Showing papers on "Stochastic process published in 1995"

Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance

Abstract: Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable stochastic integrals and harmonizable processes Self-similar processes Chentsov random fields Introduction to sample path properties Boundedness, continuity and oscillations Measurability, integrability and absolute continuity Boundedness and continuity via metric entropy Integral representation Historical notes and extensions.

The Helmholtz machine

Abstract: Discovering the structure inherent in a set of patterns is a fundamental aim of statistical inference or learning. One fruitful approach is to build a parameterized stochastic generative model, independent draws from which are likely to produce the patterns. For all but the simplest generative models, each pattern can be generated in exponentially many ways. It is thus intractable to adjust the parameters to maximize the probability of the observed patterns. We describe a way of finessing this combinatorial explosion by maximizing an easily computed lower bound on the probability of the observations. Our method can be viewed as a form of hierarchical self-supervised learning that may relate to the function of bottom-up and top-down cortical processing pathways.

Global Optimization Methods in Geophysical Inversion

Abstract: Part 1 Preliminary statistics: random variables random nunmbers probability probability distribution, distribution function and density function joint and marginal probability distributions mathematical expectation, moments, variances and covariances conditional probability Monte Carlo integration importance sampling stochastic processes Markov chains homogeneous, inhomogeneous, irreducible and aperiodic Markov chains the limiting probability. Part 2 Direct, linear and iterative-linear inverse methods: direct inversion methods model based inversion methods linear/linearized inverse methods iterative linear methods for quasi-linear problems Bayesian formulation solution using probabilistic formulation. Part 3 Monte Carlo methods: enumerative or grid search techniques Monte Carlo inversion hybrid Monte Carlo-linear inversion directed Monte Carlo methods. Part 4 Simulated annealing methods: metropolis algorithm heat bath algorithm simulated annealing without rejected moves fast simulated annealing very fast simulated reannealing mean field annealing using SA in geophysical inversion. Part 5 Genetic algorithms: a classical GA schemata and the fundamental theorem of genetic algorithms problems combining elements of SA into a new GA a mathematical model of a GA multimodal fitness functions, genetic drift, GA with sharing, and repeat (parallel) GA uncertainty estimates evolutionary programming - a variant of GA. Part 6 Geophysical applications of SA and GA: 1-D seismic waveform inversion pre-stack migration velocity estimation inversion of resistivity sounding data for 1-D earth models inversion of resistivity profiling data for 2-D earth models inversion of magnetotelluric sounding data for 1-D earth models stochastic reservoir modelling seismic deconvolution by mean field annealing and Hopfield network. Part 7 Uncertainty estimation: methods of numerical integration simulated annealing - the Gibbs' sampler genetic algorithm - the parallel Gibbs' sampler numerical examples.

Option pricing by Esscher transforms.

Abstract: The Esscher transform is a time-honored tool in actuarial science. This paper shows that the Esscher transform is also an efficient technique for valuing derivative securities if the logarithms of the prices of the primitive securities are governed by certain stochastic processes with stationary and independent increments. This family of processes includes the Wiener process, the Poisson process, the gamma process, and the inverse Gaussian process. An Esscher transform of such a stock-price process induces an equivalent probability measure on the process. The Esscher parameter or parameter vector is determined so that the discounted price of each primitive security is a martingale under the new probability measure. The price of any derivative security is simply calculated as the expectation, with respect to the equivalent martingale measure, of the discounted payoffs. Straightforward consequences of the method of Esscher transforms include, among others, the celebrated Black-Scholes optionpricing formula, the binomial option-pricing formula, and formulas for pricing options on the maximum and minimum of multiple risky assets. Tables of numerical values for the prices of certain European call options (calculated according to four different models for stock-price movements) are also provided.

Simulation and Chaotic Behavior of α-Stable Stochastic Processes

Abstract: Preliminary remarks Brownian motion, poisson process, alpha-stable Levy motion computer simulation of alpha-stable random variables stochastic integration spectral representations of stationary processes computer approximations of continuous time processes examples of alpha-stable stochastic modelling convergence of approximate methods chaotic behaviour of stationary processes hierarchy of chaos for stable and ID stationary processes. Appendix - a guide to simulation.

