Large Deviation Properties Research Articles

Large Deviations and Stochastic Stability in the Small Noise Double Limit, II: The Logit Model

We describe the large deviations properties, stationary distribution asymptotics, and stochastically stable states of stochastic evolutionary processes based on the logit choice rule, focusing on behavior in the small noise double limit. These aspects of the stochastic evolutionary process can be characterized in terms of solutions to certain minimum cost path problems. We solve these problems explicitly using tools from optimal control theory. The analysis focuses on three-strategy coordination games that satisfy the marginal bandwagon property and that have an interior equilibrium, but our approach can be applied to other classes of games and other choice rules.

Large deviation properties for patterns

Deciding whether a given pattern is over- or under-represented according to a given background model is a key question in computational biology. Such a decision is usually made by computing some p-values reflecting the “exceptionality” of a pattern in a given sequence or set of sequences. In the simplest cases (short and simple patterns, simple background model, small number of sequences), an exact p-value can be computed with a tractable complexity. The realistic cases are in general too complicated to get such an exact p-value. Approximations are thus proposed (Gaussian, Poisson, Large deviation approximations). These approximations are applicable under some conditions: Gaussian approximations are valid in the central domain while Poisson and Large deviation approximations are valid for rare events. In the present paper, we prove a large deviation approximation to the double strands counting problem that refers to a counting of a given pattern in a set of sequences that arise from both strands of the genome. In that case, dependencies between a sequence and its reverse complement cannot be neglected. They are captured here for a Bernoulli model from general combinatorial properties of the pattern. A large deviation result is also provided for a set of small sequences.

Nonequilibrium Microcanonical and Canonical Ensembles and Their Equivalence

Generalizations of the microcanonical and canonical ensembles for paths of Markov processes have been proposed recently to describe the statistical properties of nonequilibrium systems driven in steady states. Here, we propose a theory of these ensembles that unifies and generalizes earlier results and show how it is fundamentally related to the large deviation properties of nonequilibrium systems. Using this theory, we provide conditions for the equivalence of nonequilibrium ensembles, generalizing those found for equilibrium systems, construct driven physical processes that generate these ensembles, and rederive in a simple way known and new product rules for their transition rates. A nonequilibrium diffusion model is used to illustrate these results.

A Thermodynamic Formalism for Continuous Time Markov Chains with Values on the Bernoulli Space: Entropy, Pressure and Large Deviations

Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice $\{1,\dots,d\}^{{\mathbb{N}}}$ (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator , where is a discrete time Ruelle operator (transfer operator), and $A:\{1,\dots,d\}^{{\mathbb{N}}}\to\mathbb{R}$ is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the a priori probability. Given a Lipschitz interaction $V:\{1,\dots,d\}^{{\mathbb{N}}}\to\mathbb{R}$ , we are interested in Gibbs (equilibrium) state for such V. This will be another continuous time stationary Markov chain. In order to analyze this problem we will use a continuous time Ruelle operator (transfer operator) naturally associated to V. Among other things we will show that a continuous time Perron-Frobenius Theorem is true in the case V is a Lipschitz function. We also introduce an entropy, which is negative (see also Lopes et al. in Entropy and Variational Principle for one-dimensional Lattice Systems with a general a-priori probability: positive and zero temperature. Arxiv, 2012), and we consider a variational principle of pressure. Finally, we analyze large deviations properties for the empirical measure in the continuous time setting using results by Y. Kifer (Tamsui Oxf. J. Manag. Sci. 321(2):505–524, 1990). In the last appendix of the paper we explain why the techniques we develop here have the capability to be applied to the analysis of convergence of a certain version of the Metropolis algorithm.

Typical and large-deviation properties of minimum-energy paths on disordered hierarchical lattices

We perform numerical simulations to study the optimal path problem on disordered hierarchical graphs with effective dimension d=2.32. Therein, edge energies are drawn from a disorder distribution that allows for positive and negative energies. This induces a behavior which is fundamentally different from the case where all energies are positive, only. Upon changing the subtleties of the distribution, the scaling of the minimum energy path length exhibits a transition from self-affine to self-similar. We analyze the precise scaling of the path length and the associated ground-state energy fluctuations in the vincinity of the disorder critical point, using a decimation procedure for huge graphs. Further, using an importance sampling procedure in the disorder we compute the negative-energy tails of the ground-state energy distribution up to 12 standard deviations away from its mean. We find that the asymptotic behavior of the negative-energy tail is in agreement with a Tracy-Widom distribution. Further, the characteristic scaling of the tail can be related to the ground-state energy flucutations, similar as for the directed polymer in a random medium.

