On the Form of the Large Deviation Rate Function for the Empirical Measures
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Large Deviations and Exit Time Asymptotics for Diffusions and Stochastic Resonance
Large deviations and exit time asymptotics for diffusions and stochastic resonance DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Mathematik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät II Humboldt-Universität zu Berlin von Diplom-Mathematiker Dierk Peithmann geboren am 29.06.1972 in Rahden Präsident der Humboldt-Universität zu Berlin: Prof. Dr. Christoph Markschies Dekan der Mathematisch-Naturwissenschaftlichen Fakultät II: Prof. Dr. Wolfgang Coy Gutachter: 1. Prof. Dr. Peter Imkeller 2. Prof. Dr. Ludwig Arnold 3. Prof. Dr. Michael Scheutzow Tag der mündlichen Prüfung: 13. August 2007 Abstract In this thesis, we study the asymptotic behavior of exit and transition times of certain weakly time inhomogeneous diffusion processes. Based on these asymptotics, a prob- abilistic notion of stochastic resonance is investigated. Large deviations techniques play the key role throughout this work. In the first part we recall the large deviations theory for time homogeneous diffusions. Chapter1 gives a brief overview of the classical results due to Freidlin und Wentzell. In Chapter2 we present an extension of their theory to stochastic differential equa- tions with locally Lipschitz coefficients that depend on the noise amplitude. Kramers’ exit time law, which makes up the foundation stone for the results obtained in this thesis, is recalled in Chapter3. The second part deals with the phenomenon of stochastic resonance. That is, we study periodicity properties of diffusion processes. First, in Chapter4 we explain the paradigm of stochastic resonance. Afterwards, physical notions of measuring pe- riodicity properties of diffusion trajectories are discussed, and their drawbacks are pointed out. The latter suggest to follow an alternative probabilistic approach, which is introduced in Section 4.3 and discussed in subsequent chapters. -
Large-Deviation Principles, Stochastic Effective Actions, Path Entropies
Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions Eric Smith Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA (Dated: February 18, 2011) The meaning of thermodynamic descriptions is found in large-deviations scaling [1, 2] of the prob- abilities for fluctuations of averaged quantities. The central function expressing large-deviations scaling is the entropy, which is the basis for both fluctuation theorems and for characterizing the thermodynamic interactions of systems. Freidlin-Wentzell theory [3] provides a quite general for- mulation of large-deviations scaling for non-equilibrium stochastic processes, through a remarkable representation in terms of a Hamiltonian dynamical system. A number of related methods now exist to construct the Freidlin-Wentzell Hamiltonian for many kinds of stochastic processes; one method due to Doi [4, 5] and Peliti [6, 7], appropriate to integer counting statistics, is widely used in reaction-diffusion theory. Using these tools together with a path-entropy method due to Jaynes [8], this review shows how to construct entropy functions that both express large-deviations scaling of fluctuations, and describe system-environment interactions, for discrete stochastic processes either at or away from equilibrium. A collection of variational methods familiar within quantum field theory, but less commonly applied to the Doi-Peliti construction, is used to define a “stochastic effective action”, which is the large-deviations rate function for arbitrary non-equilibrium paths. We show how common principles of entropy maximization, applied to different ensembles of states or of histories, lead to different entropy functions and different sets of thermodynamic state variables. -
Arxiv:1909.00374V3 [Math.PR] 1 Apr 2021 2010 Mathematics Subject Classification
