Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble

It is shown that the Kolmogorov distance between the expected spectral distribution function of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n ^−2/3+v ). Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, by RFBF grant N 02-01-00233, by RFBR-DFG grant N 04-01-04000, by RF grant of the leading scientific schools NSh-4222.2006.1.     

An estimate of stability of the reconstruction of the normal multiplicative type

An estimate of the characterization stability of the normal distribution type is obtained for the case of the multiplicative type. Under some conditions, the estimate has the orderɛ ^1/3 L(ɛ), where L(∈) is a slowly varying function. Supported by the Russian Founation for Basic Research (grant Nos. 97-01-00273, 98-01-00621, and 98-01-00926) and by INTAS-RFBR (grant No. IR-97-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

The rate of convergence for spectra of GUE and LUE matrix ensembles

We obtain optimal bounds of order O(n ⁻¹) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively. Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, by RFBR grant N 02-01-00233, and by RFBR-DFG grant N 04-01-04000.     

Statistical problems of the neyman type and decomposable statistics

We consider estimators of distribution parameters, obtained by the moment method and the maximum likelihood method from a sample of specially dependent random variables; namely, we assume that the sample forms a scheme of decomposable statistics. Supported by the Russian Foundation for Fundamental Research (grant No. 94-01-01273) and by the Grant Center on Mathematical Research of the Novosibirsk State University. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996. Part II.     

Some inequalities for multivariate characteristic functions

We obtain some new lower and upper bounds for characteristic functions of multivariate distributions that can be useful in various applications. Supported by the Russian Foundation for Basic Research (grant Nos. 97-01-00273, 98-01-00621, and 98-01-00926) and by INTAS-RFBR (grant No. IR-97-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Symptotic estimates of insurance tariffs in the individual risk model

In the paper, asymptotic estimates of insurance tariffs that are least admissible for the insurer are obtained; the individual risk model and factorization model of an insurance claim for various distributions of insurance portfolio volume are considered. Supported by the Russian Humanitarian Scientific Foundation (grant No. 97-02-02235) and by the Russian Foundation for Fundamental Research (grant No. 93-01-00271). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Absolute continuity of measures in the class of Markov and semi-Markov processes of diffusion type

The property of absolute continuity of measures in the class of semi-Markov processes of diffusion type is investigated. The measure of such a process can be represented in the form of a composition of two measures. The first one is the distribution of a random track, and the second one is a conditional distribution of the time run along the track. The desired density (if it exists) is represented in the form of the product of the corresponding two densities. The first density is based on the asymptotic of the distribution density of the first exit point for the process exiting from an ellipsoidal neighborhood of its initial point. In terms of the associated Markov process and the induced Wiener process, this formula coincides with the known formula for the density of a diffusion-type Markov process measure. The second density is based on the semi-Markov property, which implies that the conditional distribution of the time run along a given track is the distribution of a monotone process with independent increments. Bibliography: 6 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 216–244.This research was supported by the Russian Foundation for Basic Research, grant 01-01-00613, and by the Program Leading Scientific Schools, grant 00-15-96019.Translated by B. P. Harlamov.     

On the rate of convergence in a limit theorem for random symmetric polynomials

In terms of the mean metric, a convergence rate estimate is obtained in a limit theorem for random symmetric polynomials. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920) and by the Hungarian National Foundation for Scientific Research (grant No. T 16665). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part 11.     

Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy

We investigate differences in isomorphism types for Rogers semilattices of computable numberings of families of sets lying in different levels of the arithmetical hierarchy. Supported by RFBR grant No. 05-01-00819 and by INTAS grant No. 00-499. Supported by NSFC grant No. 60310213. __________ Translated from Algebra i Logika, Vol. 45, No. 6, pp. 637–654, November–December, 2006.     

Robust convex quadratically constrained programs

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. In contrast to [4], where it is shown that such robust problems can be formulated as semidefinite programs, our focus in this paper is to identify uncertainty sets that allow this class of problems to be formulated as second-order cone programs (SOCP). We propose three classes of uncertainty sets for which the robust problem can be reformulated as an explicit SOCP and present examples where these classes of uncertainty sets are natural. Research partially supported by DOE grant GE-FG01-92ER-25126, NSF grants DMS-94-14438, CDA-97-26385, DMS-01-04282 and ONR grant N000140310514.Research partially supported by NSF grants CCR-00-09972, DMS-01-04282 and ONR grant N000140310514.     

