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.     

A general theorem on the limit behavior of superpositions of independent random processes with applications to Cox processes

A general theorem is presented which gives necessary and sufficient conditions of the convergence of one-dimensional distributions of appropriately centered and normalized superpositions of independent stochastic processes. This theorem is used as a tool for obtaining limit theorems for doubly stochastic Poisson processes (Cox processes). Supported in part by the Russian Foundation for Fundamental Research (grant No. 93-01-01446) and also by the International Science Foundation and the government of Russia (projects NFW000 and NFW300). Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Exponential bounds for generalized poisson distributions

Exponential estimates are constructed for the probabilities of large deviations of Poisson random sums. 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. Moscow, Russia, 1996, 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.     

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.     

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.     

On the stability of characterization by the distribution of any order statistic

Some upper and lower estimates of stability of characterization of distribution by the distribution of any order statistic are obtained. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-01919 and 97-01-00273) and INTAS-RFBR (grant No. IR-98-0537). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

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.     

Natural estimates of convergence rate in the central limit theorem

The concept of natural convergence rate estimates in the central limit theorem is proposed connecting convergence criteria and convergence rate. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló. Hungary. 1997, Part II.     

Weight of faces in plane maps

Precise upper bounds are obtained for the minimum weight of minor faces in normal plane maps and 3-polytopes with specified maximum vertex degree. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 648–657, November, 1998. The research of the first named author was supported in part by the Visiting Fellowship Research Grant GR/K00561 from the Engineering and Physical Sciences Research Council and by the Russian Foundation for Basic Research under grant No. 96-01-01614 and No. 97-01-01075.     

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.     

Regular sets in residual classes

Sets regular modulo a fixed odd prime power are explicitly constructed under the condition that their cardinalities do not exceed an arbitrarily small positive power of the modulus.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 224–228, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00721 and by the Professor B. Novak grant (Karlov University, Prague).     

Convergence of multidimensional random sequences with independent random indices

The paper present necessary and sufficient conditions for the weak convergence of R^m-valued random sequences with independent random indices under some additional assumptions. Operator normalization is considered. Supported by the Russian Foundation for Fundamental Research (grant No. 93-01-01446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.     

Reliability of a system with regular inspection times

A method of calculating upper and lower bounds of the reliability function and mean life time is proposed for a system subjected to regular inspections. With the help of minimal cut sets, a lower bound of the reliability function is obtained; the corresponding numerical procedure requires minor calculations. Supported in part by the Russian Foundation for Fundamental Research (grant No. 95-01-00023), the International Science Foundation and the Russian Government (grant J76100), and the EEC (grant INTAS-93-893). Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

Coverings by classes of permutable equivalence relations

Supported by the Russian Foundation for Fundamental Research, grant No. 93-01-16011.     

On the rate of conveergence in the local limit theorem for densities

The aim of this paper is to improve the structure of the convergence rate estimates in the local limit theorem for densities and to give asymptotic estimates of the absolute constants involved in uniform, nonuniform, and L_p-estimates. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part II.     

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.     

Use of the Lagrange method in nonlinear analysis

The problem of the spectrum of a nonlinear system and its relation to ellipsoid transform widths are studied.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 334–342, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Critical cases of stability. Converse implicit function theorem for dynamical systems with cosymmetry

Stability criteria of boundary equilibria for dynamical systems in the three critical cases, (n, k)=(3, 0), (2, 1), and (1, 1), are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 572–578, April, 1998.The author wishes to thank V. I. Yudovich for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01791 and by the International Science Foundation under grant NRQ000.