On the false pole problem

We prove that if a convex body has an interior false pole with respect to some hyperplane, then the body is an ellipsoid. This research was partially carried out during the postdoctoral visit of this author at University College London, and it was supported by CONACYT, México.     

Large deviations without principle: join the shortest queue

We develop a methodology for studying “large deviations type” questions. Our approach does not require that the large deviations principle holds, and is thus applicable to a large class of systems. We study a system of queues with exponential servers, which share an arrival stream. Arrivals are routed to the (weighted) shortest queue. It is not known whether the large deviations principle holds for this system. Using the tools developed here we derive large deviations type estimates for the most likely behavior, the most likely path to overflow and the probability of overflow. The analysis applies to any finite number of queues. We show via a counterexample that this system may exhibit unexpected behavior Work of the first author was performed in part while visiting the Technion. Work of the second author was performed in part while visiting the Vrije Universiteit, Amsterdam, and was supported in part by Fund for the promotion of research at the Technion.     

The lagrange approach to infinite linear programs

In this paper, we use the Lagrange multipliers approach to study a general infinite-dimensionalinequality-constrained linear program IP. The main problem we are concerned with is to show that thestrong duality condition for IP holds, so that IP and its dual IP* are both solvable and their optimal values coincide. To do this, we first express IP as a convex program with a Lagrangian function L, say. Then we show that the strong duality condition implies the existence of a saddle point for L, and that, under an additional, mild condition, theconverse is also true. Moreover, the saddle point gives optimal solutions for IP and IP*. Thus, our original problem is essentially reduced to prove the existence of a saddle point for L, which is shown to be the case under suitable assumptions. We use this fact to studyequality-constrained programs, and we illustrate our main results with applications to thegeneral capacity and themass transfer problems. This research was partially supported by the Consejo Nacional de Ciencia y Tecnología (CONACYT) grants 32299-E and 37355-E. It was also supported by CONACYT (for JRG and RRLM) and PROMEP (for JRG) scholarships.     

LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR SUMS OF NEGATIVELY DEPENDENT RANDOM VARIABLES

In this article, we obtain the large deviations and moderate deviations for negatively dependent (ND) and non-identically distributed random variables defined on (-∞, +∞). The results show that for some non-identical random variables, precise large deviations and moderate deviations remain insensitive to negative dependence structure.     

Spectra of Toeplitz Operators and Compositions of Muckenhoupt Weights with Blaschke Products

We discuss the optimality of a sufficient condition from [12] for a point to belong to the essential spectrum of a Toeplitz operator with a bounded measurable coefficient. Our approach is based on a new sufficient condition for a composition of a Muckenhoupt weight with a Blaschke product to belong to the same Muckenhoupt class. The first author was partially supported by CONACYT project U46936-F, Mexico.     

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??In this paper, precise large deviations of nonnegative, non-identical distributions and negatively associated random variables are investigated. Under certain conditions, the lower bound of the precise large deviations for the non-random sum is solved and the uniformly asymptotic results for the corresponding random sum are obtained. At the same time, we deeply discussed the compound renewal risk model, in which we found that the compound renewal risk model can be equivalent to renewal risk model under certain conditions. The relative research results of precise large deviations are applied to the more practical compound renewal risk model, and the theoretical and practical values are verified. In addition, this paper also shows that the impact of this dependency relationship between random variables to precise large deviations of the final result is not significant.     

Nonparametric adaptive control of discounted stochastic systems with compact state space

We are concerned with discrete-time stochastic control models for which the random disturbances are independent with a common unknown distribution. When the state space is compact, we prove that mild continuity conditions are sufficient to obtain adaptive policies which are asymptotically optimal with respect to the discounted reward criterion.This research was supported in part by the Consejo Nacional de Ciencia y Tecnología (CONACYT), Mexico City, Mexico, under Grant No. PCEXCNA-050156.     

Upper Large Deviations for Mixing Random Sequence

In this article, we prove upper large deviations for the empirical measure generated by stationary mixing random sequence under some suitable assumptions and upper large deviations for the mixing random sequence.     

Analysis of an interacting particle method for rare event estimation

We present a large deviations analysis for the performance of an interacting particle method for rare event estimation. The analysis is restricted to a one-dimensional setting, though even in this restricted setting a number of new techniques must be developed. In contrast to the large deviations analyses of related algorithms, for interacting particle schemes it is an occupation measure analysis that is relevant, and within this framework many standard assumptions (stationarity, Feller property) can no longer be assumed. The methods developed are not limited to the question of performance analysis, and in fact give the full large deviations principle for such systems.     

