Precise Large Deviations for Long-Tailed Distributions

In this paper, we investigate the precise large deviations for sums of independent identically distributed random variables with heavy-tailed distributions. We prove asymptotic relations for non-random sums and for random sums of random variables with long-tailed distributions. We apply the results on two useful counting processes, namely, renewal and compound-renewal processes.     

On the limit shape of elements of an arithmetic semigroup with an exponentially growing counting function of basis elements

We consider an arithmetic semigroup with exponential growth of the counting function of abstract primes. The Bose–Einstein statistics provides the most probable mean occupation numbers in the sense that large deviations of a sum of occupation numbers from the corresponding sum for the Bose–Einstein statistics have small probabilities. The probabilities of large deviations are estimated.     

On asymptotic behavior for probabilities of large deviations of compound Cox processes

We derive logarithmic asymptotics for probabilities of large deviations for compound Cox processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables and processes with independent increments. When these conditions fail, the asymptotics of large deviations probabilities for compound Cox processes are quite different. Bibliography: 5 titles. Translated from Zapiski Nauehnykh, Seminarov POMI, Vol. 361, 2008, pp. 167–181.     

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.     

On asymptotic behavior of probabilities of large and moderate deviations for some iterated stochastic processes

We derive logarithmic asymptotics for probabilities of large deviations for some iterated processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables. When these conditions do not hold, the asymptotics of large deviations for iterated processes are quite different. When the iterated process is a homogeneous process with independent increments in which time is replaced by random one, the behavior of large and moderate deviations is studied in the case of finite variance. For this case, the following one-sided moment restriction are considered: the Cramèr condition, the Linnik condition, and the existence of moment of order p > 2 for a positive part. Bibliography: 6 titles.     

Large deviation for pinned covering diffusion

We shall establish large deviations for the pinned motions of a periodic diffusion process on n dimensional Euclidean space and the Brownian motion on 2 dimensional Lobachevsky space. It will be shown that the rate functions for the large deviations are corresponding to the infinitesimal generators of the diffusion processes obtained from the above processes through a kind of harmonic transform by positive principal eigenfunctions.     

Large deviations for the radial processes of the Brownian motions on hyperbolic spaces

Using the heat kernel estimates by Davies (1989) and Anker et al. (1996), we show large deviations for the radial processes of the Brownian motions on hyperbolic spaces.     

Rough asymptotics for tandem non-homogeneous $$M/G/\infty$$ queues via Poissonized empirical processes

We obtain a large deviations principle and moderate deviations principle for the joint distribution of the queue length processes and departure process for tandem queues. The results are obtained by applying a result providing necessary and sufficient conditions on a class of functions for a large deviations principle and moderate deviations principle to hold for a Poissonized empirical process over the class of functions. As an application, we examine how large queue lengths and numbers of departures are built up. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some asymptotic results for nonlinear Hawkes processes

Hawkes process is a class of simple point processes with self-exciting and clustering properties. Hawkes process has been widely applied in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study fluctuations, large deviations and moderate deviations nonlinear Hawkes processes in a new asymptotic regime, the large intensity function and the small exciting function regime. It corresponds to the large baseline intensity asymptotics for the linear case, and can also be interpreted as the asymptotics for the mean process of Hawkes processes on a large network.     

Large deviations for statistics of the Jacobi process

This paper aims to derive large deviations for statistics of the Jacobi process already conjectured by M. Zani in her thesis. To proceed, we write in a simpler way the Jacobi semi-group density. Being given by a bilinear sum involving Jacobi polynomials, it differs from Hermite and Laguerre cases by the quadratic form of its eigenvalues. Our attempt relies on subordinating the process using a suitable random time change. This gives a Mehler-type formula whence we recover the desired semi-group density. Once we do, an adaptation of Zani’s result [M. Zani, Large deviations for squared radial Ornstein–Uhlenbeck processes, Stochastic. Process. Appl. 102 (1) (2002) 25–42] to the non-steep case will provide the required large deviations principle.     

Large deviations behavior of counting processes and their inverses

We show, under regularity conditions, that a counting process satisfies a large deviations principle in or the Gärtner-Ellis condition (convergence of the normalized logarithmic moment generating functions) if and only if its inverse process does. We show, again under regularity conditions, that embedded regenerative structure is sufficient for the counting process or its inverse process to have exponential asymptotics, and thus satisfy the Gärtner-Ellis condition. These results help characterize the small-tail asymptotic behavior of steady-state distributions in queueing models, e.g., the waiting time, workload and queue length.     

