Moment inequality for sums of multi-indexed dependent random variables

We study a real random field defined on an integer lattice. Its dependence is described by certain covariance inequalities. We obtain an upper bound of absolute moments of appropriate order for particular sums (generated by a given field) taken over finite sets of arbitrary configuration.     

Analyzing the finite buffer batch arrival queue under Markovian service process: GIX/MSP/1/N

We consider a batch arrival finite buffer single server queue with inter-batch arrival times are generally distributed and arrivals occur in batches of random size. The service process is correlated and its structure is presented through Markovian service process (MSP). The model is analyzed for two possible customer rejection strategies: partial batch rejection and total batch rejection policy. We obtain steady-state distribution at pre-arrival and arbitrary epochs along with some important performance measures, like probabilities of blocking the first, an arbitrary, and the last customer of a batch, average number of customers in the system, and the mean waiting times in the system. Some numerical results have been presented graphically to show the effect of model parameters on the performance measures. The model has potential application in the area of computer networks, telecommunication systems, manufacturing system design, etc.      

Catalan monoids,monoids of local endomorphisms,and their presentations

The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of finite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Aîzen?tat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a finite symmetric chain.     

关于齐次树上随机场的一类强偏差定理

通过引进渐近对数似然比作为齐次树上任意Markov随机场逼近的一种度量,通过构造鞅的方法,建立了关于随机场的一类强偏差(也称小偏差)定理.所得结论推广了一个已知的结果.     

有限连通图上随机扩张森林和连通子图的边负相关性

设G为有限连通图．本文研究图G的子图空间G上的三类概率测度,它们分别刻画图的随机扩张树,随机扩张森林和随机连通子图．基于G上均匀扩张树的边负相关性,我们构造G上的一族边负相关的非平凡随机扩张森林和随机连通子图．此外,我们还给出一定条件下图上均匀扩张森林的边负相关性．     

Independent random cascades on Galton-Watson trees

Consider an independent random cascade acting on the positive Borel measures defined on the boundary of a Galton-Watson tree. Assuming an offspring distribution with finite moments of all orders, J. Peyrière computed the fine scale structure of an independent random cascade on Galton-Watson trees. In this paper we use developments in the cascade theory to relax and clarify the moment assumptions on the offspring distribution. Moreover a larger class of initial measures is covered and, as a result, it is shown that it is the Hölder exponent of the initial measure which is the critical parameter in the Peyrière theory.

     

局部有限无穷树上的Mandelbrot瀑布混沌算子

本文研究了由局部有限无穷树上的独立不同分布的随机变量所生成的Mandelbrot瀑布混沌算子.通过构造随机测度,得到了判别一般测度的正则性和奇异性的充分条件;计算出了混沌算子的像测度的Hausdorff维数;并举例证明了充分条件不是必要的.     

General linear-fractional branching processes with discrete time

We study a linear-fractional Bienaymé–Galton–Watson process with a general type space. The corresponding tree contour process is described by an alternating random walk with the downward jumps having a geometric distribution. This leads to the linear-fractional distribution formula for an arbitrary observation time, which allows us to establish transparent limit theorems for the subcritical, critical and supercritical cases. Our results extend recent findings for the linear-fractional branching processes with countably many types.     

On the GI/M/1/N queue with multiple working vacations—analytic analysis and computation

We consider finite buffer single server GI/M/1 queue with exhaustive service discipline and multiple working vacations. Service times during a service period, service times during a vacation period and vacation times are exponentially distributed random variables. System size distributions at pre-arrival and arbitrary epoch with some important performance measures such as, probability of blocking, mean waiting time in the system etc. have been obtained. The model has potential application in the area of communication network, computer systems etc. where a single channel is allotted for more than one source.     

An analogue of the big <Emphasis Type="Italic">q</Emphasis>-Jacobi polynomials in the algebra of symmetric functions

It is well known how to construct a system of symmetric orthogonal polynomials in an arbitrary finite number of variables from an arbitrary system of orthogonal polynomials on the real line. In the special case of the big q-Jacobi polynomials, the number of variables can be made infinite. As a result, in the algebra of symmetric functions, there arises an inhomogeneous basis whose elements are orthogonal with respect to some probability measure. This measure is defined on a certain space of infinite point configurations and hence determines a random point process.     