CONTRACTION PRINCIPLE FOR TRAJECTORIES OF RANDOM WALKS AND CRAMER'S´ THEOREM FOR KERNEL-WEIGHTED SUMS VLADISLAV VYSOTSKY Abstract. In 2013 A.A. Borovkov and A.A. Mogulskii proved a weaker-than-standard \metric" large deviations principle (LDP) for trajectories of random walks in Rd whose increments have the Laplace transform finite in a neighbourhood of zero. We prove that general metric LDPs are preserved under uniformly continuous mappings. This allows us to transform the result of Borovkov and Mogulskii into standard LDPs. We also give an explicit integral representation of the rate function they found. As an application, we extend the classical Cram´ertheorem by proving an LPD for kernel-weighted sums of i.i.d. random vectors in Rd. 1. Introduction The study of large deviations of trajectories of random walks was initiated by A.A. Borovkov in the 1960's. In 1976 A.A. Mogulskii [24] proved a large deviations result for the trajectories of a multidimensional random walk under the assumption that the Laplace transform of its increments is finite. In [24] Mogulskii also studied the large deviations under the weaker Cram´ermoment assumption, i.e. when the Laplace transform is finite in a neighbourhood of zero, but these results appear to be of a significantly limited use. The further progress was due to the concept of metric large deviations principles (LDPs, in short) on general metric spaces introduced by Borovkov and Mogulskii in [5] in 2010. The upper bound in such metric LDPs is worse than the conventional one { the infimum of the rate function is taken over shrinking "-neighbourhoods of a set rather than over its closure as in standard LDPs; compare definitions (1) and (2) below. -
Long Term Risk: a Martingale Approach
Long Term Risk: A Martingale Approach Likuan Qin∗ and Vadim Linetskyy Department of Industrial Engineering and Management Sciences McCormick School of Engineering and Applied Sciences Northwestern University Abstract This paper extends the long-term factorization of the pricing kernel due to Alvarez and Jermann (2005) in discrete time ergodic environments and Hansen and Scheinkman (2009) in continuous ergodic Markovian environments to general semimartingale environments, with- out assuming the Markov property. An explicit and easy to verify sufficient condition is given that guarantees convergence in Emery's semimartingale topology of the trading strate- gies that invest in T -maturity zero-coupon bonds to the long bond and convergence in total variation of T -maturity forward measures to the long forward measure. As applications, we explicitly construct long-term factorizations in generally non-Markovian Heath-Jarrow- Morton (1992) models evolving the forward curve in a suitable Hilbert space and in the non-Markovian model of social discount of rates of Brody and Hughston (2013). As a fur- ther application, we extend Hansen and Jagannathan (1991), Alvarez and Jermann (2005) and Bakshi and Chabi-Yo (2012) bounds to general semimartingale environments. When Markovian and ergodicity assumptions are added to our framework, we recover the long- term factorization of Hansen and Scheinkman (2009) and explicitly identify their distorted probability measure with the long forward measure. Finally, we give an economic interpre- tation of the recovery theorem of Ross (2013) in non-Markovian economies as a structural restriction on the pricing kernel leading to the growth optimality of the long bond and iden- tification of the physical measure with the long forward measure. -
Sample Path Large Deviations for Lévy Processes and Random
Sample Path Large Deviations for L´evyProcesses and Random Walks with Regularly Varying Increments Chang-Han Rhee1;3 Jose Blanchet2;4 Bert Zwart1;3 December 10, 2017 Abstract Let X be a L´evyprocess with regularly varying L´evymeasure ν. We ¯ obtain sample-path large deviations for scaled processes Xn(t) , X(nt)=n and obtain a similar result for random walks. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the incre- ment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connec- tions with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold. Keywords Sample Path Large Deviations · Regular Variation · M- convergence · L´evyProcesses Mathematics Subject Classification 60F10 · 60G17 · 60B10 1 Introduction In this paper, we develop sample-path large deviations for one-dimensional L´evy processes and random walks, assuming the jump sizes are heavy-tailed. Specif- ically, let X(t); t ≥ 0; be a centered L´evyprocess with regularly varying L´evy measure ν. Assume that P(X(1) > x) is regularly varying of index −α, and that P(X(1) < −x) is regularly varying of index −β; i.e. there exist slowly varying functions L+ and L− such that −α −β P(X(1) > x) = L+(x)x ; P(X(1) < −x) = L−(x)x : (1.1) Throughout the paper, we assume α; β > 1. We also consider spectrally one- sided processes; in that case only α plays a role. -
