High Dimensional Nonlinear Diffusion Stochastic Processes

Series on Advances in Mathematics for Applied Sciences - Vo!. 56

MBH-BiMENSMNAL NONUNEAR MFFUSMN STGCHASHC PROCESSES Modeling for Engineering Appiications

Yevgeny Mamontov Magnus WiHander

Worid Scientific

H!GH-0!MENSiONAL NONUNEAR OiFFMStON STOCHASTiC PROCESSES Modeling for Engineering Appiications

Series on Advances in Mathematics for Apptied Sciences - Vo). 56

H!GH-0!MENSiONAL NONUNEAR R!FFUS!0N STOCHASHC PROCESSES ModeMing for Engineering Appiications

Yevgeny Mamontov Magnus Wiiiander Department of Physics, M C 2 Gothenburg University and Chaimers University of Technoiogy Sweden

A W o r i d Scientific In

S/ngapore * MewJersey * London * Hong Kong

PtiMisneo' oy World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 M R S ofice.' Suite IB, 1060 Main Street, River Edge, NJ 07661 M f oj9!ce.' 57 Shelton Street, Covent Garden, London W C 2 H 9 H E

Library of Congress Cataloging-in-Publication Data Mamontov, Yevgeny, 1955High-dimensional nonlinear diffusion stochastic processes : modelling for engineering applications / Yevgeny Mamontov and Magnus Willander. p. cm. — (Series on advances in mathematics for applied sciences ; vol. 56) Includes bibliographical references and index. ISBN 9810243855 (alk. paper) 1. Engineering — Mathematical models. 2. Stochastic processes. 3. Diffusion processes. 4. Differential equations, Nonlinear. 1. Willander, M . II. Title. III. Series. T A 3 4 2 .M35 2001 620'.001'5118-dc21

00-053437

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To oMr /a??M^es

Preface

Almost every book concerning diffusion stochastic processes (DSPs) is devoted to the problems arising wstde stochastic theory in the course of its logical development. The corresponding results serve inherent needs of m a thematics. This w a y of research is related to what is usually called pure mathematics. Those of its achievements which w prmctpJe allow quantitative treatments can also be applied to the stochastic problems arising oM^stde pure mathematics. Such applications always assume practical implementation. They are addressed to scientists or engineers, both mathematicians and nonmathematicians, w h o are not necessarily specialists in stochastic theory. In so doing, the forms of the recipes granted by pure mathematics are not discussed since the main attention is paid to the corresponding numerical techniques. However, the advantage to allow quantitative analysis in principle does not always m e a n to allow it in reality. In reality one has to deal with: (1) people w h o work in dissimilar sciences or branches of engineering and perhaps do not consider pure mathematics as the main field of their activities; some of the above recipes m a y be recognized by these people as not very transparent; (2) tools such as computer systems or software environments that, alas, are not of unlimited capabilities; indeed, great efforts are in

Pre/bee

VH1

m a n y cases m a d e to cram the above recipes (not always aimed at any practical applications) into tight frames of the existing tools. These circumstances constitute the problems which are out of a customary stream of pure mathematics and, in view of this, form a n e w and very stimulating field of research. The present book is intended (see Section 1.13) to facilitate solution of these two problems in the field of nonlinear diffusion stochastic processes (DSPs) of a large number of variables. More precisely, the book considers D S P s in Euclidean space of high dimension (much greater than a few units) with the coefficients nonlinearly dependent on the space coordinates. M a n y complex stochastic systems in science and engineering lead to such DSPs. A s is well-known, D S P s are closely related to Ito's stochastic differential equations (ISDEs). This fact explains w h y D S P theory is usually interpreted from the viewpoint of these eqautions. However, such treatment is not only unnecessary but also presumes reader's expertise in ISDEs. To eliminate this complication means to contribute to solution of the above problem (1). Accordingly, the present book derives (see Chapters 2-4) analysis of high-dimensional D S P s only from the well-known expressions for drift vector and diffusion matrix of the processes and from Kolmogorov's forward equations. N o techniques directly related to I S D E theory are involved. The corresponding prerequisites for reading (Section 1.1) are merely awareness of some basic facts associated with probability theory. The outcome of the purely D S P treatment is two-fold: *

compared with the ISDE-based alternative, the developed treatment is simpler, compact and uniform; it can be more willingly accepted by nonspecialists in stochastic integral and differential calculuses, in particular, by nonmathematicians;

*

the derived representations have not been revealed in stochastic theory yet and are also of purely mathematical interest; they provide a deeper insight into behavior of the basic characteristics of D S P s that can contribute to a greater success of the subsequent analyses.