On the accuracy of a first-order Markov model for data transmission on fading channels

Abstract: Transmission of data blocks on a fading mobile radio channel is considered. The binary process of the success and failure of the data blocks is investigated, and the accuracy of a Markov approximation is studied. It is shown through analysis and simulation that a first-order Markov process is a good approximation. Data-link performance of ARQ protocols is considered as an example of application.

Probability, Stochastic Processes, and Queueing Theory: The Mathematics of Computer Performance Modeling

Abstract: This textbook provides a comprehensive introduction to probability and stochastic processes, and shows how these subjects may be applied in computer performance modelling. The author's aim is to derive the theory in a way that combines its formal, intuitive, and applied aspects so that students may apply this indispensable tool in a variety of different settings. Readers are assumed to be familiar with elementary linear algebra and calculus, including the concept of limit, but otherwise this book provides a self-contained approach suitable for graduate or advanced undergraduate students. The first half of the book covers the basic concepts of probability including expectation, random variables, and fundamental theorems. In the second half of the book the reader is introduced to stochastic processes. Subjects covered include renewal processes, queueing theory, Markov processes, and reversibility as it applies to networks of queues. Examples and applications are drawn from problems in computer performance modelling.

Dynamic Processes and Equilibrium in Transportation Networks: Towards a Unifying Theory

Abstract: Traditionally, traffic assignment models, both for within-day static and dynamic demand, have been formulated following an equilibrium approach in which a state ensuring internal consistency between demand (flows) and costs is sought. However, equilibrium analysis is significant under some assumptions on its “representativeness” (coincidence or closeness with the actual attractor of the system) and analytical properties, such as existence, uniqueness, and stability. Moreover, transients due to modifications of demand and/or supply cannot be simulated through equilibrium models, nor can a statistical description of the state of the system, i.e. means, modes, moments and, more generally, frequency distributions of flows over time be obtained. In this paper, interperiodic (day-to-day) dynamic modeling of transportation networks is addressed following two different approaches, namely deterministic and stochastic processes. In both cases several theoretical results are shown by making use of a formal framework...

An intermediate course in probability

Abstract: The purpose of this book is to provide the reader with a solid background and understanding of the basic results and methods in probability theory before entering into more advanced courses. The first six chapters focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, transforms, order variables, the multivariate normal distribution, convergence. A final chapter is devoted to the Poisson process as a means both to introduce stochastic processes and to apply many of the techniques introduced earlier in the text. Students are assumed to have taken a first course in probability, though no knowledge of measure theory is assumed. Throughout, the presentation is thorough and includes many examples that are discussed in detail. Thus, students considering more advanced research in probability theory will benefit from this wide-ranging survey of the subject that provides them with a foretaste of the subjects many treasures. The present second edition offers updated content, one hundred additional problems for solution, and a new chapter that provides an outlook on further areas and topics, such as stable distributions and domains of attraction, extreme value theory and records, and martingales. The main idea is that this chapter may serve as an appetizer to the more advanced theory.

Quantized energy cascade and log-Poisson statistics in fully developed turbulence.

Abstract: It is proposed that the statistics of the inertial range of fully developed turbulence can be described by a quantized random multiplicative process. We then show that (i) the cascade process must be a log-infinitely divisible stochastic process (i.e., stationary independent log-increments); (ii) the inertial-range statistics of turbulent fluctuations, such as the coarse-grained energy dissipation, are log-Poisson; and (iii) a recently proposed scaling model [Z.-S. She and E. Leveque 72, 336 (1994)] of fully developed turbulence can be derived. A general theory using the Levy-Khinchine representation for infinitely divisible cascade processes is presented, which allows for a classification of scaling behaviors of various strongly nonlinear dissipative systems.

