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Oleksandr Mokliachuk

Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

**Part of the book: **Stochastic Processes: Fundamentals and Emerging Applications

The results of the study of the problem of modeling of stochastic processes in various spaces with given reliability and accuracy are proposed. In the first part of the chapter, results on modeling of stochastic processes in D_{V},_{W} spaces of random variables are presented. These quasi-Banach spaces with a probability norm are convenient for estimating the reliability and accuracy of modeling of stochastic processes that do not have moments; for example, processes with L`evy distribution or Cauchy distribution. Note that the classical methods use the values of the moments of stochastic processes for estimating the reliability and accuracy of the constructed model. When using quasi-Banach spaces of random variables, these obstacles may be overcome. In the second part of the chapter properties of stochastic processes in Subϕ(Ω) spaces are described. The Karhunen-Lo’eve representation of stochastic processes with the help of systems of orthogonal polynomials are exploited. In cases where polynomials for this representation cannot be found explicitly, approximations are used for the description of the model. However, each approximation introduces its own errors. The impact of approximations errors on the reliability and accuracy of models of stochastic processes constructed using this method is analyzed.

**Keywords:** stochastic process, models of stochastic processes, sub-Gaussian spaces of

random variables, Kσ-spaces of random variables, D_{V},_{W} spaces of random variables,

Karhunen-Lo`eve representation, accuracy and reliability of simulation

[1] Kolmogorov A. N., On certain asymptotic characteristics of completely bounded metric spaces, Dokl. Akad. Nauk SSSR, vol. 108, no 3, pp. 385–388, 1956.

[2] Kolmogorov A. N., Selected works by Kolmogorov A. N., Vol. II: Probability theory

and mathematical statistics. Ed. by A. N. Shiryayev. Mathematics and its Applications. Soviet Series. 26. Dordrecht etc. Kluwer Academic Publishers, 1992.

[3] Yadrenko M. I., Local properties of sample funtions of stochastic fields, Visn., Mat.

Mekh., Kyiv. Univ. Im. Tarasa Shevchenka, vol. 9, 1967.

[4] Yadrenko M. I., Spectral Theory of Random Fields, New York: Optimization Software, Inc., Publications Division, Springer-Verlag, New York etc., 259 p., 1983.

[5] Kozachenko Yu. V. and Yadrenko M. I., Local properties of sample functions of random fields, Teor. Veroyatn. Mat. Stat., vol. 14, pp. 53–66; vol. 15, pp. 82–98, 1976.

[6] Skorokhod A. V., A theorem of continuity of a random function on the compact in

Hilbert space, Prob. Theory and Applic., vol. 18, no. 4, pp. 809–811, 1973.

[7] Yadrenko M. I., On continuity of the sample functions of the Gaussian random fields

on the Hilbert space, Dopov. Akad. Nauk Ukr. RSR, Ser. A, pp. 734–737, 1968.

[8] Belyaev Yu. K., Local properties of sample functions of stationary stochastic processes, Teor. Veroyatn. Primen., vol. 5, no. 1, pp. 128–131, 1960.

[9] Belyaev Yu. K., Continuity and Holder’s conditions for sample functions of stationary

Gaussian processes, Proc. Fourth Berkeley Symp. on Math. and Probability, vol.2, pp.

23–33, 1961.

[10] Dudley R. M., Gaussian processes on several parameters, Ann. Math. Statist., vol. 36,

no. 3, pp. 771–788, 1965.

[11] Delport J., Fonctions aleatoires presque surement continues sur un intervalle ferme

[Random functions almost surely continuous on a closed interval], Theres presentees

a la faculte des sciences d l’universite de Lill. Paris, 215 p., 1967.

[12] Skorokhod A. V., A note on Gaussian measures in a Banach space, Theory Probab.

Appl., vol. 15, no.3, pp. 519–520, 1970.

[13] Landau H. J., Shepp L. A., On the supremum of Gaussian processes, Sankhya, Ser. A,

vol. 32, no. 4, pp. 369 – 378, 1970.

[14] Fernique X., Int´egrabilit´e des vecteurs gaussiens [Integrability of Gaussian vectors],

C. R. Acad. Sci. A. vol. 270, no. 7, pp. 1698–1699, 1970.

[15] Ledoux M. and Talagrand M., Probability in Banach spaces. Isoperimetry and processes, Springer Verlag, Berlin etc., 480 p., 1991.