Large Deviations for Markov Sequences
Chapter 32 Large Deviations for Markov Sequences This chapter establishes large deviations principles for Markov sequences as natural consequences of the large deviations principles for IID sequences in Chapter 31. (LDPs for continuous-time Markov processes will be treated in the chapter on Freidlin-Wentzell theory.) Section 32.1 uses the exponential-family representation of Markov sequences to establish an LDP for the two-dimensional empirical dis- tribution (“pair measure”). The rate function is a relative entropy. Section 32.2 extends the results of Section 32.1 to other observ- ables for Markov sequences, such as the empirical process and time averages of functions of the state. For the whole of this chapter, let X1, X2, . be a homogeneous Markov se- quence, taking values in a Polish space Ξ, with transition probability kernel µ, and initial distribution ν and invariant distribution ρ. If Ξ is not discrete, we will assume that ν and ρ have densities n and r with respect to some reference measure, and that µ(x, dy) has density m(x, y) with respect to that same ref- erence measure, for all x. (LDPs can be proved for Markov sequences without such density assumptions — see, e.g., Ellis (1988) — but the argument is more complicated.) 32.1 Large Deviations for Pair Measure of Markov Sequences It is perhaps not sufficiently appreciated that Markov sequences form expo- nential families (Billingsley, 1961; Ku¨chler and Sørensen, 1997). Suppose Ξ is 226 CHAPTER 32. LARGE DEVIATIONS FOR MARKOV SEQUENCES 227 discrete. Then t 1 n t − P X1 = x1 = ν(x1) µ(xi, xi+1) (32.1) i=1 ! " #t 1 − log µ(xi,xi+1) = ν(x1)e i=1 (32.2) P t x,y Ξ2 Tx,y (x1) log µ(x,y) = ν(x1)e ∈ (32.3) P t where Tx,y(x1) counts the number of times the state y follows the state x in the t sequence x1, i.e., it gives the transition counts. -
Large Deviations of Poisson Cluster Processes 1
Stochastic Models, 23:593–625, 2007 Copyright © Taylor & Francis Group, LLC ISSN: 1532-6349 print/1532-4214 online DOI: 10.1080/15326340701645959 LARGE DEVIATIONS OF POISSON CLUSTER PROCESSES Charles Bordenave INRIA/ENS, Départment d’Informatique, Paris, France Giovanni Luca Torrisi Istituto per le Applicazioni del Calcolo “Mauro Picone” (IAC), Consiglio Nazionale delle Ricerche (CNR), Roma, Italia In this paper we prove scalar and sample path large deviation principles for a large class of Poisson cluster processes. As a consequence, we provide a large deviation principle for ergodic Hawkes point processes. Keywords Hawkes processes; Large deviations; Poisson cluster processes; Poisson processes. Mathematics Subject Classification 60F10; 60G55. 1. INTRODUCTION Poisson cluster processes are one of the most important classes of point process models (see Daley and Vere-Jones[7] and Møller; Waagepetersen[24]). They are natural models for the location of objects in the space, and are widely used in point process studies whether theoretical or applied. Very popular and versatile Poisson cluster processes are the so-called self-exciting or Hawkes processes (Hawkes[12,13]; Hawkes and Oakes[14]). From a theoretical point of view, Hawkes processes combine both a Poisson cluster process representation and a simple stochastic intensity representation. Poisson cluster processes found applications in cosmology, ecology and epidemiology; see, respectively, Neyman and Scott[25], Brix and Chadoeuf[5], and Møller[19,20]. Hawkes processes are particularly appealing for seismological applications. Indeed, they are widely used as statistical models for the standard activity of earthquake series; see the papers by Received October 2006; Accepted August 2007 Address correspondence to Charles Bordenave, INRIA/ENS, Départment d’Informatique, 45 rue d’Ulm, F-75230 Paris Cedex 05, France; E-mail: [email protected] 594 Bordenave and Torrisi Ogata and Akaike[28], Vere-Jones and Ozaki[32], and Ogata[26,27]. -