Regarding the above problem (2), i.e. to better adapt the DSP-theory results to limited power of majority of computing systems, the present book concentrates on the analytical part of a combined, analytical-numerical approach. The main advantage of this approach is that it presents a reasonable compromise between accuracy of the applied models and complexity of

Pre/ace

IX

their numerical analysis. The above analytical part is constructed (Chapters 2-4) with the help of the purely D S P treatment. It grants the benefit to derive such analytical approximations which allow for the nonlinearities of the D S P coefficients and, in conjunction with proper, parallel simulation techniques, can lead to realistic computing expenses even if the process is of a high dimension. These features enable one to solve m u c h wider family of high-dimensional stochastic problems by m e a n s of numerical method and m a k e this analysis more accessible for scientific and engineering community. The present book concerns m a n y aspects to better elucidate the involved notions and techniques. In particular, it includes a separate chapter (Chapter 3) devoted to nonstationary invariant processes. This topic is of importance in both qualitative and quantitative analyses. Nevertheless, it is regrettably missed out in most of the works on applications of DSPs. Another special feature of the present book is the chapter (Chapter 5) on modelling non-Markov phenomena with various ISDEs and their connection to high-dimensional DSPs. This connection is provided by m e a n s of the well-known stochastic adaptive interpolation (SAI) method. The chapter equips this method with bases of function Banach spaces and shows h o w it can involve the proposed analytical-numerical treatment to approximately solve Ito's stochastic partial (integro-)differential equations. The specific examples concerning the above chapter are considered in Chapter 6. It is devoted to Ito's stochastic partial differential equations of semiconductor theory which are based on the fluid-dynamics model. The first part of the chapter emphasizes the capabilities of c o m m o n , nonrandom semiconductor equations to describe the electron (or hole) fluid not only in macroscale but also in mesoscale and microscale domains. This topic is considered in connection with a certain limit case of a proper random walk. The second part of the chapter discusses Ito's stochastic generalization of these equations for macroscale and weakly mesoscale domains and s u m m a rizes some of the corresponding results on noise in semiconductors. This also includes related topics for future development. Chapter 7 focuses on the distinguishing features of engineering applications. A combined, analytical-numerical approach, its advantages and disadvantages as well as the connection to c o m m o n analytical D S P results are discussed in Chapter 8. For reader's convenience, the book includes the introductory chapter (Chapter 1) that lists some basic facts and methods of D S P theory and some

X

Pre/ace

of the related practical problems. This chapter leads to the detailed formulation (Section 1.13) of the purpose of the book. The book includes m a n y appendixes. They present the proofs of the lemmata and theorems, descriptions of the examples and the technical details which are needed in the course of the consideration, can contribute to better understand some stochastic features of the nonrandom models or can be helpful in practical implementation of the described methods. The theorems are numbered with the two-position numbers in the form (K.L) where L is the cardinal number of the theorem in Chapter K. The definitions, lemmata, corollaries and remarks are numbered similarly. The formulas in the book are numbered with the two- or three-position numbers in the forms (K.L) or (K.M.L) respectively. In so doing, position L is the cardinal number of the formula in Chapter K or, if the chapter is divided into sections, in Section K.M, i.e. the M t h section of Chapter K. The end of Index includes the list of all the definitions, lemmata, theorems, corollaries and remarks. The application field which the book aims the developed models and m e thods at can include different subjects but can be formulated very briefly. This field is complex systems, no matter which specific science or engineering they are associated with. The book considers various aspects related to applied research. For instance, it: * * *

* *

stresses the mathematical reading and practical meaning of the signal-to-noise ratio (Appendix B); explains h o w a common, nonrandom equation for fluid concentration includes not very simple stochastic phenomena (Appendix F); presents a /M^Jy ?:77te-c?07Mam, approximate analytical model to evaluate the particle-velocity covariance for a fairly general class of fluids (Section 4.10 and Appendix C); shows h o w to describe the long, non-exponential "tails" of this covariance by means of the I S D E with a fMwJwear friction (Section 4.10.3); derives the nonrandom model which suggests an explanation of stochastic resonance and other s^ocAas^cs-wc^Mced effects in the m e a n responses of the systems governed by nonJmear D S P s (Section 2.4).

