Solution of partial differential equations by a modified random walk

A new Monte Carlo technique is applied to solve difference equations of elliptic and parabolic partial differential equations with given boundary values. Fixed random walk is extended to modified random walk, whereby a random walk is made on a maximum square. The average number of steps and the computational time in a modified random walk is much less than in a fixed random walk. Numerical examples support the utility of this method.     

Convergence rates for reversible Markov chains without the assumption of nonnegative definite matrices

Explicit convergence rates in geometric and strong ergodicity for denumerable discrete time Markov chains with general reversible transition matrices are obtained in terms of the geometric moments or uniform moments of the hitting times to a fixed point. Another way by Lyapunov’s drift conditions is also used to derive these convergence rates. As a typical example, the discrete time birth-death process (random walk) is studied and the explicit criteria for geometric ergodicity are presented.     

Restricted random walks on a graph

The problem of a restricted random walk on graphs, which keeps track of the number of immediate reversal steps, is considered by using a transfer matrix formulation. A closed-form expression is obtained for the generating function of the number ofn-step walks withr reversal steps for walks on any graph. In the case of graphs of a uniform valence, we show that our result has a probabilistic meaning, and deduce explicit expressions for the generating function in terms of the eigenvalues of the adjacency matrix. Applications to periodic lattices and the complete graph are given.Supported in part by National Science Foundation Grant DMR-9614170.     

The Weak Convergence Theorem for the Distribution of the Maximum of a Gaussian Random Walk and Approximation Formulas for its Moments

In this study, asymptotic expansions of the moments of the maximum (M(β)) of Gaussian random walk with negative drift (???β), β?>?0, are established by using Bell Polynomials. In addition, the weak convergence theorem for the distribution of the random variable Y(β)?≡?2?β?M(β) is proved, and the explicit form of the limit distribution is derived. Moreover, the approximation formulas for the first four moments of the maximum of a Gaussian random walk are obtained for the parameter β?∈?(0.5, 3.2] using meta-modeling.     

Stellar Hydrodynamic Equations for a Thin Disk Galaxy

Assuming that the motions of stars depart only slightly from circular, finite closed systems of stellar hydrodynamic equations are obtained for a disk galaxy by taking moments of the collisionless Boltzmann equation. The usual closure problem is avoided in that explicit expressions are obtained for the highest moments arising.     

组合分布在求解直线上n步随机游动首次通过问题上的应用

基于组合过程模型给出其轨道对目标集的首次通过概率及首中点的分布函数 ,并由此给出直线上n步随机游动的首次通过概率及首中点分布函数的一类显式 .     

The Distribution of the Sojourn Time for the Brownian Excursion

The solutions of various problems in the theories of queuing processes, branching processes, random graphs and others require the determination of the distribution of the sojourn time (occupation time) for the Brownian excursion. However, no standard method is available to solve this problem. In this paper we approximate the Brownian excursion by a suitably chosen random walk process and determine the moments of the sojourn time explicitly. By using a limiting approach, we obtain the corresponding moments for the Brownian excursion. The moments uniquely determine the distribution, enabling us to derive an explicit formula.     

Stochastic perturbation approach to the wavelet‐based analysis

The wavelet‐based decomposition of random variables and fields is proposed here in the context of application of the stochastic second order perturbation technique. A general methodology is employed for the first two probabilistic moments of a linear algebraic equations system solution, which are obtained instead of a single solution projection in the deterministic case. The perturbation approach application allows determination of the closed formulas for a wavelet decomposition of random fields. Next, these formulas are tested by symbolic projection of some elementary random field. Copyright © 2004 John Wiley & Sons, Ltd.     

Large Deviations Approximations to Distributions of the Total Distance of Compound Random Walks with von Mises Directions

This article considers the planar random walk where the direction taken by each consecutive step follows the von Mises distribution and where the number of steps of the random walk is determined by the class of inhomogeneous birth processes. Saddlepoint approximations to the distribution of the total distance covered by the random walk, i.e. of the length of the resultant vector of the individual steps, are proposed. Specific formulae are derived for the inhomogeneous Poisson process and for processes with linear contagion, which are the binomial and the negative binomial processes. A numerical example confirms the high accuracy of the proposed saddlepoint approximations.     

Random points associated with rectangles

The study of the distribution and moments of the distance between random points within a rectangle or in two coplanar rectangles is required in a wide variety of fields. Formulae for the distributions and arbitrary moments of the distance between two random points associated with one or two rectangles in various situations are given here explicitly. These explicit formulae will be helpful to those who work in various applied areas for the computations required in their problems. The third Author has partially been supported by C.N.R..     

Brownian bridge asymptotics for random mappings

Uniform random mappings of an n-element set to itself have been much studied in the combinatorial literature. We introduce a new technique, which starts by specifying a coding of mappings as walks with ± 1 steps. The uniform random mapping is thereby coded as a nonuniform random walk, and our main result is that as n→∞ the random walk rescales to reflecting Brownian bridge. This result encompasses a large number of limit theorems for “global” characteristics of uniform random mappings. © 1994 John Wiley & Sons, Inc.     

