带干扰的积分高斯过程的破产概率

本文研究了带干扰的积分高斯过程的破产概率.利用经典大偏差的方法,在一定的条件下,得到了相应概率的对数渐近式及测度族的大偏差原理.结果表明在不带干扰的情形下与已有结果一致.     

Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems

A classical damping Hamiltonian system perturbed by a random force is considered. The locally uniform large deviation principle of Donsker and Varadhan is established for its occupation empirical measures for large time, under the condition, roughly speaking, that the force driven by the potential grows infinitely at infinity. Under the weaker condition that this force remains greater than some positive constant at infinity, we show that the system converges to its equilibrium measure with exponential rate, and obeys moreover the moderate deviation principle. Those results are obtained by constructing appropriate Lyapunov test functions, and are based on some results about large and moderate deviations and exponential convergence for general strong-Feller Markov processes. Moreover, these conditions on the potential are shown to be sharp.     

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

Summary For Gibbsian systems of particles inR ^d , we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.     

Asymptotic Expansions for Large Deviation Probabilities of Noncentral Generalized Chi-Square Distributions

Asymptotic expansions for large deviation probabilities are used to approximate the cumulative distribution functions of noncentral generalized chi-square distributions, preferably in the far tails. The basic idea of how to deal with the tail probabilities consists in first rewriting these probabilities as large parameter values of the Laplace transform of a suitably defined function f_k; second making a series expansion of this function, and third applying a certain modification of Watson's lemma. The function f_k is deduced by applying a geometric representation formula for spherical measures to the multivariate domain of large deviations under consideration. At the so-called dominating point, the largest main curvature of the boundary of this domain tends to one as the large deviation parameter approaches infinity. Therefore, the dominating point degenerates asymptotically. For this reason the recent multivariate asymptotic expansion for large deviations in Breitung and Richter (1996, J. Multivariate Anal.58, 1–20) does not apply. Assuming a suitably parametrized expansion for the inverse g⁻¹ of the negative logarithm of the density-generating function, we derive a series expansion for the function f_k. Note that low-order coefficients from the expansion of g⁻¹ influence practically all coefficients in the expansion of the tail probabilities. As an application, classification probabilities when using the quadratic discriminant function are discussed.     

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.     

Mixtures of Global and Local Edgeworth Expansions and Their Applications

Edgeworth expansions which are local in one coordinate and global in the rest of the coordinates are obtained for sums of independent but not identically distributed random vectors. Expansions for conditional probabilities are deduced from these. Both lattice and continuous conditioning variables are considered. The results are then applied to derive Edgeworth expansions for bootstrap distributions, for Bayesian bootstrap distribution, and for the distributions of statistics based on samples from finite populations. This results in a unified theory of Edgeworth expansions for resampling procedures. The Bayesian bootstrap is shown to be second order correct for smooth positive “priors,” whenever the third cumulant of the “prior” is equal to the third power of its standard deviation. Similar results are established for weighted bootstrap when the weights are constructed from random variables with a lattice distribution.     

Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times

This paper considers a stable GI/GI/1 queue with subexponential service time distribution. Under natural assumptions we derive the tail behaviour of the busy period of this queue. We extend the results known for the regular variation case under minimal conditions. Our method of proof is based on a large deviations result for subexponential distributions.     

Functional continuity and large deviations for the behavior of single-class queueing networks

We consider a single-class queueing network in which the functional network primitives describe the cumulative exogenous arrivals, service times and routing decisions of the queues. The behavior of the network consisting of the cumulative total arrival, cumulative idle time, and queue length developments for each node is specified by conditions which relate the network primitives to the network behavior. For a broad class of network primitives, including discrete customer and fluid models, a network behavior exists, but need not be unique. Nevertheless, the mapping from network primitives to the set of associated network behavior is upper semicontinuous at network primitives with continuous routing. As an application we consider a sequence of random network primitives satisfying a sample path large deviation principle. We take advantage of the partial functional set-valued upper semicontinuity in order to derive a large deviation principle for the sequence of associated random queue length processes and to identify the rate function. This extends the results of Puhalskii (Markov Process. Relat. Fields 13(1), 99–136, 2007) about large deviations for the tail probabilities of generalized Jackson networks. Since the analysis is carried out on the doubly-infinite time axis ?, we can directly treat stationary situations.     

