Large Deviation Estimates Research Articles

Large Deviations in Dynamical Systems and Stochastic Processes

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.

On evaluating the rate function of large deviations for jump processes

This is a sequel to our joint paper[4] in which upper bound estimates for large deviations for Markov chains are studied. The purpose of this paper is to characterize the rate function of large deviations for jump processes. In particular, an explicit expression of the rate function is given in the case of the process being symmetrizable.

Large deviation estimates in the stochastic quantization ofϕ 2 4

We study in the small noise limit the behaviour of field trajectories for the process constructed by the authors in connection with the stochastic quantization ofϕ24. Due to the presence of infinite renormalization the usual large deviation techniques do not apply immediately and a new strategy has to be developed. We prove some estimates analogous to the Freidlin-Ventzel inequalities. From these it follows that the field trajectories suitably smeared in space over a scaler0 behave, when the noise is small, as the projection on the same scale of a field obeying a regularized stochastic equation with a large cut-off. However the estimates are not uniform in the cut-off and an interesting feature of the problem is that the scale over which the field is smeared determines whether the noise is sufficiently small for the estimates to apply.

Some Large Deviation Results for Dynamical Systems

We prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics, rate of escape problems, and to shift spaces.

Large deviations in dynamical systems and stochastic processes

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.

Large deviations in dynamical systems

We prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics, rate of escape problems, and to shift spaces.

Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures

Let (X i ,Y i ) ∈ℤ d , be independent identically distributed random variables with arbitrary distribution. We show that, for almost every(Y i ) i , the conditional law of the empirical field given(Y i ) i satisfies to large deviation inequalities. This applies to the study of Gibbs measures with random interaction, in the case of some mean-field models as well as of short range summable interaction. We show that the pressure is nonrandom, and is given by a variational formula. These random Gibbs measures have the same large deviation rate, which does not depend on the particular realization of the interaction; their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula.

Expected Power System Production Costs Using Large Deviation and Mixture of Normals Approximations

This paper proposes a possible enhancement to the computational performance of the large deviation method for determining expected system production costs. With the use of a mixed normal distribution to approximate the initial load duration curve, a faster algorithm which results in considerable amount of savings in computation time is developed. The proposed method is tested using a synthetic system with two different load duration curves and the results are compared to several other approximation methods. Depending on the accuracy with which a mixture of normals can be obtained to represent the load duration curve, the modified large deviation approximation method is comparable in speed and accuracy to the method of cumulants.

Approximate exit probabilities for a Brownian bridge on a short time interval, and applications

Let T be the first exit time of Brownian motion W(t) from a region ℛ in d-dimensional Euclidean space having a smooth boundary. Given points ξ0 and ξ1 in ℛ, ordinary and large-deviation approximations are given for Pr{T < ε |W(0) = ξ0, W(ε) = ξ 1} as ε → 0. Applications are given to hearing the shape of a drum and approximating the second virial coefficient.

Approximate exit probabilities for a Brownian bridge on a short time interval, and applications

LetTbe the first exit time of Brownian motionW(t) from a region ℛ ind-dimensional Euclidean space having a smooth boundary. Given points ξ0and ξ1in ℛ, ordinary and large-deviation approximations are given for Pr{T < ε|W(0) = ξ0,W(ε)=ξ1} asε→ 0. Applications are given to hearing the shape of a drum and approximating the second virial coefficient.

Reliability Computations for Interconnected Generating Systems via Large Deviation Approximation

Reliability Computations for Interconnected Generating Systems via Large Deviation Approximation

Reliability computations for interconnected generating systems via large deviation approximation

The use of the large-deviation approximation is discussed for computing the reliability of two generating systems having correlated loads and interconnected by a transmission tie-line of a given capacity. The necessary formulas are derived and applied to a prototype two-area system model. Numerical results show that the proposed approximation is quite accurate. Information is provided on the speed of the algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Lyapunov Exponent and Rotation Number of Two-Dimensional Linear Stochastic Systems with Small Diffusion

By means of a singular perturbation scheme and large deviation estimates we expand the Lyapunov exponent and rotation number in terms of the (small) noise intensity $\varepsilon $ for all $2 \times 2$ systems $dX_t^\varepsilon = AX_t^\varepsilon dt + \sqrt \varepsilon \sum\nolimits_{k = 1}^r {B_k } X_t^\varepsilon \circ dW_t^k $ for which A is complex diagonalizable. Examples are given.