[16] Liphshits V. A., Gaussian Random Functions, Kyiv: TPMS, 246 p., 1995.

[17] Yurinsky V., Sums and Gaussian Vectors, Lecture Notes in Mathematics, vol. 1617,

Springer Verlag, Berlin, Heidelberg, 305 p., 1995.

[18] Belyaev Yu. K., On the number of exits across the boundary of a region by a vector

stochastic process, Teor. Veroyatn. Primen., vol. 13, no. 2, pp. 333–337, 1968.

[19] Berman S. M., Excursions of stationary Gaussian processes above high moving barriers, Ann. Probab., vol. 1, pp. 365–387, 1973.

[20] Pickands J., Upcrossing probabilities for stationary Gaussian processes, Trans. Am.

Math. Soc., vol. 145, pp. 51–73, 1969.

[21] Piterbarg V. I., Large deviations of random processes close to Gaussian ones, Theory

Probab. Appl., vol. 27, no 3, pp. 504–524, 1982.

[22] Kahane J. P., Propri´et´es locales des fonctions `a s´eries de Fourier al´eatoires [Local

properties of functions with random Fourier series], Stud. Math., vol. 19, no. 1, pp.

1–25, 1960.

[23] Buldygin V. V. and Kozachenko Yu. V., On sub-Gaussian random variables, Ukr. Mat.

Zh., vol 32, no. 6, pp. 723–230, 1980.

[24] Buldygin V. V., Convergence of random elements in topological spaces, Kyiv:

Naukova Dumka, 240 p., 1980.

[25] Buldygin V. V. and Kozachenko Yu. V., Sub-Gaussian random vectors and processes,

Teor. Veroyatn. Mat. Stat., vol. 36, pp. 10–22, 1987.

[26] Buldygin V. V. and Kozachenko Yu. V., Metric characterization of random variables

and random processes, Transl. Math. Monogr. Providence, RI: AMS, American Mathematical Society, 2000.

[27] Jain N. C. and Marcus M. B., Continuity of sub-Gaussian processes, Adv. Probab.,

vol. 4, pp. 81–196, 1978.

[28] Juliano Antonini R., Kozachenko Yu. V. and Tegza A. M., Inequalities for norms

of sub-Gaussian vectors and accuracy of modeling of stochastic processes, Theory

Probab. Mat. Stat., vol. 66, pp. 63–72, 2003.

[29] Marcus M. B. and Pisier G., Random Fourier series with applications to harmonic

analysis, Ann. Math. Stud.– Vol. 101. – Princeton University Press, Princeton, 150 p.,

1981.

[30] Kozachenko Yu. V., Sufficient conditions for continuity with probability one of subGaussian stochastic processes, Dopov. Akad. Nauk Ukr. RSR, Ser. A, no. 2, pp. 113–

115, 1968.

[31] Buldygin V. V., Sub-Gaussian processes and convergence of random series in functional spaces, Ukr. Mat. Zh., vol, 32, no. 6, pp. 723–730, 1977.

[32] Kahane J. P., Some random series of functions, Heath Mathematical Monographs.

Lexington, Mass.: D.C. Heath and Company, a Division of Raytheon Education Company. VIII, 184 p., 1968.

[33] Jain N. C. and Marcus M. B., Integrability of infinite sums of independent vectorvalued random variables, Trans. Amer. Math. Soc., vol. 212, no. 1, pp. 1–36, 1975.

[34] Mikushin V. I. and Ostrovskij E. I., Estimates of the distribution of the maximum of

discontinuous random fields., Theory Probab. Math. Stat., vol. 41, pp. 57–63, 1990.

[35] Fukuda R., Exponential integrability of sub-Gaussian vectors, Probab. Theory Relat.

Fields, vol. 85, no. 4, pp. 505–521, 1990.

[36] Kozachenko Yu. V. and Ostrovskij E. I., Banach spaces of random variables of subGaussian type, Theory Probab. Math. Stat., vol. 32, pp. 45–56, 1986.

[37] Antonini Giuliano R., Kozachenko Yu. V. and Nikitina T., Spaces of ϕ-sub-Gaussian

random variables, Rendiconti Accademia Nazionale delle Scienze detta dei XL, Memorie di Matimatica e Applicazioni, vol. 121, pp. 95-124, 2003.

