Convergence to the Asymptotic Large Deviation Limit.

Large Deviations of Realized Volatility

The Poincaré recurrence statistic (PRS) measures the probability that a trajectory initiated in a phase space region will first return to that region, as a function of time. For deterministic, area-preserving maps with a mixture of regular and chaotic orbits, the stickiness of invariant tori and islands is responsible for a power-law decay in the PRS. We show that noise perturbations allow trajectories to access the interior of islands, enhancing their trapping effect and causing many orbits to take longer to return to a neighborhood of their initial conditions. The noisy PRS can then exhibit an extended tail on an intermediate time scale, which is eventually followed by an asymptotic exponential decay. We study a typical example, the Harper map, and compare distributions of trapping and visit times to islands with recurrence times to show the importance of noise in creating tails in the PRS. A simple finite-state Markov model of the dynamics confirms how this slower decay can be caused by noise permitting entry to previously inaccessible regions.

An asymptotic thin shell condition and large deviations for random multidimensional projections

In Section II.4, we stated three levels of large deviation properties for i.i.d. random vectors taking values in ℝ d . In Section II.6 we also stated a large deviation theorem for random vectors [Theorem 11.6.1] which generalized the level-1 property. In this chapter, Theorem 11.6.1 will be proved [Sections VII.2–VII.4] and the level-1 large deviation property will be derived as a corollary [Section VII.5]. The results on exponential convergence of random vectors stated in Theorem 1I.6.3 will be proved in Section VII.6.

This paper studies large deviations properties of vectors of empirical means and measures generated as follows. Consider a sequence X1,X2,…,Xn of independent and identically distributed random variables partitioned into d-subgroups with sizes n1,…,nd. Further, consider a d-dimensional vector mn whose coordinates are made up of the empirical means of the subgroups. We prove the following. (1) The sequence of vector of empirical means mn satisfies large deviations principle with rate n and rate function I, when the sequence X1,X2,…,Xn is Rl valued, with l≥1. (2) Similar large deviations results hold for the corresponding sequence of vector of empirical measures Ln if Xi’s, i=1,2,…,n, take on finitely many values. (3) The rate functions for the above large deviations principles are convex combinations of the corresponding rate functions arising from the large deviations principles of the coordinates of mn and Ln. The probability distributions used in the convex combinations are given by α=(α1,…,αd)=limn→∞1/n(n1,…,nd). These results are consequently used to derive variational formula for the thermodynamic limit for the pressure of multipopulation Curie-Weiss (I. Gallo and P. Contucci (2008), and I. Gallo (2009)) and mean-field Pott’s models, via a version of Varadhan’s integral lemma for an equicontinuous family of functions. These multipopulation models serve as a paradigm for decision-making context where social interaction and other socioeconomic attributes of individuals play a crucial role.

We consider the one-dimensional stochastic heat and wave equations driven by Gaussian noises with constant initial conditions.We study the spatial average of the solutions on an interval of length R and show that the family of laws of the spatial average satisfies the large deviation principle as R goes to infinity.We also present the large deviation principle in the space of continuous functions.We prove these results using the tools of Malliavin calculus to evaluate the covariance of nonlinear functionals of the solution.

We show, under regularity conditions, that a nonnegative nondecreasing real-valued stochastic process satisfies a large deviation principle (LDP) with nonlinear scaling if and only if its inverse process does. We also determine how the associated scaling and rate functions must be related. A key condition for the LDP equivalence is for the composition of two of the scaling functions to be regularly varying with nonnegative index. We apply the LDP equivalence to develop equivalent characterizations of the asymptotic decay rate in nonexponential asymptotics for queue-length tail probabilities. These alternative characterizations can be useful to estimate the asymptotic decay constant from systems measurements.

A simple and efficient numerical analysis is proposed for the Kosterlitz-Thouless (KT) phase transition. The nonequilibrium relaxation method is applied to it. The two-dimensional ferromagnetic XY models are investigated to show the efficiency. At the KT transition point as well as inside the KT phase, the nonequilibrium relaxation of magnetization from the all-aligned state shows an asymptotic power-law decay, m(t) approximately t(-lambda(T)). Only outside the KT phase, an asymptotic single exponential decay is observed. Using a standard scaling form m(t)=tau(-lambda)(-)m(t/tau) in this regime, where tau is the relaxation time at each temperature, we find a simple and efficient numerical estimation of the KT transition point and dynamical exponent. This method can be applied to various kinds of models which show the KT-like behavior.

The paper exhibits a unified approach to large deviations of dynamical systems and stochastic processes based on the existence of a pressure functional and on the uniqueness of equilibrium states for certain dense sets of functions. This enables us to generalize recent results from [OP, Y, and D] on large deviations for dynamical systems, as well, as to recover Donsker-Varadhan’s [DV2] large deviation estimates for Markov processes.

