Accuracy and reliability of a model of an isotropic and homogeneous Gaussian random field in the space $C(\mathbb {T})$
Author:
N. V. Troshki
Translated by:
S. V. Kvasko
Journal:
Theor. Probability and Math. Statist. 97 (2018), 201-209
MSC (2010):
Primary 60G15; Secondary 60G07
DOI:
https://doi.org/10.1090/tpms/1057
Published electronically:
February 21, 2019
MathSciNet review:
3746008
Full-text PDF
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Abstract: The accuracy and reliability of a model of an isotropic homogeneous random field are studied in the space $C(\mathbb {T})$.
References
- V. V. Buldygin and Yu. V. Kozachenko, Metric characterization of random variables and random processes, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR 1743716
- R. Giuliano Antonini, Yu. Kozachenko, and T. Nikitina, Spaces of $\phi $-sub-Gaussian random variables, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27 (2003), 95–124 (English, with English and Italian summaries). MR 2056414
- B. V. Dovhai, Yu. V. Kozachenko, and H. I. Slyvka-Tylyshchak, The Boundary Value Problems of Mathematical Physics with Random Factors, Kiev University, 2008. (Ukrainian)
- Yu. V. Kozachenko and L. F. Kozachenko, On the accuracy of modeling stationary Gaussian random processes in $L^2(0,T)$, Vychisl. Prikl. Mat. (Kiev) 75 (1991), 108–115 (Russian); English transl., J. Math. Sci. 72 (1994), no. 3, 3137–3143. MR 1168858, DOI https://doi.org/10.1007/BF01259486
- Yu. V. Kozachenko and A. O. Pashko, The accuracy of modeling random processes in norms of Orlicz spaces. II, Teor. Ĭmovīr. Mat. Stat. 59 (1998), 75–90 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 59 (1999), 77–92 (2000). MR 1793766
- Yu. V. Kozachenko, A. O. Pashko, and I. V. Rozora, Modeling of Random Processes and Fields, “Zadruga”, Kyiv, 2007. (Ukrainian)
- Yu. V. Kozachenko, O. O. Pogoriliak, and A. M. Tegza, Modelling of Gaussian Random Processes and Cox Processes, Karpaty, Uzhgorod, 2012. (Ukrainian)
- Yuriy Kozachenko, Oleksandr Pogorilyak, Iryna Rozora, and Antonina Tegza, Simulation of stochastic processes with given accuracy and reliability, Mathematics and Statistics Series, ISTE, London; Elsevier, Inc., Oxford, 2016. MR 3644192
- Yu. V. Kozachenko and G. Ī. Slivka, Justification of the Fourier method for a hyperbolic equation with random initial conditions, Teor. Ĭmovīr. Mat. Stat. 69 (2003), 63–78 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 69 (2004), 67–83 (2005). MR 2110906, DOI https://doi.org/10.1090/S0094-9000-05-00615-0
- Y. Kozachenko and A. Slyvka-Tylyshchak, The Cauchy problem for the heat equation with a random right part from the space $\operatorname {Sub}_{\varphi }(\Omega )$, Appl. Math. 5 (2014), 2318–2333.
- Yu. V. Kozachenko and N. V. Troshki, Accuracy and reliability of a model of Gaussian random process in $C(\mathbb {T})$ space, Int. J. Stat. Manag. Syst. 10 (2015), no. 1–2, 1–15.
