Large deviations for empirical measures of Markov chains

In this paper we obtain large-deviation upper and lower bounds for the empirical measure of a Markov chain with general state space, as well as for the associated multivariate empirical measure and empirical process. In each of these instances we improve in various ways the results in the literature.     

The Brownian loop soup

We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to chronologically add Brownian loops to simple curves in the plane.Cornell University; Research supported in part by the National Science FoundationUniversité Paris-Sud and IUFMathematics Subject Classification (2000):60J65, 81T40     

Essential spectral radius for Markov semigroups (I): discrete time case

Using two new measures of non-compactness (P) and _w (P) for a positive kernel P on a Polish space E, we obtain a new formula of Nussbaum-Gelfand type for the essential spectral radius r _ess (P) on b. Using that formula we show that different known sufficient conditions for geometric ergodicity such as Doeblins condition, drift condition by means of Lyapunov function, geometric recurrence etc lead to variational formulas of the essential spectral radius. All those can be easily transported on the weighted space b _u . Some related results on L ² () are also obtained, especially in the symmetric case. Moreover we prove that for a strongly Feller and topologically transitive Markov kernel, the large deviation principle of Donsker-Varadhan for occupation measures of the associated Markov process holds if and only if the essential spectral radius is zero; this result allows us to show that the sufficient condition of Donsker-Varadhan for the large deviation principle is in fact necessary. The knowledge of r _ess (P) allows us to estimate eigenvalues of P in L ² in the symmetric case, and to estimate the geometric convergence rate by means of that in the metric of Wasserstein. Applications to different concrete models are provided for illustrating those general results. Mathematics Subject Classification (2000):60J05, 60F10, 47A10, 47D07     

Controllability and extremality in nonconvex differential inclusions

LetF:[0, T]×R ⁿ 2 ^{R n} be a set-valued map with compact values; let :R ⁿ R ^m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion . We prove sufficient conditions for (z(T))intR and establish necessary conditions in maximum principle form for (z(T))(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.The author is grateful to an anonymous referee for his valuable remarks and comments which have helped to improve the paper.The paper was written while the author was visiting the laboratory of Prof. S. Suzuki, Department of Mechanical Engineering, Sophia University, Tokyo, Japan.     

Large deviations estimates for self-intersection local times for simple random walk in {mathbb{Z}}^3

We obtain large deviations estimates for the self-intersection local times for a simple random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length n, comes from sites visited less than some power of log(n). This is opposite to the situation in dimensions larger or equal to 5. Finally, we present an application of our estimates to moderate deviations for random walk in random sceneries.      

Stability Tests for Two Common Classes of Linear Time‐Delay and Hybrid Systems

The stability of linear timedelay systems with point internal delays is difficult to deal with in practice because of the fact that their characteristic equation is usually of transcendent type rather than of polynomial type. This feature usually causes the system to possess an infinite number of poles. In this paper, stability tests for this class of systems are obtained based either on extensions of classical tests applicable to delayfree systems or on approaches within the framework of twodimensional digital filters. Some of those twodimensional stability tests are also proved to be useful for stability testing of a common class of linear hybrid systems which involve coupled continuous and digital substates after a slightadhoc adaptation of the tests for that situation.     

(φ,ξ,η,λ)‐Structures of Parabolic Type

We consider (,,,)structures of parabolic type on hypersurfaces of dual spaces and study the rank of the affinor . We consider almost contact metric structures of parabolic type of the first kind on hypersurfaces of 4dimensional dual metric space. We study the properties of these structures and give examples of normal, integrable, and Sasakian parabolic structures.     

The Newton modified barrier method for QP problems

The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues, to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a hot start. Starting at such a hot start, at mostO(In In ^-1) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, >0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that onlyO(In ^-1)O(In In ^-1) Newton steps are required to reach an -approximation to the solution from any hot start. In order to reach the hot start, one has to perform Newton steps, wherem characterizes the size of the problem andC>0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.Partially supported by NASA Grant NAG3-1397 and National Science Foundation Grant DMS-9403218.     

Adhesion between Epoxy-Polysulfone Blends and Fibers

The adhesive strength of a fiber-polymer interface is determined, where epoxy resin blends and linear heat-resistant thermoplastics - polysulfone (PSF) and polyetherimide (Ultem) - are used as matrices. Steel wire and polyamide (nylon-6) fibers are taken as reinforcing fillers. It is shown that the addition of PSF to epoxy resin results in a maximum on the concentration curve corresponding to a 10% PSF content. It is also found that the adhesive strength of the ED-20+Ultem-steel wire interface is practically independent of the modifier content under low (up to 10%) Ultem concentrations.     