1P274 Large deviation properties of population averages: An indicator of gene expression dynamics in a single cell(24. Mathematical biology,Poster)

1P274 Large deviation properties of population averages: An indicator of gene expression dynamics in a single cell(24. Mathematical biology,Poster)

Stochastic First- and Zeroth-Order Methods for Nonconvex Stochastic Programming

In this paper, we introduce a new stochastic approximation type algorithm, namely, the randomized stochastic gradient (RSG) method, for solving an important class of nonlinear (possibly nonconvex) stochastic programming problems. We establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem. We also show that this method possesses a nearly optimal rate of convergence if the problem is convex. We discuss a variant of the algorithm which consists of applying a postoptimization phase to evaluate a short list of solutions generated by several independent runs of the RSG method, and we show that such modification allows us to improve significantly the large-deviation properties of the algorithm. These methods are then specialized for solving a class of simulation-based optimization problems in which only stochastic zeroth-order information is available.

Non-classical large deviations for a noisy system with non-isolated attractors

We study the large deviations of a simple noise-perturbed dynamical system havingcontinuous sets of steady states, which mimic those found in some partial differentialequations related, for example, to turbulence problems. The system is a two-dimensionalnonlinear Langevin equation involving a dissipative, non-potential force, which has theessential effect of creating a line of stable fixed points (attracting line) touching a line ofunstable fixed points (repelling line). Using different analytical and numericaltechniques, we show that the stationary distribution of this system satisfies, inthe low-noise limit, a large deviation principle containing two competing terms:(i) a ‘classical’ but sub-dominant large deviation term, which can be derivedfrom the Freidlin–Wentzell theory of large deviations by studying the fluctuationpaths or instantons of the system near the attracting line, and (ii) a dominantlarge deviation term, which does not follow from the Freidlin–Wentzell theory, asit is related to fluctuation paths of zero action, referred to as sub-instantons,emanating from the repelling line. We discuss the nature of these sub-instantons, andshow how they arise from the connection between the attracting and repellinglines. We also discuss in a more general way how we expect these to arise in moregeneral stochastic systems having connected sets of stable and unstable fixedpoints, and how they should determine the large deviation properties of thesesystems.

1/f Spectrum and 1-Stable Law in One-Dimensional Intermittent Map with Uniform Invariant Measure and Nekhoroshev Stability

We investigate ergodic-theoretical quantities and large deviation properties of one-dimensional intermittent maps, that have not only an indifferent fixed point but also a singular structure such that the uniform measure is invariant under the mapping. The probability density of the residence time and the correlation function are found to behave polynomially: $f(m) \sim m^{-(\kappa+1)}$ and $C(\tau) \sim \tau^{-(\kappa-1)}$ $(\kappa > 1)$. Using the Doeblin-Feller theorems in probability theory, we derive the conjecture that the rescaled fluctuations of the time average of some observable functions obey the stable distribution with the exponent $1 < \alpha \le 2$. Some exponents of the stable distribution are precisely determined by numerical simulations, and the conjecture is verified numerically. The polynomial decay of large deviations is also discussed, and it is found that the entropy function does not exist, because the moment generating function of the stable distribution can not be defined.

Large deviation properties of weakly interacting processes via weak convergence methods

We study large deviation properties of systems of weakly interacting particles modeled by It\^{o} stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean-Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay.

Large deviation properties of weakly interacting processes via weak convergence methods

Large deviation properties of weakly interacting processes via weak convergence methods

Large deviation analysis for layered percolation problems on the complete graph

AbstractWe analyze the large deviation properties for the (multitype) version of percolation on the complete graph – the simplest substitutive generalization of the Erd&amp;0151;s‐Rènyi random graph that was treated in article by Bollobás et al. (Random Structures Algorithms 31 (2007), 3–122). Here the vertices of the graph are divided into a fixed finite number of sets (called layers) the probability of {u,v} being in our edge set depends on the respective layers of u and v. We determine the exponential rate function for the probability that a giant component occupies a fixed fraction of the graph, while all other components are small. We also determine the exponential rate function for the probability that a particular exploration process on the random graph will discover a certain fraction of vertices in each layer, without encountering a giant component.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 460–492, 2012

Large deviations of realized volatility

This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clarifies that there exists a trade-off between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment.

