Problem Of Large Deviations Research Articles

Large charge fluctuations in classical Coulomb systems

The typical fluctuation of the net electric chargeQ contained in a subregionΛ of an infinitely extended equilibrium Coulomb system is expected to grow only as √S, whereS is the surface area ofΛ. For some cases it has been previously shown thatQ/√S has a Gaussian distribution as ¦Λ¦→∞. Here we study the probability law for larger charge fluctuations (large-deviation problem). We discuss the case when both ¦Λ¦ andQ are large, but now withQ of an order larger than √S. For a given value ofQ, the dominant microscopic configurations are assumed to be those associated with the formation of a double electrical layer along the surface ofΛ. The probability law forQ is then determined by the free energy of the double electrical layer. In the case of a one-component plasma, this free energy can be computed, for large enoughQ, by macroscopic electrostatics. There are solvable two-dimensional models for which exact microscopic calculations can be done, providing more complete results in these cases. A variety of behaviors of the probability law are exhibited.

Large deviation problem for some parabolic itǒ equations

AbstractThe large deviation problem of Wentzell‐Freidlin type is considered for a class of semilinear parabolic equations perturbed by a small multiplicative noise. In a Hilbert space setting, it is proved that, under suitable conditions, the associated family of solution measures, depending on a small parameter, obeys the large deviation principle with respect to a certain rate function.

A Note on Lower Bounds of Large Deviation Probabilities

In this note, a lower bound of large deviation probabilities is derived; this extends the main results of Bahadur, Gupta and Zabell (1980). An application to the Sonov large deviation problem is considered.

A PDE Approach to Certain Large Deviation Problems for Systems of Parabolic Equations

We prove an exponential decay estimate for solutions of certain scaled systems of parabolic PDE. Our techniques employ purely PDE methods, mainly the theory of viscosity solutions of Hamilton-Jacobi equations, and provide therefore an alternate approach to the probabilistic large deviation methods of Freidlin-Wentzell, Donsker-Varadhan, etc.

Large deviations and stochastic homogenization

A general theorem is stated providing large deviations estimates for a family of measures on a topological vector space. Applications are given in the second part, where large deviations problems arising in stochastic homogenization are discussed. Another application is given in similar problems connected with Donsker's invariance principle.

Exit Time Problems in Optimal Control and Vanishing Viscosity Method

The authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first-order Hamilton-Jacobi (HJ) equation up to the boundary. This equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary. When these solutions are equal, they can be obtained through the vanishing viscosity method. Finally, when the HJ equation has a continuous solution, it is proved to be the value function for the first exit time of the domain. It is also the vanishing viscosity limit arising, in particular, in some large deviations problems.

Estimating Tails of Probability Distributions

We study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold. A key idea is the use of Pickands' generalised Pareto distribution and its fitting, in most cases, by the method of maximum likelihood. The results cover all three limiting types of extreme value theory. We propose a new estimator for an index of regular variation and show that it often performs better than Hill's estimator. We give new results for estimating the endpoint of a distribution, extending earlier work by Hall and by Smith and Weissman. Finally, we give detailed results for the domain of attraction of $\exp(-e^{-x})$ and show that, in most cases, our proposed estimator is more efficient than two others, one based on the exponential distribution and the other due to Davis and Resnick. We also touch briefly on the problem of large deviations from a statistical point of view. The results make extensive use of existing work on rates of convergence.

11. Sequential methods : Problem of large deviations in the sequential case

11. Sequential methods : Problem of large deviations in the sequential case

On the Asymmetric Problem of Large Deviations When the Limit Law is Stable

On the Asymmetric Problem of Large Deviations When the Limit Law is Stable

On the Asymmetrical Problem of Large Deviations

On the Asymmetrical Problem of Large Deviations

On Large Deviation Problems for Sums of Random Variables which are not Attracted to the Normal Law

On Large Deviation Problems for Sums of Random Variables which are not Attracted to the Normal Law