These examples demonstrate not only the capabilities of the developed techniques but also emphasize the usefulness of the complex-system-related approaches to solve some problems which have not been solved with the traditional, statistical-physics methods yet. F r o m this veiwpoint, the book

Pre/ace

xi

can be regarded as a kind of complement to such books as "ZMfrodMcf:o7t fo fAe PAys:cs o/Complex Sys^ews. TAe Afesoseoptc ApproacA ^o F^MC^Mot^ows, NonJmear^y a%d <Sey-Orga7Mzafton" (Serra, Andretta, Compiani and Zanarini; 1986), "S^ocAas^:c Dy^awtcctZ -Sys^ews. Con.cep^s, NMwertcaJ Afe^Aoc^s, Da^a AMaJysts" (Honerkamp, 1994), "S^a^s^caJ PAysMS.' A n Atft^anced Approach w ^ A App^ca?t07ts" (Honerkamp, 1998) which deal with physics of complex systems, some of the corresponding analysis methods and an innovative, stochastics-based vision of theoretical physics. W h o should read this book? W e believe that any reader whose work or study concerns nonlinear high-dimensional stochastic systems will find in it something that could really help and m a k e the painstaking but interesting job more enjoyable and fruitful. To be specific in this issue, w e would point out the following groups of readers: * * * * *

nonmathematicians (e.g.,theoretical physicists, engineers in industry, specialists in models for finance or biology, computing scientists); mathematicians; undergraduate and postgraduate students of the corresponding specialties; managers in applied sciences and engineering dealing with the advancements in the related fields; any specialists w h o use diffusion stochastic processes to model highdimensional (or large-scale) nonlinear stochastic systems.

and other communities w h o are interested in the topic of the book. To what extent is this book suitable and useful for you personally? The best w a y to get the answer is to look through the "Contents" and Section 1.1 "Prerequisites for Reading". They present the information that helps you to m a k e your decision.

Contents

Preface

vii

C h a p t e r 1 Introductory Chapter 1.1 Prerequisites for Reading 1.2 R a n d o m Variable. Stochastic Process. R a n d o m Field. High-Dimensional Process. One-Point Process 1.3 Two-Point Process. Expectation. Markov Process. Example of Non-Markov Process Associated with Multidimensional Markov Process 1.4 Preceding, Subsequent and Transition Probability Densities. The Chapman-Kolmogorov Equation. Initial Condition for Markov Process 1.4.1 The Chapman-Kolmogorov equation 1.4.2 Initial condition for Markov process 1.5 Homogeneous Markov Process. Example of Markov Process: The Wiener Process 1.6 Expectation, Variance and Standard Deviations of Markov Process 1.7 Invariant and Stationary Markov Processes. Covariance. Spectral Densities 1.8 Diffusion Process

xiii

1 1 3

10

16 19 20 23 26 30 37

xiv

1.9 1.10 1.11 1.12

1.13

Con^enfs

Example of Diffusion Processes: Solutions of Ito's Stochastic Ordinary Differential Equation The Kolmogorov Backward Equation Figures of Merit. Diffusion Modelling of High-Dimensional Systems C o m m o n Analytical Techniques to Determine Probability Densities of Diffusion Processes. The Kolmogorov Forward Equation 1.12.1 Probability density 1.12.2 Invariant probability density 1.12.3 Stationary probability density The Purpose and Content of This Book

Chapter 2 Diffusion Processes 2.1 Introduction 2.2 Time-Derivatives of Expectation and Variance 2.3 Ordinary Differential Equation Systems for Expectation 2.3.1 The first-order system 2.3.2 The second-order system 2.3.3 Systems of the higher orders 2.4 Models for Noise-Induced Phenomena in Expectation 2.4.1 The case of stochastic resonance 2.4.2 Practically efficient implementation of the second-order system 2.5 Ordinary Differential Equation System for Variance 2.5.1 Damping matrix 2.5.2 The uncorrelated-matrixes approximation 2.5.3 Nonlinearity of the drift function 2.5.4 Fundamental limitation of the state-spaceindependent approximations for the diffusion 2.6

and damping matrixes The Steady-State Approximation for The Probability Density

Chapter 3 Invariant Diffusion Processes 3.1 Introduction 3.2 Preliminary Remarks 3.3 Expectation. The Finite-Equation Method

40 46 48

51 51 54 57 60

. .