Stochastic Processes in Polymeric Fluids: Tools and Examples for Developing Simulation Algorithms

Abstract: 1 Stochastic Processes, Polymer Dynamics, and Fluid Mechanics.- 1.1 Approach to Kinetic Theory Models.- 1.2 Flow Calculation and Material Functions.- 1.2.1 Shear Flows.- 1.2.2 General Extensional Flows.- 1.2.3 The CONNFFESSIT Idea.- References.- I Stochastic Processes.- 2 Basic Concepts from Stochastics.- 2.1 Events and Probabilities.- 2.1.1 Events and ?-Algebras.- 2.1.2 Probability Axioms.- 2.1.3 Gaussian Probability Measures.- 2.2 Random Variables.- 2.2.1 Definitions and Examples.- 2.2.2 Expectations and Moments.- 2.2.3 Joint Distributions and Independence.- 2.2.4 Conditional Expectations and Probabilities.- 2.2.5 Gaussian Random Variables.- 2.2.6 Convergence of Random Variables.- 2.3 Basic Theory of Stochastic Processes.- 2.3.1 Definitions and Distributions.- 2.3.2 Gaussian Processes.- 2.3.3 Markov Processes.- 2.3.4 Martingales.- References.- 3 Stochastic Calculus.- 3.1 Motivation.- 3.1.1 Naive Approach to Stochastic Differential Equations.- 3.1.2 Criticism of the Naive Approach.- 3.2 Stochastic Integration.- 3.2.1 Definition of the Ito Integral.- 3.2.2 Properties of the Ito Integral.- 3.2.3 Ito's Formula.- 3.3 Stochastic Differential Equations.- 3.3.1 Definitions and Basic Theorems.- 3.3.2 Linear Stochastic Differential Equations.- 3.3.3 Fokker-Planck Equations.- 3.3.4 Mean Field Interactions.- 3.3.5 Boundary Conditions.- 3.3.6 Stratonovich's Stochastic Calculus.- 3.4 Numerical Integration Schemes.- 3.4.1 Euler's Method.- 3.4.2 Mil'shtein's Method.- 3.4.3 Weak Approximation Schemes.- 3.4.4 More Sophisticated Methods.- References.- II Polymer Dynamics.- 4 Bead-Spring Models for Dilute Solutions.- 4.1 Rouse Model.- 4.1.1 Analytical Solution for the Equations of Motion.- 4.1.2 Stress Tensor.- 4.1.3 Material Functions in Shear and Extensional Flows.- 4.1.4 A Primer in Brownian Dynamics Simulations.- 4.1.5 Variance Reduced Simulations.- 4.2 Hydrodynamic Interaction.- 4.2.1 Description of Hydrodynamic Interaction.- 4.2.2 Zimm Model.- 4.2.3 Long Chain Limit and Universal Behavior.- 4.2.4 Gaussian Approximation.- 4.2.5 Simulation of Dumbbells.- 4.3 Nonlinear Forces.- 4.3.1 Excluded Volume.- 4.3.2 Finite Polymer Extensibility.- References.- 5 Models with Constraints.- 5.1 General Bead-Rod-Spring Models.- 5.1.1 Philosophy of Constraints.- 5.1.2 Formulation of Stochastic Differential Equations.- 5.1.3 Generalized Coordinates Versus Constraint Conditions.- 5.1.4 Numerical Integration Schemes.- 5.1.5 Stress Tensor.- 5.2 Rigid Rod Models.- 5.2.1 Dilute Solutions of Rod-like Molecules.- 5.2.2 Liquid Crystal Polymers.- References.- 6 Reptation Models for Concentrated Solutions and Melts.- 6.1 Doi-Edwards and Curtiss-Bird Models.- 6.1.1 Polymer Dynamics.- 6.1.2 Stress Tensor.- 6.1.3 Simulations in Steady Shear Flow.- 6.1.4 Efficiency of Simulations.- 6.2 Reptating-Rope Model.- 6.2.1 Basic Model Equations.- 6.2.2 Results for Steady Shear Flow.- 6.3 Modified Reptation Models.- 6.3.1 A Model Related to "Double Reptation".- 6.3.2 Doi-Edwards Model Without Independent Alignment.- References.- Landmark Papers and Books.- Solutions to Exercises.- References.- Author Index.