The local index and the Swan conductor

In this paper we study local indices of systems of p-adic linearly differential equations which arise from p-adic representations of the absolute Galois group of local field of characteristic p with finite monodromy. We show the induction formula of the local index of p-adic differential equations and prove the equality between the local index of differential equations and the Swan conductor of p-adic Galois representations by inductive methods.     

On the Representation of Approximate Subdifferentials for a Class of Generalized Convex Functions

Given a family of real-valued functions defined in a normed vector space X, we study a class of -convex functions having a simpler representation for the --subdifferential. The case =X^* with X being a Banach space (the Fenchel case) is particularly analysed, and we find that the sublinear lower semicontinuous functions satisfy the simpler representation with respect to X^*. As a side result, we provide various new subdifferential-type charaterizations of positively homogeneous functions among those which are lower semicontinuous and convex. In addition, we also discuss that family related to the the so-called prox-bounded functions. In this more general framework our simpler representation may give rise to a new notion of enlargement of the subdifferential.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.This work is based on research material supported in part by CONICYT-Chile through FONDECYT 101-0116 and FONDAP-Matemáticas Aplicadas II.     

Estimates for distances between first exit times via parabolic equations in unbounded cylinders

First exit times and their dependence on variations of parameters are studied for diffusion processes with non-stationary coefficients. Estimates of L_p-distances and some other distances between two exit times are obtained. These estimates are based on some new prior estimates for solutions of parabolic Kolmogorovs equations with infinite horizon without Cauchy conditions.Mathematics Subject Classifications (2000): 60G17, 60G40, 60J50, 60J60, 60J65     

A large deviation principle for (r,p)-capacities on the Wiener space

Summary We formulate and prove a large deviation principle for the (r, p)-capacity on an abstract Wiener space. As an application, we obtain a sharpening of Strassen's law of the iterated logarithm in terms of the capacity.     

Inequality of Petrović and Giaccardi for Convex Function of Higher Order

In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrovis inequality.     

Large deviations for local time fractional Brownian motion and applications

Let be a fractional Brownian motion of Hurst index H∈(0,1) with values in R, and let be the local time process at zero of a strictly stable Lévy process of index 1<α?2 independent of WH. The α-stable local time fractional Brownian motion is defined by ZH(t)=WH(Lt). The process ZH is self-similar with self-similarity index and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps [P. Becker-Kern, M.M. Meerschaert, H.P. Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004) 730-756; M.M. Meerschaert, H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004) 623-638]. However, ZH does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process ZH. As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for ZH.     

On the eigenvalues and eigenvectors of an overlapping Markov chain

Starting with a sequence of i.i.d. [uniform] random variables with m possible values, we consider the overlapping Markov chain formed by sliding a window of size k through the i.i.d. sequence. We study the limiting covariance matrix B_k of this Markov chain and give algorithms for constructing the eigenvectors of B_k. We also discuss the applicability of the results in strengthening Pearsons ² test as well as the relation to approximate entropy and the usefulness in the area of testing the hypothesis of uniformity of random number generators.Mathematics Subject Classification (2000):Primary: 60J10; Secondary: 11K45     

Uniqueness for isotropic diffusions with a linear drift

This paper proves that-valued solutions to the SDE are unique in distribution, when D^d is convex and open, D, c>0, is positive and locally Lipschitz on D and zero on D, and {xD:g(x)r} is convex for r sufficiently small. The proof (for =0) is based on the transformation X_te^ctX_t, which removes the drift, and a random time change. Although the set-up is rather specialized the result gives uniqueness for some SDEs that cannot be treated by any of the conventional techniques.Mathematics Subject Classification (2000):60J60, 60H10     

μ-Statistically Convergent Function Sequences

In the present paper we are concerned with convergence in -density and -statistical convergence of sequences of functions defined on a subset D of real numbers, where is a finitely additive measure. Particularly, we introduce the concepts of -statistical uniform convergence and -statistical pointwise convergence, and observe that -statistical uniform convergence inherits the basic properties of uniform convergence.     

Connections between Interval and Unit Circle for Sobolev Orthogonal Polynomials. Strong Asymptotics on the Real Line

In this paper we show the connection between Sobolev orthogonal Laurent polynomials on the unit circle and Sobolev orthogonal polynomials on a bounded interval of the real line. As a consequence we deduce the strong outer asymptotics for Sobolev orthogonal polynomials with respect to the inner product