Integro-Local Limit Theorems for Compound Renewal Processes under Cramér’S Condition. I

We obtain integro-local limit theorems in the phase space for compound renewal processes under Cramér’s moment condition. These theorems apply in a domain analogous to Cramér’s zone of deviations for random walks. It includes the zone of normal and moderately large deviations. Under the same conditions we establish some integro-local theorems for finite-dimensional distributions of compound renewal processes.     

On the small time behavior of Ornstein–Uhlenbeck processes with unbounded linear drifts

In this paper, we study the small time behavior of Ornstein–Ulenbeck processes with unbounded linear drifts on Hilbert spaces. The large deviations estimates are obtained. The general theory of Dirichlet forms, Lyons–Zheng′s decompositions and the convolution representation of the processes play an important role. Received: 12 February 1997 / Revised version: 27 November 1998     

Neural correlates of counting large numerosity

The mastery of counting numerosities larger than those correctly estimated by infants or non-human species is an important foundation for the development of higher level calculation skills. The cognitive processes involved in counting are related to spatial attention, language, and number processing. However, the respective involvement of language- and/or visuo-spatial-based brain systems during counting is still under debate. In the present functional magnetic resonance imaging study, we asked 27 right-handed participants to perform an enumeration task on visual arrays of bars that varied in numerosity. Each enumeration condition was contrasted to a color-detection condition that was numerically and spatially matched to the counting condition. The results showed a behavioral discontinuity in response time slopes between large (6–10) and small (1–5) numerosities during enumeration, suggesting that during large enumeration, participants engaged counting processes. Comparing brain regional activity during the enumeration of large numerosity to the enumeration of smaller numerosity, we found increased activation in the bilateral fronto-parietal attentional network, the inferior parietal gyri/intraparietal sulci, and the left ventral premotor and left inferior temporal areas. These results indicated that in adults who master enumeration, counting more than five items requires the strong involvement of spatial attention and eye movements, as well as numerical magnitude processes. Counting large numerosity also recruited verbal working memory areas, subtending a subvocal articulatory code and a visual representation of numbers.     

A thermodynamic study of the Benedix effect in the framework of the gradient theory

The methods of the thermodynamics of nonequilibrium processes are used to describe the Benedix effect under the assumption that the hypothesis of local equilibrium is applicable at large deviations of the system from its equilibrium position.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 11–14.     

A Large Deviation Principle of Retarded Ornstein-Uhlenbeck Processes Driven by Lévy Noise

In this article, we develop a large deviation principle (LDP) for a class of retarded Ornstein-Uhlenbeck processes driven by Lévy processes. We first present a LDP result for time delay systems driven by cylindrical Wiener processes based on the large deviations of Gaussian processes. By using a contraction technique and passing on a finite-dimensional approximation, an LDP is obtained for stochastic time delay evolution equations driven by additive Lévy noise, whose solutions are generally not Lévy processes any more.     

Inequalities between some large deviation rates

In some cases it is interesting to have inequalities between asymptotic rates. Here, we consider asymptotic rates in the fashion of large deviations. We present inequalities between large deviation rate functions for telegrapher processes in the first part, and inequalities between Lundberg parameters for Markov‐additive processes in the second part. Copyright © 2007 John Wiley & Sons, Ltd.     

Large deviations for renewal processes

We investigate large deviations for the empirical measure of the forward and backward recurrence time processes associated with a classical renewal process with arbitrary waiting-time distribution. The Donsker-Varadhan theory cannot be applied in this case, and indeed it turns out that the large deviations rate functional differs from the one suggested by such a theory. In particular, a non-strictly convex and non-analytic rate functional is obtained.     

Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion processes

We prove the hydrostatics of boundary driven gradient exclusion processes, Fick’s law and we present a simple proof of the dynamical large deviations principle which holds in any dimension.     

暂留对称扩散过程全占据时的大偏差

本文讨论暂留对称扩散过程的全占据时的大偏差．我们证明在一致椭圆条件下,暂留对称扩散过程的全占据时满足大偏差原理．