63 63 64 66 66 68 70 71 71 73 76 76 77 80

81 82 85 85 85 86

Con^enfs

3.4 3.5

3.6

3.7

Explicit Expression for Variance The Simplified Detailed-Balance Approximation for Invariant Probability Density 3.5.1 Partial differential equation for logarithm of the density 3.5.2 Truncated equation for the logarithm and the detailed-balance equation 3.5.3 Case of the detailed balance 3.5.4 The detailed-balance approximation 3.5.5 The simplified detailed-balance approximation. Theorem on the approximating density Analytical-Numerical Approach to Non-Invariant and Invariant Diffusion Processes 3.6.1 Choice of the bounded domain of the integration 3.6.2 Evaluation of the multifold integrals. The Monte Carlo technique 3.6.3 Summary of the approach Discussion

Chapter 4 Stationary Diffusion Processes 4.1 Introduction 4.2 Previous Results Related to Covariance and Spectral-Density Matrixes 4.3 Time-Separation Derivative of Covariance in the Limit Case of Zero Time Separation 4.4 Flicker Effect 4.5 Time-Separation Derivative of Covariance in the General Case 4.6 Case of the Uncorrelated Matrixes 4.7 Representations for Spectral Density in the Uncorrelated-Matrixes Case 4.8 Example: Comparison of the Dampings for a Particle Near the Minimum of Its Potential Energy 4.9 The Deterministic-Transition Approximation 4.10 Example: Non-exponential Covariance of Velocity of a Particle in a Fluid 4.10.1 Covariance in the general case

xv

88 90 90 91 93 95 96

..

99 100 102 104 105 107 107 108 109 Ill 112 114 117 118 123 126 126

xvi

Confers

4.10.2

4.11

4.12

Covariance and the quatities related to it in a simple fluid 4.10.3 Case of the hard-sphere fluid 4.10.4 Summary of the procedure in the general case . . Analytical-Numerical Approach to Stationary Process .... 4.11.1 Practical issues 4.11.2 Summary of the approach Discussion

128 130 133 134 134 136 137

Chapter 5

5.1 5.2 5.3

5.4 5.5

Ito's Stochastic Partial Differential 141 Equations as Non-Markov Models Leading to High-Dimensional Diffusion Processes Introduction 141 Various Types of Ito's Stochastic Differential Equations . . . 142 Method to Reduce ISPIDE to System of ISODEs 144 5.3.1 Projection approach 148 5.3.2 Stochastic collocation method 151 5.3.3 Stochastic-adaptive-interpolation method 153 Related Computational Issues 159 Discussion 160

Chapter 6

163

6.1 6.2

163

6.3

Ito's Stochastic Partial Differential Equations for Electron Fluids in Semiconductors Introduction Microscopic Phenomena in Macroscopic Models of Multiparticle Systems 6.2.1 Microscopic random walks in deterministic macroscopic models of multiparticle systems 6.2.2 Macroscale, mesoscale and microscale domains in terms of the wave-diffusion equation 6.2.3 Stochastic generalization of the deterministic macroscopic models of multiparticle systems The ISPDE System for Electron Fluid in M-Type Semiconductor 6.3.1 Deterministic model for electron fluid in semiconductor

165 165 171 176 177 177

Con?en?s

6.3.2

6.4

Mesoscopic wave-diffusion equations in the deterministic fluid-dynamic model 6.3.3 Stochastic generalization of the deterministic fluid-dynamic model. The semiconductor-fluid ISPDE system Semiconductor Noises and the SF-ISPDE System: Discussion and Future Development 6.4.1 The SF-ISPDE system in connection with semiconductor noises 6.4.2 Some directions for future development

xvii

Chapter 7 7.1 7.2 7.3 7.4

Distinguishing Features of Engineering Applications

High-Dimensional Diffusion Processes Efficient Multiple Analysis Reasonable Amount of the Main Computer Memory Real-Time Signal Transformation by Diffusion Process

Chapter 8

8.1 8.2

Analytical-Numerical Approach to Engineering Problems and C o m m o n Analytical Techniques Analytical-Numerical Approach to Engineering Problems Severe Practical Limitations of C o m m o n Analytical Techniques in the High-Dimensional Case. Possible Alternatives

180

187 192 192 195 197 197 198 198 199 201

201

203

Appendix A

Example of Markov Processes: Solutions of the Cauchy Problems for Ordinary Differential Equation System