Lattices That are Embeddable in Suborder Lattices

Various types of lattices are embedded in suborder lattices of posets possessing certain properties. In particular, it is shown that the class of lattices isomorphic to sublattices of suborder lattices of posets of length at most n is a variety, for any n < ω.Supported by INTAS-YSF grant 2001/1-65, by RFBR-DFG grant No. 01-01-04003, by RF Ministry of Education grant No. E02-1.0-32, by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1, and by a grant from the Russian Science Support Foundation.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 483–511, July–August, 2005.     

On the separation of spectra in the analysis of Berezin kernels

The problem of restricting a highest weight representation of the group U(p, q) to the subgroup O(p, q) is considered. This restriction has an intricate spectrum that contains representations of different types. We construct a decomposition of this representation into reducible representations each of which has a single-type spectrum. Some integrals over classical groups are also calculated; these integrals generalize those of Hua. Partially supported by grant RFBR 98-01-00303 and the Russian program of support of leading scientific schools (grant RFBR 96-01-96249). Moscow Institute of Electronics and Mathematics. Translated from Funktsional'nyi analiz i Ego Prilozheniya, Vol. 34, No. 3, pp. 49–62, July–September, 2000. Translated by Yu. A. Neretin     

On the rate of convergence and aymptotic expansions forU-statistics under alternatives

In this paper, we consider asymptotic expansions and the rate of convergence for the distribution function of asymptotically efficient U-statistics under alternatives in the one-sample problem. Section 1 is an introduction. Section 2 contains the theorem concerning the rate of convergence for U-statistics; in Sec. 3, we formulate sets of sufficient conditions under which Edgeworth-type asymptotic expansions for U-statistics under alternatives will be constructed (see Theorem 2). Finally, these theorems are proved in Sec. 4. Supported by the Russian Foundation for Basic Research (grant No. 96-01-01919). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

Guaranteed bounds for insurance premium rates for the insurance portfolio of factorizable claims

In this paper, we obtain guaranteed upper bounds for the minimal possible insurance premium in the individual risk model with factorizable claims; we consider cases where the volume of the insurance portfolio is fixed and where that volume is distributed by the Poisson law. Supported by the Russian Foundation for Fundamental Research (grant No. 97-01-00271) and the Russian Humanitarian Science Foundation (grant No. 97-02-02235). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.     

Asymptotic behavior of compound nonordinary Cox processes with nonzero means

We present necessary and sufficient conditions for the weak convergence of compound, nonordinary, doubly stochastic Poisson processes (also called compound nonordinary Cox processes) when the jumps have nonzero means and finite variances, describe the class of limit laws, give the convergence rate estimate, and construct the asymptotic expansions for the distributions of compound nonordinary Cox processes and the estimates for their concentration functions. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-01919 and 97-01-00271) and by the Russian Humanitarian Scientific Foundation (grant No. 97-02-02235). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Irrationality of certain numbers that contain values of the di- and trilogarithm

We prove that, for z ∈ {1/2, 2/3, 3/4, 4/5}, at least one of the two numbers is irrational. Kh. Hessami Pilehrood was supported in part by Institute for Studies in Theoretical Physics and Mathematics (IPM) grant no. 83110021. T. Hessami Pilehrood was supported in part by IPM grant no. 83110020. W. Zudilin was supported in part by Russian Foundation for Basic Research grant no. 03-01-00359.     

On Classes of Infinite Groups with Involutions

We present a short review of the recent results about infinite groups with given strongly insolated, strongly embedded subgroups and Zassenhaus groups. This research is supported by Krasnoyarsk Science Foundation, grant 11F202C, the Russian Foundation for Basic Research, grant 03-01-00356.Mathematics Subject Classifications (2000) 20B07, 20B22.     

Convergence rate estimates in local limit theorems for poisson random sums

The asymptotic properties of Poisson random sums are investigated. A series of convergence-rate estimates in local limit theorems for those sums are obtained. Supported by the Russian Foundation for Basic Research (grant Nos. 96-01-01919, 97-01-00271, and 97-01-00273). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

The asymptotic behavior of extrema of compound Cox processes with nonzero means

We give necessary and sufficient conditions for the weak convergence of one-dimensional distributions of extrema of compound Cox processes with jumps having finite variances and nonzero expectations. No moment-type restrictions are imposed on the controlling process. As corollaries we obtain criteria of the normal convergence of extrema of compound Cox processes with nonzero means. Convergence rate estimates are also presented. Supported by the Russian Foundation for Basic Research (grant Nos. 96-01-01919 and 97-01-00271), the Russian Humanitarian Sciences Foundation (grant No. 97-02-02235), and the Committee on Knowledge Extension Research (CKER) of the American Society of Actuaries. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.