Applied non-Gaussian processes : examples, theory, simulation, linear random vibration, and MATLAB solutions

Abstract: Non-Gaussian data and probalilistic methods classes of non-Gaussian processes simulation of non-Gaussian processes response of linear systems to non-Gaussian inputs.

Performance of optimum and suboptimum receivers in the presence of impulsive noise modeled as an alpha-stable process

Abstract: Impulsive noise bursts in communication systems are traditionally handled by incorporating in the receiver a limiter which clips the received signal before integration. An empirical justification for this procedure is that it generally causes the signal-to-noise ratio to increase. Recently, very accurate models of impulsive noise were presented, based on the theory of symmetric /spl alpha/-stable probability density functions. We examine the performance of optimum receivers, designed to detect signals embedded in impulsive noise which is modeled as an infinite variance symmetric /spl alpha/-stable process, and compare it against the performance of several suboptimum receivers. As a measure of receiver performance, we compute an asymptotic expression for the probability of error for each receiver and compare it to the probability of error calculated by extensive Monte-Carlo simulation. >

Stochastic Ordering and Dependence in Applied Probability

Abstract: 1 Univariate Ordering.- 1.1 Construction of iid random variables.- 1.2 Strong ordering.- 1.3 Convex ordering.- 1.4 Conditional orderings.- 1.5 Relative inverse function orderings.- 1.6 Dispersive ordering.- 1.7 Compounding.- 1.8 Integral orderings for queues.- 1.9 Relative inverse orderings for queues.- 1.10 Loss systems.- 2 Multivariate Ordering.- 2.1 Strassen's theorem.- 2.2 Coupling constructions.- 2.3 Conditioning.- 2.4 Markov processes.- 2.5 Point processes on R, martingales.- 2.6 Markovian queues and Jackson networks.- 2.7 Poissonian flows and product formula.- 2.8 Stochastic ordering of Markov processes.- 2.9 Stochastic ordering of point processes.- 2.10 Renewal processes.- 2.11 Comparison of replacement policies.- 2.12 Stochastically monotone networks.- 2.13 Queues with MR arrivals.- 3 Dependence.- 3.1 Association.- 3.2 MTP2.- 3.3 A general theory of positive dependence.- 3.4 Multivariate orderings and dependence.- 3.5 Negative association.- 3.6 Independence via uncorrelatedness.- 3.7 Association for Markov processes.- 3.8 Dependencies in Markovian networks.- 3.9 Dependencies in Markov renewal queues.- 3.10 Associated point processes.- A.- A.1 Probability spaces.- A.2 Distribution functions.- A.3 Examples of distribution functions.- A.4 Other characteristics of probability measures.- A.5 Random variables equal in distribution.- A.6 Bibliography.

Stochastic analysis of fractional brownian motions

Abstract: In this article some of the important ideas in ordinary stochastic analysis are applied to fractional Brownian motions (fBm's). First we give a simple and elementary proof of the fact that any fBm has zero quadratic variation. This fact leads to the non-semimartingale structure of fBm's. Another consequence is that we can integrate (in probability) the functionals of fBm's with fBm differentials. With the same integrator, we then develop the L2 integration theory of bounded sure processes based on of K. Bichteler's integral extension theory. Finally, we investigate the corresponding stochastic differential equations with fractional Brownian noise

Area-interaction point processes

Abstract: We introduce a new Markov point process that exhibits a range of clustered, random, and ordered patterns according to the value of a scalar parameter. In contrast to pairwise interaction processes, this model has interaction terms of all orders. The likelihood is closely related to the empty space functionF, paralleling the relation between the Strauss process and Ripley'sK-function. We show that, in complete analogy with pairwise interaction processes, the pseudolikelihood equations for this model are a special case of the Takacs-Fiksel method, and our model is the limit of a sequence of auto-logistic lattice processes.