Large Deviations Theory and Emergent Landscapes in Biological Dynamics
The mathematical theory of large deviations provides a nonequilibrium thermodynamic description of complex biological systems that consist of heterogeneous individuals. In terms of the notions of stochastic elementary reactions and pure kinetic species, the continuous-time, integer- valued Markov process dictates a thermodynamic structure that generalizes (i) Gibbs’ macroscopic chemical thermodynamics of equilibrium matters to nonequilibrium small systems such as living cells and tissues; and (ii) Gibbs’ potential function to the landscapes for biological dynamics, such as that of C. H. Waddington’s and S. Wright’s. Large Deviations Theory and Emergent Landscapes in Biological Dynamics Hong Qian Department of Applied Mathematics University of Washington The biological world is stochastic The maximum entropy principle (MEP) is a consequence of the theory of large deviations (Cover and van Campenhout, 1981). It is a twin of the Gibbs’ statistical mechanics, which applies to only thermodynamic equilibrium. How to represent (e.g., describe) a biochemical or a biological system that consists of populations of individuals? (i) classifying biochemical (or biological) individuals into populations of kinetic species; (ii) counting the number of individuals in each and every “pure kinetic species”; (iii) representing changes in terms of “stochastic elementary processes”. A stochastic elementary process + + + + + = , , … , 2 3 ( + ) = + | = + ∆, = = 1 + , = 0, ∆ . � − ∆ 표 instantaneous rate function stoichiometric coefficients -
Large Deviation Theory
Large Deviation Theory J.M. Swart October 19, 2016 2 3 Preface The earliest origins of large deviation theory lie in the work of Boltzmann on en- tropy in the 1870ies and Cram´er'stheorem from 1938 [Cra38]. A unifying math- ematical formalism was only developed starting with Varadhan's definition of a `large deviation principle' (LDP) in 1966 [Var66]. Basically, large deviation theory centers around the observation that suitable func- tions F of large numbers of i.i.d. random variables (X1;:::;Xn) often have the property that −snI(x) P F (X1;:::;Xn) 2 dx ∼ e as n ! 1; (LDP) where sn are real contants such that limn!1 sn = 1 (in most cases simply sn = n). In words, (LDP) says that the probability that F (X1;:::;Xn) takes values near a point x decays exponentially fast, with speed sn, and rate function I. Large deviation theory has two different aspects. On the one hand, there is the question of how to formalize the intuitive formula (LDP). This leads to the al- ready mentioned definition of `large deviation principles' and involves quite a bit of measure theory and real analysis. The most important basic results of the ab- stract theory were proved more or less between 1966 and 1991, when O'Brian en Verwaat [OV91] and Puhalskii [Puk91] proved that exponential tightness implies a subsequential LDP. The abstract theory of large deviation principles plays more or less the same role as measure theory in (usual) probability theory. On the other hand, there is a much richer and much more important side of large deviation theory, which tries to identify rate functions I for various functions F of independent random variables, and study their properties. -
Probability (Graduate Class) Lecture Notes
Probability (graduate class) Lecture Notes Tomasz Tkocz∗ These lecture notes were written for the graduate course 21-721 Probability that I taught at Carnegie Mellon University in Spring 2020. ∗Carnegie Mellon University; [email protected] 1 Contents 1 Probability space 6 1.1 Definitions . .6 1.2 Basic examples . .7 1.3 Conditioning . 11 1.4 Exercises . 13 2 Random variables 14 2.1 Definitions and basic properties . 14 2.2 π λ systems . 16 − 2.3 Properties of distribution functions . 16 2.4 Examples: discrete and continuous random variables . 18 2.5 Exercises . 21 3 Independence 22 3.1 Definitions . 22 3.2 Product measures and independent random variables . 24 3.3 Examples . 25 3.4 Borel-Cantelli lemmas . 28 3.5 Tail events and Kolmogorov's 0 1law .................. 29 − 3.6 Exercises . 31 4 Expectation 33 4.1 Definitions and basic properties . 33 4.2 Variance and covariance . 35 4.3 Independence again, via product measures . 36 4.4 Exercises . 39 5 More on random variables 41 5.1 Important distributions . 41 5.2 Gaussian vectors . 45 5.3 Sums of independent random variables . 46 5.4 Density . 47 5.5 Exercises . 49 6 Important inequalities and notions of convergence 52 6.1 Basic probabilistic inequalities . 52 6.2 Lp-spaces . 54 6.3 Notions of convergence . 59 6.4 Exercises . 63 2 7 Laws of large numbers 67 7.1 Weak law of large numbers . 67 7.2 Strong law of large numbers . 73 7.3 Exercises . 78 8 Weak convergence 81 8.1 Definition and equivalences . -
