Large deviations for multi-dimensional reflected fractional Brownian motion

By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening know sample path large deviation principles for fractional Brownian motion to this topology, we obtain large deviation principles for the image of multi-dimensional fractional Brownian motions under Skorokhod maps as an immediate consequence of the contraction principle. As an application, we explicitly calculate large deviation decay rates for steady-state tail probabilities of certain queueing systems in multi-dimensional heavy traffic models driven by fractional Brownian motions.     

Large and Moderate Deviations of Random Upper Semicontinuous Functions

We firstly discuss the topological properties of the space of upper semicontinuous functions, and then we obtain large deviation principles for random upper semicontinuous functions under various topologies. Finally, we prove moderate deviation principles for random sets and random upper semicontinuous functions.     

A large deviation principle form-variate von Mises-statistics and U-statistics

A large deviation principle form-variate von Mises-statistics and U-statistics with a kernel function satisfying natural moment conditions is proved. Sanov's large deviation result for the empirical distribution function and two fundamental conservation principles in large deviation theory are the main tools. The rate functions are “drawback”-entropy functionals.     

Large deviation principle for stochastic integrals and stochastic differential equations driven by infinite-dimensional semimartingales

The paper concerns itself with establishing large deviation principles for a sequence of stochastic integrals and stochastic differential equations driven by general semimartingales in infinite-dimensional settings. The class of semimartingales considered is broad enough to cover Banach space-valued semimartingales and the martingale random measures. Simple usable expressions for the associated rate functions are given in this abstract setup. As illustrated through several concrete examples, the results presented here provide a new systematic approach to the study of large deviation principles for a sequence of Markov processes.     

Large and moderate deviation principles for susceptible-infected-removed epidemic in a random environment

We are concerned with SIR epidemics in a random environment on complete graphs, where edges are assigned with i.i.d. weights. Our main results give large and moderate deviation principles of sample paths of this model. Our results generalize large and moderate deviation principles of the classic SIR models given by E. Pardoux and B. Samegni-Kepgnou [J. Appl. Probab., 2017, 54: 905-920] and X. F. Xue [Stochastic Process. Appl., 2019, 140: 49-80].     

New large deviation results for some super-Brownian processes

We give large deviation results for the super-Brownian excursion conditioned to have unit mass or unit extinction time and for super-Brownian motion with constant non-positive drift. We use a representation of these processes by a path-valued process, the so-called Brownian snake for which we state large deviation principles.     

On Laplace–Varadhan's integral lemmaUne note sur le lemme intégral de Laplace–Varadhan

In this Note we propose a complement to an integral lemma of Laplace–Varadhan arising in the literature of large deviations. We examine a situation in which the state space may depend on the rate of deviation. We have used this extension to analyze in a new way large deviation principles for the empirical measures on path space associated to interacting particle systems. We prove new large deviation principles on path space for an abstract class of discrete generation, as well as pure jump or McKean–Vlasov interacting particle systems. To cite this article: P. Del Moral, T. Zajic, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 693–698.     

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.     

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     

Large Deviations for Brownian Motion on Scale Irregular Sierpinski Gaskets

We study sample path large deviation principles for Brownian motion on scale irregular Sierpinski gaskets which are spatially homogeneous but do not have any exact self-similarity. One notable point of our study is that the rate function depends on a large deviation parameter and as such, we can only obtain an example of large deviations in an incomplete form. Instead of showing the large deviations principle we would expect to hold true, we show Varadhan’s integral lemma and exponential tightness by using an incomplete version of such large deviations.     

Large deviations for product of sums of random variables

In this paper, we obtain sample path and scalar large deviation principles for the product of sums of positive random variables. We study the case when the positive random variables are independent and identically distributed and bounded away from zero or the left tail decays to zero sufficiently fast. The explicit formula for the rate function of a scalar large deviation principle is given in the case when random variables are exponentially distributed.     

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.     

On Large Deviation Convergence of Invariant Measures

We present new results on the connection between large deviation principles for trajectories of stochastic processes and the associated invariant measures. Applications to diffusion and queuing processes are provided.     

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.     

The effect of memory on functional large deviations of infinite moving average processes

The large deviations of an infinite moving average process with exponentially light tails are very similar to those of an i.i.d. sequence as long as the coefficients decay fast enough. If they do not, the large deviations change dramatically. We study this phenomenon in the context of functional large, moderate and huge deviation principles.     

Large deviations for the Fleming–Viot process with neutral mutation and selection,II

Large deviation principles are established for the Fleming–Viot process with neutral mutation and with selection, and the associated equilibrium measures as the sampling rate approaches zero and when the state space is equipped with the weak topology. The path-level large deviation results improve the results of Dawson and Feng (1998, Stochastic Process. Appl. 77, 207–232) in three aspects: the state space is more natural, the initial condition is relaxed, and a large deviation principle is established for the Fleming–Viot process with selection. These improvements are achieved through a detailed study of the behaviour near the boundary of the Fleming–Viot process with finite types.     

LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR m-NEGATIVELY ASSOCIATED RANDOM VARIABLES

M-negatively associated random variables,which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its par- ticular case,are introduced and studied.Large deviation principles and moderate devi- ation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.     

Sample path large and moderate deviations for risk model with delayed claims

Sample path large and moderate deviation principles for Markov modulated risk models with delayed claims are proved by the exponential martingale method. As applications, asymptotic estimates and exponential bounds of the ruin probability are also studied.     

带移民超布朗运动的占位时中偏差

本文证明了当底空间维数d≥3时,一类带移民超布朗运动占位时过程的中偏差,其移民由Lebesgue 测度控制.可以清楚地看出,中偏差的规范化因子和速度函数恰好介于中心极限定理和大偏差之间,在 这个意义下,中偏差填补了中心极限定理和大偏差之间的空白.     

Asymptotic analysis of the optimal cost in some transportation problems with random locations

In this paper we provide an asymptotic analysis of the optimal transport cost in some matching problems with random locations. More precisely, under various assumptions on the distribution of the locations and the cost function, we prove almost sure convergence, and large and moderate deviation principles. In general, the rate functions are given in terms of infinite-dimensional variational problems. For a suitable one-dimensional transportation problem, we provide the expression of the large deviation rate function in terms of a one-dimensional optimization problem, which allows the numerical estimation of the rate function. Finally, for certain one-dimensional transportation problems, we prove a central limit theorem.