Discrete representations of second order random functions. II
Author:
O. I. Ponomarenko
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 87 (2013), 171-183
MSC (2010):
Primary 60G10, 60G15; Secondary 60G57
DOI:
https://doi.org/10.1090/S0094-9000-2014-00911-4
Published electronically:
March 21, 2014
MathSciNet review:
3241454
Full-text PDF Free Access
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Additional Information
Abstract: This is a continuation of the author’s 2012 paper. We study the discrete representations of a general basis type for a second order random function assuming scalar or vector values.
References
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References
- I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, vol. 1, “Nauka”, Moscow, 1971; English transl., Springer-Verlag, Berlin–Heidelberg, 2004.
- I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random processes, “Nauka”, Moscow, 1977; English transl., Saunders, Philadelphia, 1969. MR 0247660 (40:923)
- A. V. Bulinskiĭ and A. N. Shiryaev, Theory of Stochastic Processes “Fizmatgiz”, Moscow, 2003. (Russian)
- D. V. Gusak, O. G. Kukush, O. M. Kulik, Yu. S. Mishura, A. Yu. Pilipenko, Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory, Kiev University Press, Kyiv, 2008; English transl., Springer-Verlag, New York, 2010. MR 2572942 (2011f:60069)
- B. V. Dovgaĭ, Yu. V. Kozachenko, and I. V. Rozora, Modelling Stochastic Processes in Physical Systems, “Zadruga”, Kyiv, 2010. (Ukrainian)
- I. G. Petrovskiĭ, Lectures in the Theory of Integral Equations, “Nauka”, Moscow, 1965. (Russian) MR 0352904 (50:5390)
- A. M. Yaglom, Second-order homogeneous random fields, Proc. of Fourth Berkeley Symp. Math. Stat. Probab., vol. II, Univ. Calif. Press, 1961, pp. 593–622. MR 0146880 (26:4399)
- M. G. Kreĭn, Hermitian positive kernels in homogeneous spaces. I, Ukr. Mat. Zh. I (1949), no. 4, 64–98. (Russian) MR 0051438 (14:480a)
- F. Riesz and B. Szökefalvi Nagy, Leçons d’analyse fonctionnelle, sixième édition, Akadémiai Kiadó, Budapest, 1972.
- O. I. Ponomarenko and Yu. D. Perun, Multidimensional weakly stationary random functions on semigroups, Teor. Imovir. Matem. Statyst. 73 (2006), 134–145; English transl. in Theory Probab. Math. Statist. 73 (2005), 151–162. MR 2213849 (2007b:60092)
- E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R.I., 1974. Third printing of the revised edition of 1957; American Mathematical Society Colloquium Publications, Vol. XXXI. MR 0423094 (54:11077)
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- O. I. Ponomarenko, Integral representations of random functions with values in locally convex spaces, Teor. Imovir. Matem. Statyst. 46 (1992), 132–141; English transl. in Theory Probab. Math. Statist. 46 (1993), 129–136. MR 1196216 (93k:60123)
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- N. Dunford and J. T. Schwartz, Linear Operators, Part II. Spectral Theory, Wiley-Interscience, New York–London, 1963. MR 1009163 (90g:47001b)
- O. Ponomarenko and Yu. Perun, Multivariate random fields on some homogeneous spaces, Theory Stoch. Process. 14(30) (2008), no. 3–4, 104–113. MR 2498608 (2010h:60155)
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- O. I. Ponomarenko and Yu. D. Perun, Multi-dimensional additively stationary random functions on convex structures, Teor. Imovir. Matem. Statyst. 74 (2006), 118–130; English transl. in Theory Probab. Math. Statist. 74 (2007), 133–146. MR 2336784 (2008g:60103)
- A. I. Ponomarenko, Stochastic Problems of Optimization, Kiev University Press, Kiev, 1980. (Russian)
- O. I. Ponomarenko, Discrete representations of second order random functions. I, Teor. Imovir. Matem. Statyst. 86 (2012). MR 2986458
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Additional Information
O. I. Ponomarenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Keywords:
Random functions defined on a compact topological space,
generalized random functions assuming values in a Hilbert space,
Karhunen–Loéve type representations,
basis type representations,
Hilbert–Schmidt operators
Received by editor(s):
September 8, 2011
Published electronically:
March 21, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