The Stochastic Random-Cluster Process and the Uniqueness of Random-Cluster Measures

Abstract: The random-cluster model is a generalization of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn. Not only is the random-cluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated physical models. This paper serves two functions. First, we introduce and survey random-cluster measures from the probabilist's point of view, giving clear statements of some of the many open problems. Second, we present new results for such measures, as follows. We discuss the relationship between weak limits of random-cluster measures and measures satisfying a suitable DLR condition. Using an argument based on the convexity of pressure, we prove the uniqueness of random-cluster measures for all but (at most) countably many values of the parameter $p$. Related results concerning phase transition in two or more dimensions are included, together with various stimulating conjectures. The uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments. In the second part of this paper is constructed a Markov process whose level sets are reversible Markov processes with random-cluster measures as unique equilibrium measures. This construction enables a coupling of random-cluster measures for all values of $p$. Furthermore, it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first-order phase transition if and only if the cluster-weighting factor $q$ is sufficiently large.

A complete parameterization of all positive rational extensions of a covariance sequence

Abstract: In this paper we formalize the observation that filtering and interpolation induce complementary, or "dual," decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence. On the one hand, by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space, we are able to develop estimates on the asymptotic behavior of the Schur parameters (1918) of positive rational extensions. On the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing, and stochastic realization theory, this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by Georgiou (1983, 1987). Our basic result, however, also enables us to analyze the robustness of this parameterization with respect to variations in the problem data. The methodology employed is a combination of complex analysis, geometry, linear systems, and nonlinear dynamics. >

Introduction to stochastic processes

Abstract: Preface to Second Edition Preface to First Edition PRELIMINARIES Introduction Linear Differential Equations Linear Difference Equations Exercises FINITE MARKOV CHAINS Definitions and Examples Large-Time Behavior and Invariant Probability Classification of States Return Times Transient States Examples Exercises COUNTABLE MARKOV CHAINS Introduction Recurrence and Transience Positive Recurrence and Null Recurrence Branching Process Exercises CONTINUOUS-TIME MARKOV CHAINS Poisson Process Finite State Space Birth-and-Death Processes General Case Exercises OPTIMAL STOPPING Optimal Stopping of Markov Chains Optimal Stopping with Cost Optimal Stopping with Discounting Exercises MARTINGALES Conditional Expectation Definition and Examples Optional Sampling Theorem Uniform Integrability Martingale Convergence Theorem Maximal Inequalities Exercises RENEWAL PROCESSES Introduction Renewal Equation Discrete Renewal Processes M/G/1 and G/M/1 Queues Exercises REVERSIBLE MARKOV CHAINS Reversible Processes Convergence to Equilibrium Markov Chain Algorithms A Criterion for Recurrence Exercises BROWNIAN MOTION Introduction Markov Property Zero Set of Brownian Motion Brownian Motion in Several Dimensions Recurrence and Transience Fractal Nature of Brownian Motion Scaling Rules Brownian Motion with Drift Exercises STOCHASTIC INTEGRATION Integration with Respect to Random Walk Integration with Respect to Brownian Motion Ito's Formula Extensions if Ito's Formula Continuous Martingales Girsanov Transformation Feynman-Kac Formula Black-Scholes Formula Simulation Exercises Suggestions for Further Reading Index

Probability, Random Processes and Estimation Theory for Engineers.

Abstract: 1. (Note: Each chapter concludes with Problems and References)2. Introduction to Probability3. Random Variables4. Functions of Random Variables5. Averages6. Vector Random Variables7. Estimation and Decision Theory I8. Random Sequences9. Random Processes10. Mean-Square Calculus11. Stationary Processes and Sequences12. Estimation Theory II13. Appendix: Review of Relevant Mathematics14. Index15. Information for Using the Program Disk

A note on state-space decomposition methods for analyzing stochastic flow networks

Abstract: Consider a flow network with single source s and single sink t with demand d>0. Assume that the nodes do not restrict flow transmission and the arcs have finite random discrete capacities. This paper has two objectives: (1) it corrects errors in well-known algorithms by Doulliez and Jamoulle (1972) for (a) computing the probability that the demand is satisfied (or network reliability), (b) the probability that an arc belongs to a minimum cut which limits the flow below d, and (c) the probability that a cut limits the flow below d; and (2) it discusses the applicability of these procedures. The Doulliez and Jamoulle algorithms are frequently referenced or used by researchers in the areas of power and communication systems and appear to be very effective for the computation of the network reliability when the demand is close to the largest possible maximum flow value. Extensive testing is required before the Doulliez and Jamoulle algorithms are disposed in favor of alternative approaches. Such testing should compare the performance of existing methods in a variety of networks including grid networks and dense networks of various sizes. >