Arxiv:1604.06828V2 [Math.PR]
AN INTRODUCTION TO LARGE DEVIATIONS FOR RANDOM GRAPHS SOURAV CHATTERJEE Abstract. This article gives an overview of the emerging literature on large deviations for random graphs. Written for the general mathematical audience, the article begins with a short introduction to the theory of large deviations. This is followed by a description of some large deviation questions about ran- dom graphs, and an outline of the recent progress on this topic. A more elaborate discussion follows, with a brief account of graph limit theory and its application in constructing a large deviation theory for dense random graphs. The role of Szemer´edi’s regularity lemma is explained, together with a sketch of the proof of the main large deviation result and some examples. Appli- cations to exponential random graph models are briefly touched upon. The remainder of the paper is devoted to large deviations for sparse graphs. Since the regularity lemma is not applicable in the sparse regime, new tools are needed. Fortunately, there have been several new breakthroughs that man- aged to achieve the goal by an indirect method. These are discussed, together with an exposition of the underlying theory. The last section contains a list of open problems. Contents 1. Large deviations 1 2. The problem with nonlinearity 3 3. Recent developments 3 4. Graph limit theory 4 5. Largedeviationsfordenserandomgraphs 6 6. Exponential random graphs 10 7. Largedeviationsforsparserandomgraphs 12 8. The low complexity gradient condition 13 9. Nonlinear large deviations 17 arXiv:1604.06828v2 [math.PR] 27 Apr 2016 10. The naive mean field approximation 20 11. Open problems 23 Acknowledgments 24 References 24 1. -
Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains
Article Large Deviations Properties of Maximum Entropy Markov Chains from Spike Trains Rodrigo Cofré 1,* , Cesar Maldonado 2 and Fernando Rosas 3,4 1 Centro de Investigación y Modelamiento de Fenómenos Aleatorios, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso 2340000, Chile 2 IPICYT/División de Matemáticas Aplicadas, Instituto Potosino de Investigación Científica y Tecnológica, San Luis Potosí 78216, Mexico; [email protected] 3 Centre of Complexity Science and Department of Mathematics, Imperial College London, London SW7 2AZ, UK; [email protected] 4 Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, UK * Correspondence: [email protected]; Tel.: +56-32-2603625 Received: 6 June 2018; Accepted: 11 July 2018; Published: 3 August 2018 Abstract: We consider the maximum entropy Markov chain inference approach to characterize the collective statistics of neuronal spike trains, focusing on the statistical properties of the inferred model. To find the maximum entropy Markov chain, we use the thermodynamic formalism, which provides insightful connections with statistical physics and thermodynamics from which large deviations properties arise naturally. We provide an accessible introduction to the maximum entropy Markov chain inference problem and large deviations theory to the community of computational neuroscience, avoiding some technicalities while preserving the core ideas and intuitions. We review large deviations techniques useful in spike train statistics to describe properties of accuracy and convergence in terms of sampling size. We use these results to study the statistical fluctuation of correlations, distinguishability, and irreversibility of maximum entropy Markov chains. We illustrate these applications using simple examples where the large deviation rate function is explicitly obtained for maximum entropy models of relevance in this field.