Multi-dimensional additively stationary random functions on convex structures
Authors:
O. I. Ponomarenko and Yu. D. Perun
Translated by:
V. Zayats
Journal:
Theor. Probability and Math. Statist. 74 (2007), 133-146
MSC (2000):
Primary 60G10; Secondary 60G57
DOI:
https://doi.org/10.1090/S0094-9000-07-00703-X
Published electronically:
July 5, 2007
MathSciNet review:
2336784
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Some classes of the wide-sense additively stationary generalized random functions taking values in a complex Hilbert space are considered. These random functions are defined on certain types of convex cones and convex sets belonging to a real vector space that can be interpreted as commutative additive semigroups endowed with the identical involution $*$. Here, stationarity is understood as $*$-stationarity with respect to a semigroup in the sense of earlier papers by the authors. For the above-mentioned classes of additively stationary random functions, spectral expansions are obtained for both these functions and their correlation functions. Properties of these expansions are studied and the problem of the extension of the described additively stationary functions to wider sets in vector spaces is considered.
References
- Michel Loève, Fonctions aléatoires à décomposition orthogonale exponentielle, Revue Sci. 84 (1946), 159–162 (French). MR 17892
- Paul Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars & Cie, Paris, 1965 (French). Suivi d’une note de M. Loève; Deuxième édition revue et augmentée. MR 0190953
- Michel Loève, Probability theory. II, 4th ed., Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 46. MR 0651018
- Serge Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), no. 1, 1–66 (French). MR 1555269, DOI https://doi.org/10.1007/BF02547400
- D. V. Widder, Necessary and sufficient conditions for the representation of a function by a doubly infinite Laplace integral, Bull. Amer. Math. Soc. 40 (1934), 321–326.
- David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
- A. Devinatz, The representation of functions as a Laplace-Stieltjes integrals, Duke Math. J. 22 (1955), 185–191. MR 69928
- R. A. Silverman, Locally stationary random processes, Div. Electromag. Res., Inst. Math. Sci., New York Univ., 1957. Res. Rep. No. MME-2. MR 0090931
- Richard A. Silverman, A matching theorem for locally stationary random processes, Comm. Pure Appl. Math. 12 (1959), 373–383. MR 125629, DOI https://doi.org/10.1002/cpa.3160120210
- Jiří Michálek, Spectral decomposition of locally stationary random processes, Kybernetika (Prague) 22 (1986), no. 3, 244–255. MR 852324
- Jiří Michálek, Locally stationary covariances, Transactions of the Tenth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Vol. A (Prague, 1986) Reidel, Dordrecht, 1988, pp. 83–103. MR 1136266
- Olivier Perrin and Rachid Senoussi, Reducing non-stationary random fields to stationarity and isotropy using a space deformation, Statist. Probab. Lett. 48 (2000), no. 1, 23–32. MR 1767607, DOI https://doi.org/10.1016/S0167-7152%2899%2900188-1
- Werner Ehm, Marc G. Genton, and Tilmann Gneiting, Stationary covariances associated with exponentially convex functions, Bernoulli 9 (2003), no. 4, 607–615. MR 1996272, DOI https://doi.org/10.3150/bj/1066223271
- Valerie Girardin and Rachid Senoussi, Semigroup stationary processes and spectral representation, Bernoulli 9 (2003), no. 5, 857–876. MR 2047689, DOI https://doi.org/10.3150/bj/1066418881
- A. I. Ponomarenko, Banach-space-valued random fields that are homogeneous in the wide sense on semigroups and homogeneous spaces, Teor. Verojatnost. i Mat. Statist. 7 (1972), 110–121, 165 (Russian, with English summary). MR 0322940