Diffusion as a chemical reaction: Stochastic trajectories between fixed concentrations

Abstract: Stochastic trajectories are described that underly classical diffusion between known concentrations. The description of those experimental boundary conditions requires a phase space using the full Langevin equation, with displacement and velocity as state variables, even if friction entirely dominates the dynamics of diffusion, because the incoming and outgoing trajectories have to be told apart. The conditional flux, probabilities, mean first‐passage times, and contents (of the reaction region) of the four types of trajectories—the trans trajectories LR and RL and the cis trajectories LL and RR—are expressed in terms of solutions of the Fokker–Planck equation in phase space and are explicitly calculated in the Smoluchowski limit of high friction. With these results, diffusion in a region between fixed concentrations can be described exactly as a chemical reaction for any potential function in the region, made of any combination of high or low barriers or wells.

The probability of a subspace swap in the SVD

Abstract: We extend the work of Tufts, Kot, and Vaccaro (TKV) published in 1980, to improve the analytical characterization of threshold breakdown in SVD methods. Our results sharpen the TKV results by lower bounding the probability of a subspace swap in the SVD. Our key theoretical result is the characteristic function for a random variable whose probability of exceeding zero bounds the probability of a threshold breakdown. >

Spectral representation of fractional Brownian motion in n dimensions and its properties

Abstract: Fractional Brownian motion (fBm) provides a useful model for processes with strong long-term dependence, such as 1/f/sup /spl beta// spectral behavior. However, fBm's are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation. To facilitate the study of this problem, another model is provided for the construction of fBm from a white-noise-like process by means of a stochastic or Ito integral in frequency of a stationary uncorrelated random process. Also a generalized power spectrum of the nonstationary fBm process is defined. This new approach to fBm can be used to compute all of the correlations, power spectra, and other properties of fBm. In this paper, a number of these fBm properties are developed from this model such as the T/sup H/ law of scaling, the power law of fractional order, the correlation of two arbitrary fBm's, and the evaluation of the fractal dimension under various transformations. This new treatment of fBm using a spectral representation is extended also, for the first time, to two or more topological dimensions in order to analyze the features of isotropic n-dimensional fBm. >

The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case

Abstract: We study the fluctuations of free energy, energy and entropy in the high temperature regime for the Sherrington-Kirkpatrick model of spin glasses. We introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter, thus formulating the thermodynamic formalism in terms of random processes. The well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments, a generalization of a result proved by Aizenman, Lebowitz and Ruelle [1].

Identifying MIMO Wiener systems using subspace model identification methods

Abstract: In this paper we show that the multivariable output-error state-space model (MOESP) class of sub-space model identification (SMI) schemes can be extended to identify Wiener systems, a series connection of a linear dynamic system followed by a static nonlinearity. In this paper, we restrict to present these extensions for the case the Taylor series expansion of the static nonlinearity contains odd terms. It is shown that the extension allows to identity the linear part of the Wiener systems as if the static nonlinearity is not present. In this way, it is related to cross-correlation analysis techniques.

Simulation of wave propagation in three-dimensional random media

Abstract: Quantitative error analyses for the simulation of wave propagation in three-dimensional random media, when narrow angular scattering is assumed, are presented for plane-wave and spherical-wave geometry. This includes the errors that result from finite grid size, finite simulation dimensions, and the separation of the two-dimensional screens along the propagation direction. Simple error scalings are determined for power-law spectra of the random refractive indices of the media. The effects of a finite inner scale are also considered. The spatial spectra of the intensity errors are calculated and compared with the spatial spectra of intensity. The numerical requirements for a simulation of given accuracy are determined for realizations of the field. The numerical requirements for accurate estimation of higher moments of the field are less stringent.