Discrete and Continuous Time Extremes of Gaussian Processes

Joint distribution of maximums of a Gaussian stationary process in continuous time and in uniform grid on the real axis is studied. When the grid is sufficiently sparse, maxima are asymptotically independent. When the grid is sufficiently tight, the maximums asymptotically coincide. In the boundary case which we call Pickands grid, the limit distribution is non-degenerate. It calculated in terms of a Pickands type constant.AMS 2000 Subject Classification. Primary—60G70, Secondary—60G15*Partially supported by the Scientific foundation of the Netherlands, RFFI grant 0401-00700 and grant DFG 436 RUS 113/722.     

On the Rate of Decay of the Concentration Function of the Sum of Independent Random Variables

Let X₁, ... , X_n be i.i.d. integral valued random variables and S_n their sum. In the case when X₁ has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for max _{k ∊ ℤ} Pr{S_n = k} (which can be expressed in term of the Lévy Doeblin concentration of S_n), under the extra condition that X₁ is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X₁ has a very large tail (larger than a Cauchy-type distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum S_n. Proofs are constructive and enhance the connection between additive number theory and probability theory.À Jean-Louis Nicolas, avec amitié et respect2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25     

A limit theorem for multiple sums of identically distributed independent random variables

One investigates the asymptotic behavior (with respect to a small parameter) of series of probabilities of large deviations for sums of random variables with multidimensional indices.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 265–277, 1986.     

Attainable rates of convergence of maxima

Any exponential rate of convergence can be obtained for maxima of i.i.d. random variables, while faster than exponential convergence implies that the variables have an extreme value distribution.     

Acceleration Tools for Diagonal Information Global Optimization Algorithms

In this paper we face a classical global optimization problem—minimization of a multiextremal multidimensional Lipschitz function over a hyperinterval. We introduce two new diagonal global optimization algorithms unifying the power of the following three approaches: efficient univariate information global optimization methods, diagonal approach for generalizing univariate algorithms to the multidimensional case, and local tuning on the behaviour of the objective function (estimates of the local Lipschitz constants over different subregions) during the global search. Global convergence conditions of a new type are established for the diagonal information methods. The new algorithms demonstrate quite satisfactory performance in comparison with the diagonal methods using only global information about the Lipschitz constant.     

The first cohomology group of leaves and local stability of compact foliations

A foliation with all leaves compact (compact foliation) is called locally stable if every leaf has a basis of neighborhoods which are unions of leaves. We study the relationship between the first real cohomology group of leaves and the local stability of compact foliations. We show by example that the topology of the typical leaves (i.e. leaves with zero holonomy) has no influence on the local stability of the foliation while — at least for small codimensions — (less than 4 in general or less than 5 for foliations on compact minifolds) — a locally unstable foliation has a leaf F with infinite holonomy and a finite covering F' of F such that H¹(F'; IR) O. We also prove a related structural stability result for fibre bundles.     

Convergence of series of probabilities of large deviations for sums of random variables with multidimensional indices

One finds necessary and sufficient conditions for the convergence of series of weighted probabilities of large deviations for sums of independent random variables with multidimensional indices. These conditions are expressed in terms of the initial distribution.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 114–131, 1986.     

Coleman's principle for the determination of the stationary points of invariant functions

One finds broad conditions on a set X, on a collection G of operators acting on X, and on a functional f, defined on X and invariant relative to G, under which the following fact holds: a point which is stationary for f on X_o, the set of all fixed (relative to G) points from X, is stationary also for f on the entire X. One gives applications of the theorems established in the paper to various multidimensional variational problems, among others, to the Skyrme model.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 84–102, 1982.     

An Extension of the Kolmogorov–Feller Weak Law of Large Numbers with an Application to the St. Petersburg Game

The Kolmogorov–Feller weak law of large numbers for i.i.d. random variables without finite mean is extended to a larger class of distributions, requiring regularly varying normalizing sequences. As an application we show that the weak law of large numbers for the St. Petersburg game is an immediate consequence of our result.     

Existence of Multivariate Max-Universal Laws

If suitably normalized maxima of an i.i.d. sample converge in distribution, the limiting distribution is known to be max-infinitely divisible and the common distribution of the sample is said to belong to its domain of attraction. We prove the existence of max-universal distributions belonging to the domain of attraction of every max-infinitely divisible law. The proof follows in the spirit of corresponding results for normalized sums of i.i.d. random variables originated by Doeblin and shows that necessarily the sampling size has to be rapidly increasing. Restricting the growth rate of the sampling size, we prove that one necessarily deals with max-semistable distributions and their domains of attraction. 2000 Mathematics subject classification Primary—60G70 Secondary—60E99, 60F05