The First Order Asymptotics of the Extreme Eigenvectors of Certain Hermitian Toeplitz Matrices

The paper is concerned with Hermitian Toeplitz matrices generated by a class of unbounded symbols that emerge in several applications. The main result gives the third order asymptotics of the extreme eigenvalues and the first order asymptotics of the extreme eigenvectors of the matrices as their dimension increases to infinity. This work was partially supported by CONACYT projects 60160 and 80504, Mexico.     

On algebras of two dimensional singular integral operators with homogeneous discontinuities in symbols

We describe the Fredholm symbol algebra for theC ^*-algebra generated by two dimensional singular integral operators, acting onL ₂(²), and whose symbols admit homogeneous discontinuities. Locally these discontinuities are modeled by homogeneous functions having slowly oscillating (and, in particular, piecewise continuous) discontinuities on a system of rays outgoing from the origin.These results extend the well-known Plamenevsky results for the two dimensional case. We present here an alternative and much clearer approach to the problem.Rostov State University RussiaPartially supported by Russian Fund for Fundamental Investigations, RFFI-98-01-01-023, and by CONACYT project 32424-EPartially supported by CONACYT Project 27934-E, México.     

Large deviations theorems in nonparametric regression on functional data

In this Note we prove large deviations principles for the Nadaraya–Watson estimator of the regression of a real-valued variable with a functional covariate. Under suitable conditions, we show pointwise and uniform large deviations theorems.     

Uniform Boundedness of Toeplitz Matrices with Variable Coefficients

Uniform boundedness of sequences of variable-coefficient Toeplitz matrices is a delicate problem. Recently we showed that if the generating function of the sequence belongs to a smoothness scale of the H?lder type and if α is the smoothness parameter, then the sequence may be unbounded for α < 1/2 while it is always bounded for α > 1. In this paper we prove boundedness for all α > 1/2. This work was partially supported by CONACYT project U46936-F, Mexico.     

Estimation of distribution tails for normalized and self-normalized sums

A combinatorial inequality is derived. This inequality is applied to obtain new estimates for probabilities of large deviations of normalized and self-normalized sums of independent and dependent positive random values. As a consequence, an estimate from above is derived for the strong law of large numbers. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 77–87.This research was supported in part by the Ministry of Education of Russia, grant E00-1.0-45, and by the Russian Foundation for Basic Research, grant 02-01-01099a.Translated by V. A. Egorov.     

一类负相伴随机阵列部分和的精致大偏差

本文在一些适当的条件下得到了多风险模型中负相伴随机阵列的精致大偏差,推广了一些已知的结果,同时表明在多风险模型中负相伴结构对精致大偏差同样不具有敏感性.     

Multicriteria dynamic programming with an application to the integer case

Fundamental dynamic programming recursive equations are extended to the multicriteria framework. In particular, a more detailed procedure for a general recursive solution scheme for the multicriteria discrete mathematical programming problem is developed. Definitions of lower and upper bounds are offered for the multicriteria case and are incorporated into the recursive equations to aid problem solution by eliminating inefficient subpolicies. Computational results are reported for a set of 0–1 integer linear programming problems.This research was supported in part by CONACYT (Consejo Nacional de Ciencia y Technologia), Mexico City, Mexico.     

On evaluating the rate function of large deviations for jump processes

This is a sequel to our joint paper^[4] in which upper bound estimates for large deviations for Markov chains are studied. The purpose of this paper is to characterize the rate function of large deviations for jump processes. In particular, an explicit expression of the rate function is given in the case of the process being symmetrizable.     

基于R~d上核密度估计的对称检验的大偏差（英文）

设f_n是基于一个核函数K和取值于R~d的独立同分布随机变量列的一个非参数核密度估计.本文推广了在He和Gao(2008)中相应大偏差的结果,即证明统计量sup x∈Rd|f_n(x)-f_n(-x)|的大偏差.     

Rough asymptotics for tandem non-homogeneous $$M/G/\infty$$ queues via Poissonized empirical processes

We obtain a large deviations principle and moderate deviations principle for the joint distribution of the queue length processes and departure process for tandem queues. The results are obtained by applying a result providing necessary and sufficient conditions on a class of functions for a large deviations principle and moderate deviations principle to hold for a Poissonized empirical process over the class of functions. As an application, we examine how large queue lengths and numbers of departures are built up. This revised version was published online in June 2006 with corrections to the Cover Date.     

Milnor numbers and Euler obstruction*

Using a geometric approach, we determine the relations between the local Euler obstruction Eu_f of a holomorphic function f and several generalizations of the Milnor number for functions on singular spaces. *This work was partially supported by CNRS-CONACYT (12409) Cooperation Program. The first and third named authors partially supported by CONACYT grant G36357-E and DGPA (UNAM) grant IN 101 401.