[38] Vasylyk O. I., Kozachenko Yu. V. and Yamnenko R. Ye, ϕ-sub-Gaussian stochastic

processes, Ky¨ıv: Vydavnychyj Tsentr “Ky¨ıvs’ky˘ı Universytet”, 231 p., 2008.

[39] Artemiev S. S., Numerical methods of Cauchy problem solution for systems of ordinary and stochastic differential equations, Novosibirsk: VC SO RAN, 156 p., 1993.

[40] Iermakov S. I., Statistical modeling. Tutorial, M.: Nauka, 296 p, 1982.

[41] Khamitov G. P., Imitation of stochastic processes, Irkutsk Univ. Publ., 1983.

[42] Shaligin A. S. and Palagin Yu. I., Applied methods of statistical modeling, Leningrad:

‘Mashinostroenie’, 320 p., 1986.

[43] Sabelfeld K. K. and Kurbanmuradov O. A., Numerical statistical model of classical

incompressible isotropic turbulence, Sov. J. Numer. Anal. Math. Model., vol.5, no.3,

pp. 251–263, 1990.

[44] Prigarin S. M., Some problems of numerical modeling of stochastic processes and

fields, Theory and Appl. of Stat. Model., Novosibirsk, pp. 29–32, 1991.

[45] Ripley B. D., Stochastic simulation, Wiley Ser. Probab. Math. Stat. John Wiley &

Sons, Hoboken, NJ, 1987.

[46] Kanter R. R. and Prigarin S. M., Numerical modeling of sea waves to study the field

of reflected optical radiation, Novosibirsk, pp. 1–25, 1989 (pre-print AN USSR, Sib.

dep., vol. 829).

[47] Kargin B. A. and Prigarin S. M., On numerical modeling of optical characteristics of

waved sea surface, Methods of stochastic modeling. Novosibirsk, pp. 95–102, 1990.

[48] Kargin B. A. and Prigarin S. M., Imitation of surface of sea waves and study of its

properties using Monte-Carlo method, Optics of atmosphere and ocean, vol. 5, no. 3,

pp. 285–291, 1992.

[49] Kurbanmuradov O. A., Sabelfeld K. K. and Chopanov G., Statistical modeling of impurity diffusion in random velocity field. Modelf of random fields, Novosibirsk, pp.1–

28, 1988 (pre-print AN USSR, Sib. dep., vol. 775).

[50] Mikhailov G. A., Modeling of random processes and fields using Palm point flows,

Dokl. AN SSSR, vol. 262, no 3, pp. 531–535, 1982.

[51] Mikhailov G. A., Numerical construction of random field with given spectral density,

Dokl. AN SSSR, vol. 238, no 4, pp. 793–795, 1982.

[52] Prigarin S. M., Spectral models of uniform vector fields, Theor. and Appl. of Stat.

Modeling, Novosibirsk, pp. 1– 36, 1989 (pre-print AN USSR, Sib. dep., vol.945).

[53] Mikhailov G. A., On “repeating” method for modeling of random vectors and fields

(randomization of correlation matrices), Theor. Prob. and Appl., vol. 19, no. 4, pp.

873–878, 1974.

[54] Mikhailov G. A., On numerical modelling of impurity diffusion in stochastic velocity

fields, Izv. AN. SSSR, Ser. FAO, vol. 16, no. 3, pp. 229–235, 1980.

[55] Mikhailov G. A., Approximated models of random processes and fields, Journ. of

Comp. Math. and Math. Phys., vol. 23, no. 3, pp. 558–566, 1983.

[56] Troinikov V. S., Numerical modeling of random processes and fields using Palm point

flows in problems of radiation transfer in a cloudy environment, Izv. AN USSR, SER.

FAO, vol.20, no. 4, pp. 274–279, 1984.

[57] Voytishek A. V., Study on weak convergence of models of Gaussian random fields

with given spectral decomposition of correlation function, Modeling on Comp. Systems, Novosibirsk, pp. 119–129, 1982.

[58] Voytishek A. V., Randomized numerical spectral model of a stationary random function, Math. Imit. Models of Syst. Novosibirsk, pp. 17–25, 1983.