Factorial moments of point processes

We derive joint factorial moment identities for point processes with Papangelou intensities. Our proof simplifies previous combinatorial approaches to the computation of moments for point processes. We also obtain new explicit sufficient conditions for the distributional invariance of point processes with Papangelou intensities under random transformations.     

Generating a Random Signed Permutation with Random Reversals

Signed permutations form a group known as the hyperoctahedral group. We bound the rate of convergence to uniformity for a certain random walk on the hyperoctahedral group that is generated by random reversals. Specifically, we determine that O(n log n) steps are both necessary and sufficient for total variation distance and ℓ² distance to become small. This random walk arose as the result of an effort in molecular biology to model certain types of genome rearrangements.     

Saddlepoint approximations to the distribution of the total distance of the multivariate isotropic and von Mises–Fisher random walks

This article considers the random walk over R^p, with p ≥ 2, where the directions taken by the individual steps follow either the isotropic or the vonMises–Fisher distributions. Saddlepoint approximations to the density and to upper tail probabilities of the total distance covered by the random walk, i.e., of the length of the resultant, are derived. The saddlepoint approximations are onedimensional and simple to compute, even though the initial problem is p-dimensional. Numerical illustrations of the high accuracy of the proposed approximations are provided.     

Graph presentations for moments of noncentral Wishart distributions and their applications

We provide formulas for the moments of the real and complex noncentral Wishart distributions of general degrees. The obtained formulas for the real and complex cases are described in terms of the undirected and directed graphs, respectively. By considering degenerate cases, we give explicit formulas for the moments of bivariate chi-square distributions and 2 × 2 Wishart distributions by enumerating the graphs. Noting that the Laguerre polynomials can be considered to be moments of a noncentral chi-square distributions formally, we demonstrate a combinatorial interpretation of the coefficients of the Laguerre polynomials.     

The Ground State Energy of the Massless Spin-Boson Model

We provide an explicit combinatorial expansion for the ground state energy of the massless spin-Boson model as a power series in the coupling parameter. Our method uses the technique of cluster expansion in constructive quantum field theory and takes as a starting point the functional integral representation and its reduction to an Ising model on the real line with long range interactions. We prove the analyticity of our expansion and provide an explicit lower bound on the radius of convergence. We do not need multiscale nor renormalization group analysis. A connection to the loop-erased random walk is indicated.     

Moment bounds for mixing random variables useful in nonparametric function estimation

Bounds for even moments of sums of strong mixing random variables are given which extend existing bounds. The method of proof uses simple facts about strong mixing random variables and combinatorial methods. The bound is particularly useful for triangular arrays with entries decreasing in size. To illustrate this, applications are being discussed to nonparametric kernel estimation with dependent observations.     

Shortest expected delay routing for Erlang servers

The queueing problem with Poisson arrivals and two identical parallel Erlang servers is analyzed for the case of shortest expected delay routing. This problem may be represented as a random walk on the integer grid in the first quadrant of the plane. An important aspect of the random walk is that it is possible to make large jumps in the direction of the boundaries. This feature gives rise to complicated boundary behavior. Generating function approaches to analyze this type of random walk seem to be extremely complicated and have not been successful yet. The approach presented in this paper directly solves the equilibrium equations. It is shown that the equilibrium distribution of the random walk can be written as an infinite linear combination of products. This linear combination is constructed in a compensation procedure. The starting solutions for this procedure are found by solving the shortest expected delay problem with instantaneous jockeying. The results can be used for an efficient computation of performance criteria, such as the waiting time distribution and the moments of the waiting time and the queue lengths.     

A method for analysis of linear dynamic systems driven by stationary non-Gaussian noise with applications to turbulence-induced random vibration

A method is developed for approximating the properties of the state of a linear dynamic system driven by a broad class of non-Gaussian noise, namely, by polynomials of filtered Gaussian processes. The method involves four steps. First, the mean and correlation functions of the state of the system are calculated from those of the input noise. Second, higher order moments of the state are calculated based on Itô’s formula for continuous semimartingales. It is shown that equations governing these moments are closed, so that moment of any order of the state can be calculated exactly. Third, a conceptually simple technique, which resembles the Galerkin method for solving differential equations, is proposed for constructing approximations for the marginal distribution of the state from its moments. Fourth, translation models are calibrated to representations of the marginal distributions of the state as well as its second moment properties. The resulting models can then be utilized to estimate properties of the state, such as the mean rate at which the state exits a safe set. The implementation of the proposed method is demonstrated by numerous examples, including the turbulence-induced random vibration of a flexible plate.     

A second-order Monte Carlo method for the solution of the Ito stochastic differential equation

A difference approximation that is second-order accurate in the time step his derived for the general Ito stochastic differential equation. The difference equation has the form of a second-order random walk in which the random terms are non-linear combinations of Gaussian random variables. For a wide class of problems, the transition pdf is joint-normal to second order in h; the technique then reduces to a Gaussian random walk, but its application is not limited to problems having a Gaussian solution. A large number of independent sample paths are generated in a Monte Carlo solution algorithm; any statistical function of the solution (e.g., moments or pdf's) can be estimated by ensemble averaging over these paths