The expansion theorem for the deviation integra

The so-called deviation integral (functional) describes the logarithmic asymptotics of the probabilities of large deviations for random walks generated by sums of random variables or vectors. Here an important role is played by the expansion theorem for the deviation integral in which, for an arbitrary function of bounded variation, the deviation integral is represented as the sum of suitable integrals of the absolutely continuous, singular, and discrete components composing this function. The expansion theorem for the deviation integral was proved by A. A. Borovkov and the author in [9] under some simplifying assumptions. In this article, we waive these assumptions and prove the expansion theorem in the general form.     

Moderate deviation for super-Brownian motion with immigration

We prove a moderate deviation principle for a super-Brownian motion with im-migration of all dimensions, and consequently fill the gap between the central limit theoremand large deviation principle.     

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 FOR EMPIRICAL FIELD OF A SYMMETRIC MEASURE

This paper discusses the large deviation for the empirical field of a symmetricmeasure.The lower bound of the large deviation is obtained by extending the classcal Shannon-McMillan theorem. The upper bound is established by means of Legendre transformation and the minimax theorem.     

E─值独立随机变量部分和的大偏差及应用

本文给出了Ｅ－值随机变量之部分和的大偏差定理（Ｅ是一类局部凸空间）．作为其应用，解决了独立弱收敛随机变量列的经验分布的大偏差问题，从而推广了Ｄｏｎｓｋｅｒ－Ｖａｒａｄｈａｎ的结果。     

Large deviations under a viewpoint of metric geometry: Measure-valued process cases

We illustrate a metric geometry viewpoint for large deviation principles by analyzing the proof of a long-standing conjecture on an explicit Schilder-type theorem for super-Brownian motions given by the authors recently, and by understanding sample path large deviations for Fleming-Viot processes.     

Classical large deviation theorems on complete Riemannian manifolds

We generalize classical large deviation theorems to the setting of complete, smooth Riemannian manifolds. We prove the analogue of Mogulskii’s theorem for geodesic random walks via a general approach using viscosity solutions for Hamilton–Jacobi equations. As a corollary, we also obtain the analogue of Cramér’s theorem. The approach also provides a new proof of Schilder’s theorem. Additionally, we provide a proof of Schilder’s theorem by using an embedding into Euclidean space, together with Freidlin–Wentzell theory.     

Large Deviations from the Almost Everywhere Central Limit Theorem

We prove large deviation principles for the almost everywhere central limit theorem, assuming that the i.i.d. summands have finite moments of all orders. The level 3 rate function is a specific entropy relative to Wiener measure and the level 2 rate the Donsker-Varadhan entropy of the Ornstein-Uhlenbeck process. In particular, the rate functions are independent of the particular distribution of the i.i.d. process under study. We deduce these results from a large deviation theory for Brownian motion via Skorokhod's representation of random walk as Brownian motion evaluated at random times. The results for Brownian motion come from the well-known large deviation theory of the Ornstein-Uhlenbeck process, by a mapping between the two processes.     

Large deviations for sums of i.i.d. random compact sets

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets.

     

Genus Expansion for Real Wishart Matrices

We present an exact formula for moments and cumulants of several real compound Wishart matrices in terms of an Euler characteristic expansion, similar to the genus expansion for complex random matrices. We consider their asymptotic values in the large matrix limit: as in a genus expansion, the terms which survive in the large matrix limit are those with the greatest Euler characteristic, that is, either spheres or collections of spheres. This topological construction motivates an algebraic expression for the moments and cumulants in terms of the symmetric group. We examine the combinatorial properties distinguishing the leading order terms. By considering higher cumulants, we give a central-limit-type theorem for the asymptotic distribution around the expected value.     

Sample path large deviations in finer topologies

In this paper we present sufficient conditions for sample path large deviation principles to be extended to finer topologies. We consider extensions of the uniform topology by Orlicz functional and we consider Lipschitz spaces: the former are concerned with cumulative path behavior while the latter are more sensitive to extremes in local variation. We also consider sample paths indexed by the half line, where the usual projective limit topologies are not strong enough for many applications. We introduce and apply a new technique extending large deviation principles to finer topologies. We show how to apply the results to obtain large deviations for weighted statistics, to improve Schilder's theorem as well as to obtain large deviations in queueing theory     

Influence of hydrodynamic fluctuations on the phase transition in the E and F models of critical dynamics

We use the renormalization group method to study the E model of critical dynamics in the presence of velocity fluctuations arising in accordance with the stochastic Navier-Stokes equation. Using the Martin-Siggia-Rose theorem, we obtain a field theory model that allows a perturbative renormalization group analysis. By direct power counting and an analysis of ultraviolet divergences, we show that the model is multiplicatively renormalizable, and we use a two-parameter expansion in ∈ and δ to calculate the renormalization constants. Here, ∈ is the deviation from the critical dimension four, and δ is the deviation from the Kolmogorov regime. We present the results of the one-loop approximation and part of the fixedpoint structure. We briefly discuss the possible effect of velocity fluctuations on the arge-scale behavior of the model.     

Large Deviations for Additive Functionals of Symmetric Stable Processes

We establish the large deviation principle for additive functionals of symmetric α-stable processes employing the Gärtner-Ellis theorem.     

Large Deviation Theorems for Gaussian Processes and Their Applications in Information Theory

We discuss on the large deviation theorems for stationary Gaussian processes and their applications in information theory. The topics investigated here include error probability of string matching, error probabilities for random codings, and a conditional limit theorem which justifies the maximum entropy principle.     

Large deviations of Markov chains with multiple time-scales

For Markov processes evolving on multiple time-scales a combination of large component scalings and averaging of rapid fluctuations can lead to useful limits for model approximation. A general approach to proving a law of large numbers to a deterministic limit and a central limit theorem around it have already been proven in Kang and Kurtz (2013) and Kang et al. (2014). We present here a general approach to proving a large deviation principle in path space for such multi-scale Markov processes. Motivated by models arising in systems biology, we apply these large deviation results to general chemical reaction systems which exhibit multiple time-scales, and provide explicit calculations for several relevant examples.     

Large Deviations of Continuous Regular Conditional Probabilities

We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for the conditional probabilities under these measures to satisfy a large deviation principle. The arguments of these conditional probabilities are assumed to converge. A way to view regular conditional probabilities as a special case of product regular conditional probabilities is presented. This is used to derive conditions for large deviations of regular conditional probabilities. In addition, we derive a Sanov-type theorem for large deviations of the empirical distribution of the first coordinate conditioned on fixing the empirical distribution of the second coordinate.     

带小随机扰动的中偏差原理(英文)

本文研究了带小随机扰动的中偏差原理.运用收缩原理和指数逼近方法,Freidlin-Wentzell定理给出了Xε的大偏差原理,从而得到了Xε的中偏差原理.     

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and the 2D magnetic Bénard problem and also some shell models of turbulence. We state the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by a weak convergence method.     

Limit Theorems for Inverse Process T_n of Hawkes Process

We consider linear Hawkes process N_t and its inverse process T_n. The limit theorems for N_t are well known and studied by many authors. In this paper, we study the limit theorems for T_n. In particular, we investigate the law of large numbers, the central limit theorem and the large deviation principle for T_n. The main tool of the proof is based on immigration-birth representation and the observations on the relation between N_t and T_n.