The problem of communicating sensor readings over a multiaccess channel for detecting a target is considered. A natural way of communication in target detection is to let sensors simultaneously transmit one of two predetermined frequency tones indicating whether the target is detected or not. Recently, this scheme has been generalized to consider non-binary sensor observations by letting sensors simultaneously transmit orthogonal waveforms depending on the value of their observations-type-based multiple access (TBMA). TBMA was shown to be asymptotically optimal in terms of detection-error probability under the idealistic assumptions that the sensor channel gains are identical, and the sensor data are conditionally independent and identically distributed (i.i.d.). In this paper, TBMA is analyzed in a more general framework by considering non-i.i.d. data and non-identical channel gains. An asymptotically optimal detector is proposed and its error-exponents for detection probabilities are characterized using tools from large deviations theory. Numerical simulations are used to demonstrate that the error exponents provide reasonably accurate estimates of the performance of TBMA.

The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation.The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional.In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise.The approach is based on a variational representation for functionals of Brownian motion.Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.

In this work we develop stochastic control methods for the study of large deviation principles (LDP) for certain interacting particle systems. Although such methods have been well studied for analyzing large deviation properties of small noise stochastic dynamical systems [7] and of weakly interacting particle systems [6], this is the first work to implement the approach for Brownian particle systems with a local interaction. As an application of these methods we give a new proof of the large deviation principle from the hydrodynamic limit for the Ginzburg-Landau model studied in [10]. Along the way, we establish regularity properties of the densities of certain controlled Markov processes and certain results relating large deviation principles and Laplace principles in non-Polish topological spaces that are of independent interest. The proof of the LDP is based on characterizing subsequential hydrodynamic limits of controlled diffusions with nearest neighbor interaction that arise from a variational representation of certain Laplace functionals. This proof also yields a new representation for the rate function which is very natural from a control theoretic point of view. Proof techniques are very similar to those used for the law of large number analysis, namely in the proof of convergence to the hydrodynamic limit (cf. [15]). Specifically, the key step in the proof is establishing suitable bounds on relative entropies and Dirichlet forms associated with certain controlled laws. This general approach has the promise to be applicable to other interacting Brownian systems as well.

Consider the normalized partial sums of a real-valued function F of a Markov chain, $$\\phi_{n}:=n^{-1}\\sum_{k=0}^{n-1}F(\\Phi(k)),\\qquad n\\ge1.$$ The chain {Φ(k):k≥0} takes values in a general state space $\\mathsf {X}$, with transition kernel P, and it is assumed that the Lyapunov drift condition holds: $PV\\le V-W+b\\mathbb{I}_{C}$ where $V: \\mathsf {X}\\to(0,\\infty)$, $W: \\mathsf {X}\\to[1,\\infty)$, the set C is small and W dominates F. Under these assumptions, the following conclusions are obtained:\n¶ 1. It is known that this drift condition is equivalent to the existence of a unique invariant distribution π satisfying π(W)<∞, and the law of large numbers holds for any function F dominated by W: $$ϕ_n→ϕ:=π(F),\\qquad\\mathrm{a.s.}, n→∞.$$\n¶ 2. The lower error probability defined by $\\mathsf {P}\\{\\phi_{n}\\le c\\}$, for c<ϕ, n≥1, satisfies a large deviation limit theorem when the function F satisfies a monotonicity condition. Under additional minor conditions an exact large deviations expansion is obtained.\n¶ 3. If W is near-monotone, then control-variates are constructed based on the Lyapunov function V, providing a pair of estimators that together satisfy nontrivial large asymptotics for the lower and upper error probabilities.\n¶ In an application to simulation of queues it is shown that exact large deviation asymptotics are possible even when the estimator does not satisfy a central limit theorem.

In this paper we study a thermoelastic material with an internal structure which binds the materials fibres to a quadratic behaviour. Moreover, a hereditary constitutive law for heat flux is supposed. We prove results of asymptotic stability and exponential decay for the evolution problem in two‐dimensional space domain. Copyright © 2006 John Wiley & Sons, Ltd.

The fluctuation theorem for entropy production is a remarkable symmetry of the distribution of produced entropy that holds universally in non-equilibrium steady states with Markovian dynamics. However, in systems with slow degrees of freedom that are hidden from the observer, it is not possible to infer the amount of produced entropy exactly. Previous work suggested that a relation similar to the fluctuation theorem may hold at least approximately for such systems if one considers an apparent entropy production. By extending the notion of apparent entropy production to discrete bipartite systems, we investigate which criteria have to be met for such a modified fluctuation theorem to hold in the large deviation limit. We use asymptotic approximations of the large deviation function to show that the probabilities of extreme events of apparent entropy production always obey a modified fluctuation theorem and, moreover, that it is possible to infer otherwise hidden properties. For the paradigmatic case of two coupled colloidal particles on rings the rate function of the apparent entropy production is calculated to illustrate this asymptotic behavior and to show that the modified fluctuation theorem observed experimentally for short observation times does not persist in the long time limit.