- G. A. Mikhaĭlov, Modeling random processes and fields with the help of Palm processes, Doklady AN SSSR 262 (1982), no. 3, 531–535. (Russian)
- G. A. Mikhaĭlov, Nekotorye voprosy teorii metodov Monte-Karlo, Izdat. “Nauka” Sibirsk. Otdel., Novosibirsk, 1974 (Russian). MR 0405785
- G. A. Mikhaĭlov and K. K. Sabelfeld, On numerical simulation of impurity diffusion in stochastic velocity fields, Izvestiya AN SSSR Ser. Physics 16 (1980), no. 3, 229–235. (Russian)
- G. A. Mikhaĭlov, Approximate models of random processes and fields, Zh. Vychisl. Mat. i Mat. Fiz. 23 (1983), no. 3, 558–566 (Russian). MR 706881
- G. A. Mikhaĭlov and A. V. Voytishek, Numerical Statistical Modeling, “Akademia”, Moscow, 2006. (Russian)
- Andriy Ya. Olenko and Tibor K. Pogány, Direct Lagrange-Yen type interpolation of random fields, Theory Stoch. Process. 9 (2003), no. 3-4, 145–156. MR 2306067
- A. Ya. Olenko and T. K. Pogány, On sharp bounds for remainders in multidimensional sampling theorem, Sampl. Theory Signal Image Process. 6 (2007), no. 3, 249–272. MR 2445432
- N. V. Troshkī, Estimates for the supremum norms of deviations of a homogeneous and isotropic random field from its model, Teor. Ĭmovīr. Mat. Stat. 94 (2016), 150–172 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 94 (2017), 159–184. MR 3553461, DOI https://doi.org/10.1090/tpms/1016
- N. V. Troshkī, Accuracy and reliability of a model of a Gaussian homogeneous and isotropic random field in the space $L_p(\Bbb T)$, $p\ge 1$, Teor. Ĭmovīr. Mat. Stat. 90 (2014), 161–176 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 90 (2015), 183–200. MR 3242030, DOI https://doi.org/10.1090/S0094-9000-2015-00959-5
- Nataliya Troshki, Construction models of Gaussian random processes with a given accuracy and reliability in $L_p(T)$, $p\geqslant 1$, J. Class. Anal. 3 (2013), no. 2, 157–165. MR 3322266, DOI https://doi.org/10.7153/jca-03-14
- Z. O. Vyzhva, On approximation of 3-D isotropic random fields on the sphere and statistical simulation, Theory Stoch. Process. 3 (1997), no. 3–4, 463–467.
- M. Ĭ. Jadrenko, Spektral′naya teoriya sluchaĭ nykh poleĭ, “Vishcha Shkola”, Kiev, 1980 (Russian). MR 590889
- M. I. Yadrenko and A. K. Rakhimov, Statistical simulation of a homogeneous isotropic random field on the plane and estimations of simulation errors, Teor. Ĭmovīr. Mat. Stat. 49 (1993), 245–251 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 49 (1994), 177–181 (1995). MR 1445264
References
- V. V. Buldygin and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, TBiMC, Kiev, 1998; English transl. American Mathematical Society, Providence, RI, 2000. MR 1743716
- R. Guiliano Antonini, Yu. Kozachenko, and T. Nikitina, Spaces of $\varphi$-sub-Gaussian random variables, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 27 (2003), 95–124. MR 2056414
- B. V. Dovhai, Yu. V. Kozachenko, and H. I. Slyvka-Tylyshchak, The Boundary Value Problems of Mathematical Physics with Random Factors, Kiev University, 2008. (Ukrainian)
- Yu. V. Kozachenko and L. F. Kozachenko, Simulation accuracy of stationary Gaussian stochastic processes in $L^2(0,T)$, J. Math. Sci. 72 (1994), no. 3, 3137–3143. MR 1168858
- Yu. V. Kozachenko and A. O. Pashko, The accuracy of modeling random processes in norms of Orlicz spaces. I, Theory Probab. Math. Statist. 58 (2000), 51–66. MR 1793766
- Yu. V. Kozachenko, A. O. Pashko, and I. V. Rozora, Modeling of Random Processes and Fields, “Zadruga”, Kyiv, 2007. (Ukrainian)
- Yu. V. Kozachenko, O. O. Pogoriliak, and A. M. Tegza, Modelling of Gaussian Random Processes and Cox Processes, Karpaty, Uzhgorod, 2012. (Ukrainian)
- Y. Kozachenko, O. Pogorilyak, I. Rozora, and A. Tegza, Simulation of Stochastic Processes with Given Accuracy and Reliability, ISTE Press, London, Elsevier, Oxford, 2016. MR 3644192
- Yu. V. Kozachenko and G. I. Slyvka, Justification of the Fourier method for hyperbolic equations with random initial conditions, Theory Probab. Math. Statist. 69 (2004), 67–83. MR 2110906
- Y. Kozachenko and A. Slyvka-Tylyshchak, The Cauchy problem for the heat equation with a random right part from the space $\operatorname {Sub}_{\varphi }(\Omega )$, Appl. Math. 5 (2014), 2318–2333.
