Some Results for Large Closed Queueing Networks with and without Bottleneck: Up- and Down-Crossings Approach

The paper provides the up- and down-crossing method to study the asymptotic behavior of queue-length and waiting time in closed Jackson-type queueing networks. These queueing networks consist of central node (hub) and k single-server satellite stations. The case of infinite server hub with exponentially distributed service times is considered in the first section to demonstrate the up- and down-crossing approach to such kind of problems and help to understand the readers the main idea of the method. The main results of the paper are related to the case of single-server hub with generally distributed service times depending on queue-length. Assuming that the first k–1 satellite nodes operate in light usage regime, we consider three cases concerning the kth satellite node. They are the light usage regime and limiting cases for the moderate usage regime and heavy usage regime. The results related to light usage regime show that, as the number of customers in network increases to infinity, the network is decomposed to independent single-server queueing systems. In the limiting cases of moderate usage regime, the diffusion approximations of queue-length and waiting time processes are obtained. In the case of heavy usage regime it is shown that the joint limiting non-stationary queue-lengths distribution at the first k–1 satellite nodes is represented in the product form and coincides with the product of stationary GI/M/1 queue-length distributions with parameters depending on time.     

Anticipated backward stochastic differential equations driven by the Teugels martingales

In this paper, a class of anticipated backward stochastic differential equations driven by Teugels martingales associated with Lévy process is investigated. We obtain the existence and uniqueness of solutions to these equations by means of the fixed-point theorem. We show that a comparison theorem for this type of ABSDEs also holds under some slight stronger conditions.     

任意随机序列的变换及其a.s.收敛性

本文利用截尾方法构造几乎处处收敛的鞅结合无穷乘积定理，研究随机变量序列变换的局部收敛性及强大数定理，作为推论得到了关于赌博系统的若干强极限定理。     

Bootstrapping a Bayes estimator of a survival function with censored data

The large-sample frequentist property of a frequentist bootstrap for a posterior mean with respect to a Dirichlet prior of the survival function for a randomly censored data is given. The weak convergence of a bootstrap version of the Susarla-Van Ryzin estimator is established on the whole real line. An illustration of the technique and some Monte Carlo studies are also given.     

Diffusion of directed polymers in a strong random environment

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. It was shown by Imbrie and Spencer that in spatial dimensions three or above the behavior is diffusive if the directed polymer interacts weakly with the environment and if the random environment follows the Bernoulli distribution. Under the same assumption on the random environment as that of Imbrie and Spencer, we establish that in spatial dimensions four or above the behavior is still diffusive even when the directed polymer interacts strongly with the environment. More generally, we can prove that, if the random environment is bounded and if the supremum of the support of the distribution has a positive mass, then there is an integerd ₀ such that in dimensions higher thand ₀ the behavior of the random polymer is always diffusive.     

Uniform CLT for Markov chains and its invariance principle: A martingale approach

The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.⁽⁷⁾ Then we use the chaining argument with stratification adapted from that of Ossiander.⁽¹¹⁾ The results of this paper generalize those of Levental⁽¹⁰⁾ and Ossiander.⁽¹¹⁾     

On martingale limit theory and strong convergence results for stochastic approximation procedures

The analysis of asymptotic behaviour of stochastic approximation procedures rests heavily on the use of martingale limit theory, although explicit recognition of this situation is notable for its absence in the literature. This point is emphasized and in illustration a martingale iterated logarithm result is used to obtain strong convergence results of iterated logarithm type for the basic Robbins–Monro and Kiefer–Wolfowitz procedures.     

Self-dual continuous processes

The important application of semi-static hedging in financial markets naturally leads to the notion of conditionally quasi self-dual processes which is, for continuous semimartingales, related to conditional symmetry properties of both their ordinary as well as their stochastic logarithms. We provide a structure result for continuous conditionally quasi self-dual processes. Our main result is to give a characterization of continuous Ocone martingales via a strong version of self-duality.     

Generalized Backward Doubly Stochastic Differential Equations Driven by Lévy Processes with Continuous Coefficients

A new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Lévy process are investigated. We establish a comparison theorem which allows us to derive an existence result of solutions under continuous and linear growth conditions.