[59] Voytishek A. V., Moment conditions of functional convergence of numerical randomized spectral model of uniform Gaussian fields, Theor. and Appl. of Stat. Models,

Novosibirsk, pp. 41–46, 1988.

[60] Yadrenko M. I. and Rakhimov A. K., Statistical simulation of a homogeneous

isotropic random field on the plane and estimations of simulation errors, Theory

Probab. Math. Stat., vol. 49, pp. 245–251, 1993.

[61] Vyzhva Z. O. and Yadrenko M. I., Statistical simulation of isotropic random fields on

a sphere, Visn., Mat. Mekh., Ky¨ıv. Univ. Im. Tarasa Shevchenka, no.5, pp. 5–11, 2000.

[62] Vyzhva Z. O., About approximation of 3-D random fields and statistical simulation,

Random Oper. Stoch. Equ., vol. 11, no. 3, pp. 255-266, 2003.

[63] Bykov V. V., Digital modeling in statistical radiotechnocs, M.: Sov. radio, 328 p.,

1971.

[64] Dergalin N. L. and Romantsev V. V., On modeling of random fields, Trudy X vsesoyuznogo simposiuma, Section 4. Methods of representation and apparatic analysis

of random processes and fields, Leningrad, pp. 60–64, 1978.

[65] Tovstyk T. M., Modeling of uniform Gaussian fields, Trudy X vsesoyuznogo simposiuma, Section 4. Methods of representation and apparatic analysis of random processes and fields. Leningrad, pp. 75–77, 1978.

[66] Sun T. C. and Chaika Milton, On simultaion of Gaussian stationary process, Journal

of Time Series Analysis, Vol. 18, no. 1, pp. 79–93, 1997.

[67] Kozachenko Yu. V. and Kozachenko L. F., Accuracy of modeling stationary Gaussian

stochastic processes in L² (0, T), Vychisl. Prikl. Mat., vol. 75, pp. 108–115, 1991.

[68] Kozachenko Yu. V. and Kozachenko L. F., On accuracy of modeling of Gaussian

stochastic processes in L² (0, T), Vychisl. Prikl. Mat., vol. 74, pp. 88–93, 1992.

[69] Zelepugina I. N. and Kozachenko Yu. V., On a question of modeling of Gaussian

stochastic processes Some questions of the theory of stochastic processes. Kyiv, pp.

47–56, 1982.

[70] Zelepugina I. N. and Kozachenko Yu. V., On estimates of the accuracy of modeling

random fields in spaces Lp, Operational research and ASU, vol. 32, pp. 10–14, 1988.

[71] Juliano Antonini R., Kozachenko Yu. V. and Tegza A. M., Accuracy of modeling

in Lp of Gaussian stochastic processes, Visn., Mat. Mekh., Ky¨ıv. Univ. Im. Tarasa

Shevchenka, no. 5, pp. 7–14, 2002.

[72] Kozachenko Yu. V. and Tegza A. M., Application of the theory of Subϕ(Ω)-spaces

of random variables to estimation of accuracy of modeling of stationary Gaussian

processes, Theory Probab. Math. Stat., vol. 67, pp. 79–96, 2003.

[73] Kozachenko Yu. V. and Pashko A. O., Accuracy of modeling of stochastic processes

in Orlicz spaces. I, Theory Probab. Math. Stat., vol. 58, pp. 51–66, 1998.

[74] Kozachenko Yu. V. and Pashko A. O., Accuracy of modeling of stochastic processes

in Orlicz spaces.II, Theory Probab. Math. Stat., vol. 59, pp. 77–92, 1999.

[75] Kozachenko Yu. V. and Pashko A. O., Modeling of stochastic processes, Ky¨ıv: VPTs

“Ky¨ıvs’kyj Universytet”, 223 p., 1999.

[76] Kozachenko Yu. V. and Pashko A. O., On modeling of random fields. I, Theory

Probab. Math. Stat., vol. 61, pp. 61–74, 2000.

[77] Kozachenko Yu. V. and Pashko A. O., On modeling of random fields. II, Theory

Probab. Math. Stat., vol. 62, pp. 51–63, 2000.

[78] Pashko A. O., Accuracy of modeling of sub-Gaussian random processes, Visn., Mat.