- Yu. V. Kozachenko and N. V. Troshki, Accuracy and reliability of a model of Gaussian random process in $C(\mathbb {T})$ space, Int. J. Stat. Manag. Syst. 10 (2015), no. 1–2, 1–15.
- G. A. Mikhaĭlov, Modeling random processes and fields with the help of Palm processes, Doklady AN SSSR 262 (1982), no. 3, 531–535. (Russian)
- G. A. Mikhaĭlov, Some Questions of the Theory of Monte Carlo Methods, “Nauka”, Novosibirsk, 1974. (Russian) MR 0405785
- G. A. Mikhaĭlov and K. K. Sabelfeld, On numerical simulation of impurity diffusion in stochastic velocity fields, Izvestiya AN SSSR Ser. Physics 16 (1980), no. 3, 229–235. (Russian)
- G. A. Mikhaĭlov, Approximate models of random processes and fields, Zh. Vychisl. Mat. Mat. Fiz. 23 (1983), no. 3, 558–566. (Russian) MR 706881
- G. A. Mikhaĭlov and A. V. Voytishek, Numerical Statistical Modeling, “Akademia”, Moscow, 2006. (Russian)
- A. Olenko and T. Pogány, Direct Lagrange–Yen type interpolation of random fields, Theory Stoch. Process. 9(25) (2003), no. 3–4, 242–254. MR 2306067
- A. Olenko and T. Pogány, On sharp bounds for remainders in multidimensional sampling theorem, Sampl. Theory Signal Image Process. 6 (2007), no. 3, 249–272. MR 2445432
- N. V. Troshki, Upper bounds for supremums of the norms of the deviation between a homogeneous isotropic random field and its model, Theor. Probab. Math. Statist. 94 (2017), 159–184. MR 3553461
- N. V. Troshki, Accuracy and reliability of a model for a Gaussian homogeneous and isotropic random field in the space $L_p(\mathbb {T})$, $p \ge 1$, Theory Probab. Math. Stat. 90 (2015), 183–200. MR 3242030
- N. Troshki, Construction models of Gaussian random processes with a given accuracy and reliability in $L_p(\mathbb {T}), p\geq 1$, J. Classical Anal. 3 (2013), no. 2, 157–165. MR 3322266
- Z. O. Vyzhva, On approximation of 3-D isotropic random fields on the sphere and statistical simulation, Theory Stoch. Process. 3 (1997), no. 3–4, 463–467.
- M. I. Yadrenko, Spectral Theory of Random Fields, “Vyshcha Shkola”, Kiev, 1980; Optimization Software, Inc., New York, 1983. MR 590889
- M. I. Yadrenko and A. K. Rakhimov, Statistical simulation of a homogeneous isotropic random field on the plane and estimations of simulation errors, Theory Probab. Math. Stat. 49 (1994), 177–181. MR 1445264
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Additional Information
N. V. Troshki
Affiliation:
Department of Probability Theory and Mathematical Analysis, Uzhgorod National University, Universytets’ka Street, 14, Uzhgorod 88000, Ukraine
Email:
FedoryanichNatali@ukr.net
Keywords:
Gaussian random fields,
isotropic homogeneous random field,
simulation,
accuracy and reliability
Received by editor(s):
August 16, 2016
Published electronically:
February 21, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