205

Appendix B

Signal-to-Noise Ratio

209

Appendix C

Example of Application of Corollary 1.2: Nonlinear Friction and Unbounded Stationary Probability Density of the Particle Velocity in Uniform Fluid Description of the Model

213

C.l

213

xviii

C.2 C.3 C.4

Co?:?enfs

Energy-Independent Momentum-Relaxation Time. Equilibrium Probability Density Energy-Dependent Momentum-Relaxation Time: General Case Energy-Dependent Momentum-Relaxation Time: Case of Simple Fluid

220 222 227

Appendix D

Proofs of the Theorems in Chapter 2 and Other Details Proof of Theorem 2.1 Proof of Theorem 2.2 Green's Formula for the Differential Operator of Kolmogorov's Backward Equation Proof of Theorem 2.3 Quasi-Neutral Equilibrium Point Proof of Theorem 2.4

231

D.l D.2 D.3

231 232 235 237 239 241

Appendix E Proofs of the Theorems in Chapter 4 E.l Proof of Lemma 4.1 E.2 Proof of Theorem 4.2 E.3 Proof of Theorem 4.3 E.4 Proof of Theorem 4.4

243 243 243 244 245

Appendix F

Hidden Randomness in Nonrandom Equation for the Particle Concentration of Uniform Fluid and Chemical-Reaction /Generation-Recombination Noise

247

Appendix G

Example: Eigenvalues and Eigenfunctions of the Linear Differential Operator Associated with a Bounded Domain in Three-Dimensional Space

255

Appendix H

Resources for Engineering Parallel Computing under Windows 95

261

D.4 D.5 D.6

Bibliography

265

Index

281

Chapter 1

Introductory Chapter

1.1

Prerequisites for R e a d i n g

This book is addressed to all scientists and engineers w h o would like to be well-equipped for the case if they need to deal with diffusion stochastic processes in practice. This community comprises not only mathematicians but also a great number of nonmathematicians acting in the natural and social sciences and technologies. Therefore, the prerequisites for reading this book include only such elements of mathematics and probability theory which are commonly used in applications, more specifically: ^-dimensional Euclidean space R^; vectors and matrixes; ordinary and partial differential equations; the Lebesgue integrals; elementary event, space of elementary events, random event; probability of random event; stochastically independent random events; finite-dimensional random variable; stochastically independent random variables; probability density of random variable; conditional and marginal probability densities; expectation, variance, covariance of random variables. This list does not include purely theoretical topics which are not often (if at all) involved in solving applied problems. For instance, the book does 1

2

7n&*of%MC?ory C7:apfer

not involve the measure-theoretical treatment of probability theory and stochastic processes. This treatment is not very familiar to nonmathematicians. In spite of that, it is applied in almost every book to stochastic processes and stochastic differential equations. This fact is perhaps the reason that the mentioned field is still separated not only from m a n y nonmathematical applications but also from some important formalisms, e.g. classical theoretical physics and statistical mechanics (e.g., Balescu, 1975; Klimontovich, 1982; Resibois and D e Leener, 1977). Indeed, engineers or statisticians willing to acquire more knowledge in stochastic processes do have to squeeze through the measure-theoretical presentation not very transparent to them. Understandably, this road in m a n y cases remains unpassed. Another curious phenomenon is that statistical mechanics had in its own, nonmathematical w a y to obtain some results which were rigorously derived in mathematics long before and are well-known in stochastic theory. Resibois and D e Leener (1977) includes m a n y examples of this kind. The present book not involving the measure-theoretical treatment describes stochastic processes in terms of multidimensional random variables which are familiar to people working in applications. To the authors' knowledge, such approach has never been attempted before. Nevertheless, the authors hope that both nonmathematicians and mathematicians will not feel too m u c h inconvenience with it. The readers interested in the details of the rigorous, measure-theoretical presentation of probability theory and stochastic processes are referred to m a n y books on the subject, for instance: Arnold (1974), Bellomo e? a/. (1992), Feller (1968, 1971), Friedman (1975, 1976), Gard (1988), G i k h m a n and Skorokhod (1969), G i h m a n and Skorohod (1972), Gnedenko (1982), Has'minskh (1980), Kloeden and Platen (1995), Kolmogorov (1956), Ochi (1990), Prohorov and Rozanov (1969), Skorokhod (1965), W a n g and Hajek (1985). Another feature of the present book following from the above prerequisites is that it describes random variables by means of the corresponding probability densities (or, more precisely, the densities of the probability distributions of random variables). These densities exist not for every random variable. Luckily, the two following facts are in favor of the density-based approach. Ftrs?, the basic models in both classical statistical mechanics (e.g., Balescu, 1975; Klimontovich, 1982; Resibois and D e Leener, 1977) and mathematical kinetic theory (e.g., Bellomo, Palczewski and Toscani, 1988; Bellomo e? aL, 1991; Bellomo, 1995; Bellomo and Lo Schiavo, 2000) are inherently

