parametrically defined curves for this class is solved based on the assumption that these curves intersect at N (N ≥ 2) points of the partition of [0, L]. In the case of m = 2, from the obtained results follow the previous results on the approximation of plane curves with polylines in Euclidean, Hausdorff, and Hamming metrices.     

Optical local Gaussian approximation of an exponential family

Summary Under certain regularity conditions products E ⁿ of an experiment E can be locally approximated by homoschedastic Gaussian experiments G _n. G _n can be defined such that the square roots of the densities have nearly the same structure with respect to the L ²-geometry as in E ⁿ. The main result of this paper is that this choice of G _n is asymptotically optimal in the sense of minimizing the deficiency distance between E ⁿ and G if E is a one-dimensional exponential family. This work has been supported by the Deutsche Forschungsgemeinschaft     

Strong approximation for sums of asymptotically negatively dependent Gaussian sequences with applications

By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences are derived.     

Accusation probabilities in Tardos codes: beyond the Gaussian approximation

We study the probability distribution of user accusations in the q-ary Tardos fingerprinting system under the Marking Assumption, in the restricted digit model. In particular, we look at the applicability of the so-called Gaussian approximation, which states that accusation probabilities tend to the normal distribution when the fingerprinting code is long. We introduce a novel parametrization of the attack strategy which enables a significant speedup of numerical evaluations. We set up a method, based on power series expansions, to systematically compute the probability of accusing innocent users. The ‘small parameter’ in the power series is 1/m, where m is the code length. We use our method to semi-analytically study the performance of the Tardos code against majority voting and interleaving attacks. The bias function ‘shape’ parameter k{{\kappa}} strongly influences the distance between the actual probabilities and the asymptotic Gaussian curve. The impact on the collusion-resilience of the code is shown. For some realistic parameter values, the false accusation probability is even lower than the Gaussian approximation predicts.