Integration of large-deviation kernels and applications to large deviations for evolutionary games

 We investigate large-deviation properties for systems of probability kernels and these kernels when integrated with respect to a system of measures that also satisfy the large-deviation principle. The results of our analysis are then used to demonstrate the large-deviation property and derive rate functions for systems of measures derived from the distributions of evolutionary game processes. Received: 26 October 1998 / Revised version: 17 April 2001 / Published online: 11 December 2001     

On the Relation between Gradient Flows and the Large-Deviation Principle,with Applications to Markov Chains and Diffusion

Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ? that induce a flow, given by \(\mathcal{L} (\rho _{t},\dot \rho _{t})=0\) . We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ? is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.     

Two-dimensional asymptotic expansions for large deviations of spherically distributed random vectors if the dominating point degenerates asymptotically

A geometric approach to asymptotic expansions for large-deviation probabilities, developed for the Gaussian law by Breitung and Richter [J. Multivariate Anal.,58, 1–20 (1996)], will be extended in the present paper to the class of spherical measures by utilizing their common geometric properties. This approach consists of rewriting the probabilities under consideration as large parameter values of the Laplace transform of a suitably defined function, expanding this function in a power series, and then applying Watson’s lemma. A geometric representation of the Laplace transform allows one to combine the global and local properties of both the underlying measure and the large-deviation domain. A special new type of difficulty is to be dealt with because the so-called dominating points of the large-deviation domain degenerate asymptotically. As is shown in Richter and Schumacher (in print), the typical statistical applications of large-deviation theory lead to such situations. In the present paper, consideration is restricted to a certain two-dimensional domain of large-deviations having asymptotically degenerating dominating points. The key assumption is a parametrized expansion for the inverse $\bar g^{ - 1} $ of the negative logarithm of the density-generating function of the two-dimensional spherical law under consideration.     

Lifshits Tails in the Hierarchical Anderson Model

We prove that the homogeneous hierarchical Anderson model exhibits a Lifshits tail at the upper edge of its spectrum. The Lifshits exponent is given in terms of the spectral dimension of the homogeneous hierarchical structure. Our approach is based on Dirichlet–Neumann bracketing for the hierarchical Laplacian and a large-deviation argument.     

Application of large deviation methods to the pricing of index options in finance

We develop an asymptotic formula for calculating the implied volatility of European index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket. The derivation uses the steepest-descent approximation for evaluating the multivariate probability distribution function for stock prices, which is based on large-deviation estimates of diffusion processes densities by Varadhan (Comm. Pure Appl. Math. 20 (1967)). A detailed version of these results can be found in (RISK 15 (10) (2002)). To cite this article: M. Avellaneda et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Sums of Independent Random Variables with Finite Variances: Moderate Deviations and Nonuniform Bounds in the CLT

In the present note, new nonuniform bounds in the CLT for sums of independent random variables with finite variances are obtained. Some simple and sharp nonasymptotic upper bounds for large-deviation probabilities of the mentioned sums are derived. The results obtained are applied to refine the known conditions and to find new ones under which the large-deviation theorems in the series scheme are true. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 242–259.     

Large-Deviation Probabilities in Banach Spaces

Theorems concerning exact asymptotics of large-deviation probabilities of sums of independent random elements in a Banach space are proved. We consider probabilities of the following events: sums of independed elements belong to balls such that their centers deviate from the origin as the sample size increases. The results are a version of Bentkus' and Rachkauskas' theorems proved for the exteriors of balls centered at the origin. Bibliography: 11 titles.     

Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system

In this paper we determine the exact structure of the pullback attractors in non-autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria. We show that the pullback attractors of the perturbed systems inherit this structure, and are given as the union of the unstable manifolds of a set of hyperbolic global solutions which are the non-autonomous analogues of the hyperbolic equilibria. We also prove, again parallel to the autonomous case, that all solutions converge as t→+∞ to one of these hyperbolic global solutions. We then show how to apply these results to systems that are asymptotically autonomous as t→−∞ and as t→+∞, and use these relatively simple test cases to illustrate a discussion of possible definitions of a forwards attractor in the non-autonomous case.     

Sequences of capacities,with connections to large-deviation theory

Acapacity is a set function with some regularity properties on a Hausdorff spaceE. Many measures and all sup measures are examples. The set of capacities onE can be endowed with two natural topologies. The narrow topology corresponds to the weak topology for probability measures, while the vague topology corresponds to the vague topology for Radon measures. The connection between these topologies and large-deviation principles was noted in recent joint work with W. Vervaat. Here, the theory of capacities and their topologies is developed in directions which have implications for large-deviation theory.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada. Much of it was done while the author enjoyed the kind hospitality of the Delft University of Technology.     

Large Deviations of U-Statistics. II

We develop large-deviation results with explicit order terms and Cramér's series for nondegenerate U-statistics of degree m under Cramér-type conditions on the kernel. The method of the proof is based on the contraction technique of Keener, Robinson, and Weber [15], which is a natural generalization of the classical method of Cramér [10].     