Approximate Tail Probabilities for the Maxima of Some Random Fields

Abstract : Hogan and Siegmund (1986) adapt the method developed by Pickands (1969), Qualls and Watanabe (1973), and Bickel and Rosenblatt (1973) to obtain explicit large deviation approximations for the maxima of several Gaussian random fields arising in statistics. Using a special argument for one particular case, they suggest a heuristic second order approximation for that case; and they show by a Monte Carlo experiment that the second order approximation frequently gives considerably better numerical results. The purpose of this paper is to show that the method developed by Woodroofe (1976,1982) for problems in one dimensional time can be adapted to study maxima of random fields. Overall it involves simpler computations than the previous method and consequently seems potentially capable of delivering a genuine second order approximation should one seem desirable.

Large deviation for stationary processes

A quasi-local variational characterization of the entropy for stationary processes is given. This is used to establish upper and lower large deviation estimates for arbitrary stationary processes. The upper and lower rate functions are shown to coincide for all quasi-local stationary processes.

Second order large deviation estimates for ferromagnetic systems in the phase coexistence region

We consider thed-dimensional Ising model with ferromagnetic nearest neighbor interaction at inverse temperature β. Let\(M_\Lambda = |\Lambda |^{ - 1} \sum\limits_{i \in \Lambda } {\sigma _i } \) be the magnetization inside ad-dimensional hyper cube Λ, μ+ be the+Gibbs state andm*(β) be the spontaneous magnetization. For β such thatm*(β)>0 we find a sufficient condition (easily verified to hold for large β) for μ+({MΛ∈[a,b]}) to decay exponentially with |Λ|(d−1)/d when −m*<b<m*, −1≦a<b. Ford=2 this sufficient condition is the exponential decay of a connectivity function. We also prove a partial converse to this result, obtain a sharper result for the magnetization ond−1 dimensional cross sections of the model and prove a similar result ford=2, −m*<a<b<m*, and β large, when free boundary conditions are chosen outside Λ.

A Large Deviations Principle for Small Perturbations of Random Evolution Equations

We prove a result for small random perturbations of random evolution equations analogous to the Ventsel-Freidlin result on small perturbations of dynamical systems. In particular, we derive large deviations estimates and indicate how they can be used to prove an exit result. The processes we study are governed by equations of the form $dx^\varepsilon(t) = b(x^\varepsilon(t), y(t)) dt + \sqrt \varepsilon \sigma(x^\varepsilon(t)) dw(t)$, where $x^0$ is already a random process. The results include the case where $y$ is an $n$-state Markov process. In the special case $\sigma \equiv Id$, the proof of the estimates is a consequence of a generalization of the contraction principle for large deviations: We give sufficient conditions on a continuous function $F$, which ensure that if $\{X_\varepsilon: \varepsilon > 0\}$ satisfies a large deviations principle, then so does $\{F(X_\varepsilon, Y): \varepsilon > 0\}$, where $Y$ is independent of $\{X_\varepsilon: \varepsilon > 0\}$.

Sharp concentration of the chromatic number on random graphsG n, p

The distribution of the chromatic number on random graphsG n, p is quite sharply concentrated. For fixedp it concentrates almost surely in √n ω(n) consecutive integers where ω(n) approaches infinity arbitrarily slowly. If the average degreepn is less thann 1/6, it concentrates almost surely in five consecutive integers. Large deviation estimates for martingales are used in the proof.

Discussion of Dr JØrgensen’s Paper

Discussion of Dr JØrgensen’s Paper

Stochastic Approximations via Large Deviations: Asymptotic Properties

Asymptotic properties of Robbins–Munro and Kiefer–Wolfowitz type stochastic approximation algorithms are obtained via the theory of large deviations. The conditions are weak and can even yield w.p.l. convergence results. The probability of escape of the iterates from a neighborhood of a stable point of the algorithm is estimated and shown to be considerably smaller than suggested by the classical “asymptotic normality of local normalized errors” method of getting the asymptotic properties. The escape probabilities are a natural quantity of interest. In many applications, they are more useful than the “local normalized mean square errors.” Other large deviations estimates are also obtained. Typically, if ${{a_n = 1} / {n^\rho }}$, $\rho \leqq 1$, then the probability of escape from a neighborhood of a stable point in some (normalized) time interval $[n,m]:\sum_n^m {a_i \sim T} $ is $\exp - n^\rho V_\rho $, where $V_\rho $ does not depend on $\rho $ for $\rho < 1$ and is the solution to an optimal control p...