Precise Large Deviations for Sums of Negatively Associated Random Variables with Common Dominatedly Varying Tails

In this paper, we obtain results on precise large deviations for non-random and random sums of negatively associated nonnegative random variables with common dominatedly varying tail distribution function. We discover that, under certain conditions, three precise large-deviation prob- abilities with different centering numbers are equivalent to each other. Furthermore, we investigate precise large deviations for sums of negatively associated nonnegative random variables with certain negatively dependent occurrences. The obtained results extend and improve the corresponding results of Ng, Tang, Yan and Yang （J. Appl. Prob., 41, 93-107, 2004）.     

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.     

Densities of idempotent measures and large deviations

Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied. These conditions depend on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined. As an application, we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle.

     

Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks

The deviation functional (or integral) describes the logarithmic asymptotics of the probabilities of large deviations of trajectories of the random walks generated by the sums of random variables (vectors) (see [1, 2] for instance). In this article we define it on a broader function space than previously and under weaker assumptions on the distributions of jumps of the random walk. The deviation integral turns out the Darboux integral ∫ F(t, u) of a semiadditive interval function F(t, u) of a particular form. We study the properties of the deviation integral and use the results elsewhere in [3] to prove some generalizations of the large deviation principle established previously under rather restrictive assumptions.     

Precise large deviations for a customer-based individual risk model

In this paper,we propose a customer-based individual risk model,in which potential claims by customers are described as i.i.d.heavy-tailed random variables,but different insurance policy holders are allowed to have different probabilities to make actual claims.Some precise large deviation results for the prospective-loss process are derived under certain mild assumptions,with emphasis on the case of heavy-tailed distribution function class ERV(extended regular variation).Lundberg type limiting results on the finite time ruin probabilities are also investigated.     

Large deviations for empirical measures with degenerate limiting distribution

Summary This paper studies the large deviations of the empirical measure associated withn independent random variables with a degenerate limiting distribution asn. A large deviations principle — quite unlike the classical Sanov type results — is established for such empirical measures in a general Polish space setting. This result is applied to the large deviations for the empirical process of a system of interacting particles, in which the diffusion coefficient vanishes as the number of particles tends to infinity. A second way in which the present example differs from previous work on similar weakly interacting systems is that there is a singularity in the mean-field type interaction.     

Large Deviation Bounds for Single Class Queueing Networks and Their Calculation

We investigate large deviations for the behavior of single class queueing networks. The starting point is a sample large deviation principle on the path-space of network primitives describing the cumulative external arrivals, service time requirements and routing decisions. The behavior of the network, capturing the cumulative total arrivals, idle times and queue lengths, is characterized by a path-space fixed point equation containing the network primitives. The mapping from the network primitives to the set of fixed points is partially upper semicontinuous. This set-valued continuity allows us to derive large deviation bounds for the network behavior in the form of variational problems. The analysis is carried out on the doubly-infinite time axis R and can directly capture stationary and non-Markovian situations. By relaxing the fixed point equation the upper bounds and minimizing paths can be approximated with piecewise linear paths. For a class of typical rate functions we specify sequences of finite dimensional minimization problems which permit the calculation of large deviation rates and minimizing paths for the tail probabilities of queue lengths. We illustrate the approach with an example.     

Large deviations for stochastic partial differential equations driven by a Poisson random measure

Stochastic partial differential equations driven by Poisson random measures (PRMs) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equation (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative functionals of general PRMs, reduces the proof of the large deviation principle to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. As an illustration of the general theory, we consider a particular system that models the spread of a pollutant in a waterway.     

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.     

Generalized Neyman–Pearson optimality of empirical likelihood for testing parameter hypotheses

This paper studies the Generalized Neyman–Pearson (GNP) optimality of empirical likelihood-based tests for parameter hypotheses. The GNP optimality focuses on the large deviation errors of tests, i.e., the convergence rates of the type I and II error probabilities under fixed alternatives. We derive (i) the GNP optimality of the empirical likelihood criterion (ELC) test against all alternatives, and (ii) a necessary and a sufficient condition for the GNP optimality of the empirical likelihood ratio (ELR) test against each alternative.     

Moderate Deviations for Longest Increasing Subsequences: The Lower Tail

We derive a moderate deviation principle for the lower tail probabilities of the length of a longest increasing subsequence in a random permutation. It refers to the regime between the lower tail large deviation regime and the central limit regime. The present article together with the upper tail moderate deviation principle in Ref. 12 yields a complete picture for the whole moderate deviation regime. Other than in Ref. 12, we can directly apply estimates by Baik, Deift, and Johansson, who obtained a (non-standard) Central Limit Theorem for the same quantity.     

Small deviations of iterated processes in the space of trajectories

We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.