On the topological classification of dynamic equations on time scales

This paper considers the topological classification of non-autonomous dynamic equations on time scales. In this paper we show, by a counterexample, that the trivial solutions of two topologically conjugated systems may not have the same uniform stability. This is contrary to the expectation that two topologically conjugated systems should have the same topological structure and asymptotic behaviors. To counter this mismatch in expectation, we propose a new definition of strong topological conjugacy that guarantees the same topological structure, and in particular the same uniform stability, for the corresponding solutions of two strongly topologically conjugated systems. Based on the new definition, a new version of the generalized Hartman–Grobman theorem is developed. We also include some examples to illustrate the feasibility and effectiveness of the new generalized Hartman–Grobman theorem.     

Necessity of the Cramer, Linnik, and Statulevičius Conditions for Large-Deviation Probabilities

In this paper, we study the necessity of the Cramer, Linnik, and Statuleviius conditions in theorems on large-deviation probabilities. We also show that the Statuleviius condition is a generalization of the Cramer and Linnik conditions. Bibliography: 7 titles.     

Large Deviations of U-Statistics. I

We develop large-deviation results with explicit order terms and Cramér's series for nondegenerate U-statistics of degree m under Cramér-type conditions on the kernel. The method of the proof is based on the contraction technique of Keener, Robinson, and Weber [15], which is a natural generalization of the classical method of Cramér [10]. Other techniques used in the proofs include truncation, decoupling inequalities, Borell's inequality for Rademacher chaos, and a partitioning method to bound the degenerate remainder term.     

Fractional set systems with few disjoint pairs

Ahlswede (1980) [1] and Frankl (1977) [5] independently found a result about the structure of set systems with few disjoint pairs. Bollobás and Leader (2003) [3] gave an alternate proof by generalizing to fractional set systems and noting that the optimal fractional set systems are {0,1}-valued. In this paper we show that this technique does not extend to t-intersecting families. We find optimal fractional set systems for some infinite classes of parameters, and we point out that they are strictly better than the corresponding {0,1}-valued fractional set systems. We prove some results about the structure of an optimal fractional set system, which we use to produce an algorithm for finding such systems. The run time of the algorithm is polynomial in the size of the ground set.     

Large-deviation probabilities for maxima of sums of subexponential random variables with application to finite-time ruin probabilities

We establish an asymptotic relation for the large-deviation probabilities of the maxima of sums of subexponential random variables centered by multiples of order statistics of i.i.d.standard uniform random variables.This extends a corresponding result of Korshunov.As an application,we generalize a result of Tang,the uniform asymptotic estimate for the finite-time ruin probability,to the whole strongly subexponential class.     

Metriplectic structure, Leibniz dynamics and dissipative systems

A metriplectic (or Leibniz) structure on a smooth manifold is a pair of skew-symmetric Poisson tensor P and symmetric metric tensor G. The dynamical system defined by the metriplectic structure can be expressed in terms of Leibniz bracket. This structure is used to model the geometry of the dissipative systems. The dynamics of purely dissipative systems are defined by the geometry induced on a phase space via a metric tensor. The notion of Leibniz brackets is extendable to infinite-dimensional spaces. We study metriplectic structure compatible with the Euler-Poincaré framework of the Burgers and Whitham-Burgers equations. This means metriplectic structure can be constructed via Euler-Poincaré formalism. We also study the Euler-Poincaré frame work of the Holm-Staley equation, and this exhibits different type of metriplectic structure. Finally we study the 2D Navier-Stokes using metriplectic techniques.     

On the direct sum conjecture

We prove the direct sum conjecture for various sets of systems of bilinear forms. Our results depend on a priori knowledge of the complexity of at least one of the direct summands and its underlying algebraic structure. We also briefly survey some previous results concerning the complexity and structure of minimal algorithms for various direct sum systems.     

Interaction cohomology of forward or backward self-similar systems

We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings U of the invariant set, where each U consists of (backward) images of the invariant set under elements of finite word length. We give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of semigroups of polynomials. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.     

Twisted Equivariant Matter

We show how general principles of symmetry in quantum mechanics lead to twisted notions of a group representation. This framework generalizes both the classical threefold way of real/complex/ quaternionic representations as well as a corresponding tenfold way which has appeared in condensed matter and nuclear physics. We establish a foundation for discussing continuous families of quantum systems. Having done so, topological phases of quantum systems can be defined as deformation classes of continuous families of gapped Hamiltonians. For free particles, there is an additional algebraic structure on the deformation classes leading naturally to notions of twisted equivariant K-theory. In systems with a lattice of translational symmetries, we show that there is a canonical twisting of the equivariant K-theory of the Brillouin torus. We give precise mathematical definitions of two invariants of the topological phases which have played an important role in the study of topological insulators. Twisted equivariant K-theory provides a finer classification of topological insulators than has been previously available.     

Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green-Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers hq,0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak-Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers.