Large deviation principle for diffusion processes under a sublinear expectation

We represent the exponential moment of the Brownian functionals under a nonlinear expectation according to the solution to a backward stochastic differential equation.As an application,we establish a large deviation principle of the Freidlin and Wentzell type under the corresponding nonlinear probability for diffusion processes with a small diffusion coefficient.     

Large deviation on a covering graph with group of polynomial growth

We discuss a large deviation principle of a periodic random walk on a covering graph with its transformation group of polynomial volume growth in view of geometry. As we shall observe, the behavior of a random walk at infinity is closely related to the Gromov?CHausdorff limit of an infinite graph and in the case where the graph admits an action of a group of polynomial volume growth, the Carnot-Carathéodory metric shows up in its limit space.     

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.     

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. The contents of the paper is the following: 1. Introduction; 2. Notations; 3. Relative entropy of conditional expectations; 4. Relative entropy of a stationary process with respect to a covariant family of conditional expectations; 5. The role of locality and quasi-locality properties; 6. Large deviation upper estimate; 7. The Lower estimate; 8. The variational principle.     

Large deviation theorem for Hill's estimator

To estimate the exponent of a regularly varying d.f. F, the asymptotic behaviour of Hill's estimator has been extensively discussed. Under the assumption that the d.f. F is continuous, we obtain the large deviation theorem for Hill's estimator. Project supported by the National Natural Science Foundation of China     

Large deviation bounds for matrix Brownian motion

We prove large deviation bounds for the convergence of Hermitian matrix valued Brownian motion towards free Brownian motion. As a consequence, we obtain upper and lower bounds on the microstates entropy introduced by Voiculescu [24]. Oblatum 5-VIII-2002 & 18-XI-2002?Published online: 24 February 2003     

Large deviation principle for stochastic evolution equations

Summary The large deviation principle obtained by Freidlin and Wentzell for measures associated with finite-dimensional diffusions is extended to measures given by stochastic evolution equations with non-additive random perturbations. The proof of the main result is adopted from the Priouret paper concerning finite-dimensional diffusions. Exponential tail estimates for infinite-dimensional stochastic convolutions are used as main tools.     

Large deviation principle for enhanced Gaussian processes

We study large deviation principles for Gaussian processes lifted to the free nilpotent group of step N. We apply this to a large class of Gaussian processes lifted to geometric rough paths. A large deviation principle for enhanced (fractional) Brownian motion, in Hölder- or modulus topology, appears as special case.     

Large deviation probabilities forUH-statistics

ForU-statistics taking values in a Hilbert space, we obtain estimates of the rate of convergence in the central limit theorem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1611–1620, December, 1994.     

Large deviation probabilities for certain dependent processes

Certain results on large deviation probabilities for linear and m-dependent processes are considered here.     

Large deviation for stochastic Cahn-Hilliard partial differential equations

In this paper, we prove a large deviation principle for a class of stochastic Cahn-Hilliard partial differential equations driven by space-time white noises.     

Large deviation principles for the stochastic quasi-geostrophic equations

In this paper we establish the large deviation principle for the stochastic quasi-geostrophic equation with small multiplicative noise in the subcritical case. The proof is mainly based on the weak convergence approach. Some analogous results are also obtained for the small time asymptotics of the stochastic quasi-geostrophic equation.     

Large deviation theory for non-regular location shift family

We apply non-regular extensions of the large deviation theory to non-regular location shift families. Our calculation contains the location shift families generated by Beta distribution, Weibull distribution, and Gamma distribution. We point out the optimal estimator depends on the choice of our criterion in the non-regular case. The limits of relative Rényi entropies play an important role in our derivation.     

Large deviation principles for sequences of logarithmically weighted means

In this paper we consider several examples of sequences of partial sums of triangular arrays of random variables {Xn:n?1}; in each case Xn converges weakly to an infinitely divisible distribution (a Poisson distribution or a centered Normal distribution). For each sequence we prove large deviation results for the logarithmically weighted means with speed function . We also prove a sample path large deviation principle for {Xn:n?1} defined by , where σ²∈(0,∞) and {Un:n?1} is a sequence of independent standard Brownian motions.     

Large deviation theorems for extended random variables and some applications

Large deviation theorems of Chernoff type for extended random variables are proved. The large deviation principle for extended random variables is obtained, too. The obtained limit theorems are used to prove the large deviation theorems of Chernoff type for the logarithm of the likelihood ratio in general binary statistical experiments. The rate of decrease of the error probabilities is investigated for Neyman-Pearson tests, Bayes tests, and minimax tests. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Large deviation principle of occupation measure for stochastic Burgers equation

In this paper we obtain a Large Deviation Principle for the occupation measure of the solution to a stochastic Burgers equation which describes the exact rate of exponential convergence. This Markov process is strongly Feller and has a unique invariant measure. Moreover, the rate function is explicit: it is the level-2 entropy of Donsker-Varadhan.     

Large deviation principle of occupation measures for non-linear monotone SPDEs

Using the hyper-exponential recurrence criterion,we establish the occupation measures’large deviation principle for a class of non-linear monotone stochastic partial differential equations(SPDEs)driven by Wiener noise,including the stochastic p-Laplace equation,the stochastic porous medium equation and the stochastic fast-diffusion equation.We also propose a framework for verifying hyper-exponential recurrence,and apply it to study the large deviation problems for strong dissipative SPDEs.These SPDEs can be stochastic systems driven by heavy-tailedα-stable process.