Mekh., Ky¨ıv. Univ. Im. Tarasa Shevchenka, vol. 6, pp. 42–47, 2001.

[79] Pashko A. O., Estimation of accuracy of modeling in uniform metric of Gaussian

isotropic random fields on a sphere, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh.

Nauky, no. 4, pp. 31–35, 2000.

[80] Pashko A. O., An estimation of accuracy of the simulation of sub-Gaussian random

fields in a uniform metric, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, no.

2, pp. 30–36, 2001.

[81] Kozachenko Yu. V. and Rozora I. V., Accuracy and reliability of modeling of stochastic processes from Subϕ(Ω), Theory Probab. Math. Stat., vol. 71, pp. 105–117, 2005.

[82] Karhunen K., Uber lineare methoden in der wahrscheinlichkeitsrechnung [On linear

methods in probability calculus], Annales Academiae Scientarum Fennicae, vol. 37,

pp. 3–79, 1947.

[83] Lo`eve M., Fonctions aleatoires du second ordre [Second-order random functions],

Processus Stochastiques et Mouvement Brownien, P. Levy (ed.), 1948.

[84] Van Trees H. L., Detection, Estimation, and Modulation Theory, John Wiley and Sons,

Inc., New York – London – Sydney, 697 p., 1968.

[85] Kozachenko Yu. V., Rozora I. V. and Turchyn Ye. V., On expansion of random process

in series, Random Oper. Stoch. Equ., vol. 15, no.1, pp. 15–33, 2007.

[86] Kozachenko Yu. V., Distribution of the supremum of random processes from quasiBanach Kσ spaces, Ukr. Math. J., vol. 51, no. 7, pp. 1029–1043, 1999.

[87] Kozachenko Yu. V., Random processes in Orlicz spaces. I, Theory Probab. Math.

Statist., vol. 30, pp. 103 – 117, 1985.

[88] Kozachenko Yu. V., Random processes in Orlicz spaces. II, Theory Probab. Math.

Statist., vol. 31, pp. 51–58, 1985.

[89] Abzhanov E. A. and Kozachenko Yu. V., Some properties of random processes in

Banach Kσ-spaces, Ukrainian Math. J., vol. 37, no. 3, pp. 209–213, 1985.

[90] Abzhanov E. A. and Kozachenko Yu. V., Random processes in quasi-Banach Kσspaces of random variables, Probabilistic Methods for the Investigation of Systems

with Infinite Number of Degrees of Freedom, Institute of Mathematics, Academy of

Sciences of Ukraine. SSR, Kiev, pp. 4–11, 1986.

[91] Kozachenko Yu. V. and Moklyachuk O. M., Stochastic processes in the spaces DV,W ,

Theor. Probab. Math. Stat., vol. 82, pp. 43–56, 2011.

[92] Kozachenko Yu. V. and Moklyachuk O. M., Sample continuity and modeling of

stochastic processes from the spaces DV,W , Theory Probab. Math. Stat., vol. 83, pp.

95–110, 2011.

[93] Kozachenko Yu. V. and Kamenschikova O. Ye., Approximation of SSubϕ(Ω) random

processes in Lp(T), Theor. Probab. Math. Stat., vol. 79, pp. 83–88, 2008.

[94] Moklyachuk O. M., Modeling with given reliability and accuracy in spaces Lp(T)

of stochastic processes from Subϕ(Ω), that allow decompositions in series with independent elements, Applied statistics. Actuarial and financial mathematics, no. 2, pp.

13–23, 2012.

[95] Suetin P. K., Classical othogonal polynomials, Fizmatlit, Moscow, 408 p., 2005.

[96] Kamp´e de Feri´et J., Campbell R., Petio G. and Fogel T., Fonctions de la physique

math´ematique [Mathematical Physics Functions], State publishing for physics and

mathematics literature, Moscow, 102 p., 1963.

[97] Mokliachuk O. M., Modeling of stochastic processes in Lp(T) using orthogonal polynomials, Universal J. Appl. Math., vol. 2, pp. 141–147, 2014.

[98] Mokliachuk O. M., Estimation of accuracy and reliability of models of ϕ-subGaussian stochastic processes in C(T) spaces, Research Bulletin of the National Technical University of Ukraine “Kyiv Polytechnic Institute”, no. 4, pp. 17–24

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