R a n d o m VartaoZe, Process and F:eM. HtgA-D:mens:o?:a^ Process

3

associated with the probability densities. The well-known multidecade experience in application of these models (discussed, for example, by Bellomo, 1995; Bellomo and Lo Schiavo, 1997) demonstrates that the density-based treatments are sound and productive, both theoretically and practically. -Second, probability distributions present probabilities of random events as functions of the so-called Borel sets in Euclidean space. This feature is, however, impractical to describe random variables in the computing-oriented algorithms. Other related quantities should be used for this purpose. The most practically reasonable and sharp w a y is to apply the corresponding probability densities. They present not only the basic tool for the theoretical modeling (see above) but also the focus of practical statistical data processing (e.g., Cramer, 1946) in various fields of science and engineering. Potential readers of the present book include the following groups: *

*

*

*

mathematicians w h o wish to be aware of the n e w analytical results for high-dimensional diffusion stochastic processes with nonlinear coefficients; nonmathematicians (e.g., theoretical physicists, engineers in industry, specialists in models for finance or biology) w h o apply diffusion stochastic process theory to model complex, high-dimensional stochastic systems; managers in applied sciences and engineering w h o are responsible for the advanced level of the system-modelling/product-design activities of the technical personnel headed by them; undergraduate and postgraduate students of the corresponding specialties.

Such books as Papoulis (1991) or Yates and G o o d m a n (1999) can help the readers of the second, third and fourth groups to recollect the probability-related prerequisites listed at the very beginning of this section. It is hoped that the present book will be a useful and not very difficult reading for a wide audience.

1.2

R a n d o m Variable. Stochastic Process. R a n d o m Field. High-Dimensional Process. One-Point Process

Chapters 1-4 include a more systematic and detailed presentation of the results by Mamontov, Willander and Lewin (1999), M a m o n t o v and Willan-

4

7?:&*odMCfory CAapfer

der (1999) (and correct a few misprints occurred therein). S o m e applications of these results are described and discussed in Chapters 5—8. To begin with, w e consider notion of random variable. Let 6f be an integer positive number and R = (-oo,oo). R a n d o m variable in 6?-dimensional Euclidean space R*^ is real function % of elementary event ^ . This m e a n s that

* = X(Q

(1.2.1)

= (^,...,*JT(E]

(1.2.2)

where

^

^7

f Xi(Q = (X^),...,x,,(QfeR',

X(Q =

foralHeE,

(1.2.3)

x.(Qj ^ and x^, ^ = 1,...,^, are entries of the corresponding vectors, E is the space of elementary events. Every elementary event ^ can be interpreted as a specific outcome in a series of the trials in the experiment associated with random variable x Let g , be the sigma-algebra of Borel sets in R'' generated by the ^-dimensional rectangular parallelepipeds in R^ (e.g., Section 1.1 of Arnold, 1974). Non-Borel sets in R'' exist. However, it is not easy to exhibit specific examples of them. The necessary and sufficient conditions for a set to be a Borel one are k n o w n for the so-called analytic sets. They are formulated in two different forms, namely, as Suslin's criterion and Luzin's criterion. However, such criteria are a topic of set theory and are beyond the present book. In practical applications, any set in g^ can be viewed as a countable (i.e. either finite or infinitely countable) union of mutually non-intersecting ^-dimensional rectangular parallelepipeds in R^. (This interpretation follows the idea of Luzin's criterion (e.g., Luzin, 1972).) Overwhelming majority of the sets in R^ (intervals of the n u m b e r axis, plane figures, volume bodies, and m a n y others) employed in the real-world problems belongs to family g^. Note that g . by definition contains both empty set 0 and space R^, i.e.

Random Var:aMe, Process otnoJ f:eM. #:g/:-D:meHs:ofM/ Process

5

8 ^ 0 and 8