Quenched Free Energy and Large Deviations for Random Walks in Random Potentials

We study quenched distributions on random walks in a random potential on integer lattices of arbitrary dimension and with an arbitrary finite set of admissible steps. The potential can be unbounded and can depend on a few steps of the walk. Directed, undirected, and stretched polymers, as well as random walk in random environment, are covered. The restriction needed is on the moment of the potential, in relation to the degree of mixing of the ergodic environment. We derive two variational formulas for the limiting quenched free energy and prove a process‐level quenched large deviation principle (LDP) for the empirical measure. As a corollary we obtain LDPs for types of random walks in random environments not covered by earlier results. © 2012 Wiley Periodicals, Inc.     

On the recurrence of edge-reinforced random walk on ℤ×G

Let G be a finite tree. It is shown that edge-reinforced random walk on ℤ×G with large initial weights is recurrent. This includes recurrence on multi-level ladders of arbitrary width. For edge-reinforced random walk on {0,1, . . . ,n}×G, it is proved that asymptotically, with high probability, the normalized edge local times decay exponentially in the distance from the starting level. The estimates are uniform in n. They are used in the recurrence proof.     

Adaptive multi-element polynomial chaos with discrete measure: Algorithms and application to SPDEs

We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve partial differential equations with random parameters. The method is based on numerical construction of orthogonal polynomial bases in terms of a discrete probability measure. To this end, we compare the accuracy and efficiency of five different constructions. We develop an adaptive procedure for decomposition of the parametric space using the local variance criterion. We then couple the ME-PCM with sparse grids to study the Korteweg–de Vries (KdV) equation subject to random excitation, where the random parameters are associated with either a discrete or a continuous probability measure. Numerical experiments demonstrate that the proposed algorithms lead to high accuracy and efficiency for hybrid (discrete–continuous) random inputs.     

A random environment for linearly edge-reinforced random walks on infinite graphs

We consider linearly edge-reinforced random walk on an arbitrary locally finite connected graph. It is shown that the process has the same distribution as a mixture of reversible Markov chains, determined by time-independent strictly positive weights on the edges. Furthermore, we prove bounds for the random weights, uniform, among others, in the size of the graph.      

Thermodynamic limit for large random trees

We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We compute the limiting distribution explicitly and study its properties. We introduce an infinite random tree consistent with these limiting distributions and show that it satisfies a certain form of the Markov property. We also study the growth of this tree and prove several limit theorems including a diffusion approximation. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

多椭圆孔有限大复合材料层板的热弹性分析

本文利用各向异性体平面热传导，热弹性理论中的复势方法，以保角映射，Ｆａｂｅｒ级数展开以及最小二乘边界配置技术为工具，导出了内边界条件精确满足，外边界条件近似满足的多椭圆孔复合材料层板的热传导以及热弹性问题的级数解，详细探讨了层板大小，孔径，相对孔距，孔的设置方式，椭圆度以及层板的铺层比例诸参数的影响规律，得到了一些有益结论。     

Some results on first passage times in one dimensional random walks

Analytic expressions are presented for the characteristic function of the first passage time distribution for biased random walk on a finite chain (and diffusion with drift on a finite line); of the first passage time distribution for a random walk on a chain, in which the events (jumps) are governed by an arbitrary renewal process; and of the distribution of the time of escape from a bounded set of points in the latter case. A fundamental relation between the first passage time distribution and the conditional probability for random walk (or diffusion) in one dimension is analyzed and generalized.     

Robust probability updating

This paper discusses an alternative to conditioning that may be used when the probability distribution is not fully specified. It does not require any assumptions (such as CAR: coarsening at random) on the unknown distribution. The well-known Monty Hall problem is the simplest scenario where neither naive conditioning nor the CAR assumption suffice to determine an updated probability distribution. This paper thus addresses a generalization of that problem to arbitrary distributions on finite outcome spaces, arbitrary sets of ‘messages’, and (almost) arbitrary loss functions, and provides existence and characterization theorems for robust probability updating strategies. We find that for logarithmic loss, optimality is characterized by an elegant condition, which we call RCAR (reverse coarsening at random). Under certain conditions, the same condition also characterizes optimality for a much larger class of loss functions, and we obtain an objective and general answer to how one should update probabilities in the light of new information.     

Self-similarity and spectral asymptotics for the continuum random tree

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application, we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.     

On the norms of group-invariant transition operators on graphs

In this paper we consider reversible random walks on an infinite grapin, invariant under the action of a closed subgroup of automorphisms which acts with a finite number of orbits on the vertex-set. Thel ²-norm (spectral radius) of the simple random walk is equal to one if and only if the group is both amenable and unimodular, and this also holds for arbitrary random walks with bounded invariant measure. In general, the norm is bounded above by the Perron-Frobenius eigenvalue of a finite matrix, and this bound is attained if and only if the group is both amenable and unimodular.