Large deviations for the local fluctuations of random walks

We establish large deviation properties valid for almost every sample path of a class of stationary mixing processes ( X 1 , … , X n , … ) . These properties are inherited from those of S n = ∑ i = 1 n X i and describe how the local fluctuations of almost every realization of S n deviate from the almost sure behavior. These results apply to the fluctuations of Brownian motion, Birkhoff averages on hyperbolic dynamics, as well as branching random walks. Also, they lead to new insights into the “randomness” of the digits of expansions in integer bases of Pi. We formulate a new conjecture, supported by numerical experiments, implying the normality of Pi.

Large deviations of generalized method of moments and empirical likelihood estimators

Summary This paper studies large deviation properties of the generalised method of moments and generalized empirical likelihood estimators for moment restriction models. We consider two cases for the data generating probability measure: the model assumption and local deviations from the model assumption. For both cases, we derive conditions where these estimators have exponentially small error probabilities for point estimation.

Large Deviations of Realized Volatility

This paper studies large and moderate deviation properties of a realized volatility statistic of high frequency financial data. We establish a large deviation principle for the realized volatility when the number of high frequency observations in a fixed time interval increases to infinity. Our large deviation result can be used to evaluate tail probabilities of the realized volatility. We also derive a moderate deviation rate function for a standardized realized volatility statistic. The moderate deviation result is useful for assessing the validity of normal approximations based on the central limit theorem. In particular, it clarifies that there exists a trade-off between the accuracy of the normal approximations and the path regularity of an underlying volatility process. Our large and moderate deviation results complement the existing asymptotic theory on high frequency data. In addition, the paper contributes to the literature of large deviation theory in that the theory is extended to a high frequency data environment.

Large-deviation properties of largest component for random graphs

Distributions of the size of the largest component, in particular the large-deviation tail, are studied numerically for two graph ensembles, for Erdoes-Renyi random graphs with finite connectivity and for two-dimensional bond percolation. Probabilities as small as 10^-180 are accessed using an artificial finite-temperature (Boltzmann) ensemble. The distributions for the Erdoes-Renyi ensemble agree well with previously obtained analytical results. The results for the percolation problem, where no analytical results are available, are qualitatively similar, but the shapes of the distributions are somehow different and the finite-size corrections are sometimes much larger. Furthermore, for both problems, a first-order phase transition at low temperatures T within the artificial ensemble is found in the percolating regime, respectively.

Large Deviations of Generalized Method of Moments and Empirical Likelihood Estimators

This paper studies large deviation properties of the generalised method of moments and generalized empirical likelihood estimators for moment restriction models. We consider two cases for the data generating probability measure: the model assumption and local deviations from the model assumption. For both cases, we derive conditions where these estimators have exponentially small error probabilities for point estimation.

Guessing Revisited: A Large Deviations Approach

The problem of guessing a random string is revisited. A close relation between guessing and compression is first established. Then it is shown that if the sequence of distributions of the information spectrum satisfies the large deviation property with a certain rate function, then the limiting guessing exponent exists and is a scalar multiple of the Legendre-Fenchel dual of the rate function. Other sufficient conditions related to certain continuity properties of the information spectrum are briefly discussed. This approach highlights the importance of the information spectrum in determining the limiting guessing exponent. All known prior results are then re-derived as example applications of our unifying approach.

Large deviations for directed percolation on a thin rectangle

Following the recent investigations of Baik and Suidan in [Int. Math. Res. Not. (2005) 325–337] and Bodineau and Martin in [Electron. Commun. Probab. 10 (2005) 105–112 (electronic)], we prove large deviation properties for a last-passage percolation model in ℤ+ 2 whose paths are close to the axis. The results are mainly obtained when the random weights are Gaussian or have a finite moment-generating function and rely, as in [J. Baik and T.M. Suidan, Int. Math. Res. Not. (2005) 325–337] and [T. Bodineau and J. Martin, Electron. Commun. Probab. 10 (2005) 105–112 (electronic)], on an embedding in Brownian paths and the KMT approximation. The study of the subexponential case completes the exposition.