- A. I. Ponomarenko, Infinite-dimensional random fields on semigroups, Teor. Veroyatnost. i Mat. Statist. 30 (1984), 136–142 (Russian). MR 800839
- O. Ī. Ponomarenko, Integral representation of random functions with values in locally convex spaces, Teor. Īmovīr. ta Mat. Statist. 46 (1992), 132–141 (Ukrainian, with Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 46 (1993), 129–136. MR 1196216
- O. Ī. Ponomarenko and Yu. D. Perun, Multidimensional weakly stationary random functions on semigroups, Teor. Ĭmovīr. Mat. Stat. 73 (2005), 134–145 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 73 (2006), 151–162. MR 2213849, DOI https://doi.org/10.1090/S0094-9000-07-00689-8
- Helge Glöckner, Positive definite functions on infinite-dimensional convex cones, Mem. Amer. Math. Soc. 166 (2003), no. 789, xiv+128. MR 2008256, DOI https://doi.org/10.1090/memo/0789
- E. L. Lehmann, Testing statistical hypotheses, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 1997. MR 1481711
- Einar Hille and Ralph 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
References
- M. Loève, Fonctions aléatoires a decomposition orthogonale exponentielle, Rev. Sci. 84 (1946), 159–162. MR 0017892 (8:214f)
- M. Loève, Fonctions aléatoires a charactère exponentiel, Processus Stochastiques et Mouvement Brownien (P. Levy, ed.), 2nd edn., Gauthier-Villars, Paris, 1965, pp. 409–420. MR 0190953 (32:8363)
- M. Loève, Probability Theory II, Springer-Verlag, New York, 1978. MR 0651018 (58:31324b)
- S. Bernstein, Sur les fonctions absolument monotones, Acta Math. 52 (1929), 1–66. MR 1555269
- D. V. Widder, Necessary and sufficient conditions for the representation of a function by a doubly infinite Laplace integral, Bull. Amer. Math. Soc. 40 (1934), 321–326.
- D. V. Widder, The Laplace Transform, Princeton University Press, New York, 1946. MR 0005923 (3:232d)
- A. Devinatz, The representation of functions as Laplace–Stieltjes integrals, Duke Math. J. 22 (1955), 185–191. MR 0069928 (16:1102d)
- R. A. Silverman, Locally stationary random processes, IER Trans. Inform. Theory 3 (1957), 182–187. MR 0090931 (19:893b)
- R. A. Silverman, A matching theorem for locally stationary random processes, Comm. Pure Appl. Math. 12 (1959), 373–383. MR 0125629 (23:A2928)
- Y. Michálek, Spectral decomposition of locally stationary random processes, Kybernetika 22 (1986), 244–255. MR 852324 (87k:60101)
- Y. Michálek, Locally stationary covariances, Information Theory, Statistical Decision Functions, Random Processes (S. Kubik, ed.), Reidel, Dordrecht, 1988, pp. 115–129. MR 1136266 (92i:60068)
- O. Perrin and R. Senoussi, Reducing non-stationary stochastic fields to stationary and isotropy using a space deformation, Statist. Probab. Lett. 48 (2000), 23–32. MR 1767607 (2001i:60087)
- W. Ehm, M. G. Genton, and T. Gneiting, Stationary covariances associated with exponentially convex functions, Bernoulli 9 (2003), 607–615. MR 1996272 (2004c:62151)
- V. Gerardin and R. Senoussi, Semigroup stationary processes and spectral representation, Bernoulli 9 (2003), 857–876. MR 2047689 (2005d:60054)
- A. I. Ponomarenko, Wide sense homogeneous random fields on semigroups and homogeneous spaces with values in a Banach space, Teor. Veroyatnost. i Mat. Statist. 7 (1972), 110–121; English transl. in Theory Probab. Math. Statist. 7 (1973), 104–114. MR 0322940 (48:1300)
- A. I. Ponomarenko, Infinite-dimensional random fields on semigroups, Teor. Veroyatnost. i Mat. Statist. 30 (1984), 136–142; English transl. in Theory Probab. Math. Statist. 30 (1985), 153–158. MR 800839 (87b:60077)
- O. Ī. Ponomarenko, Integral representation of random functions with values in locally convex spaces, Teor. Ĭmovīr. Mat. Stat. 46 (1992), 132–141; English transl. in Theory Probab. Math. Statist. 46 (1993), 129–136. MR 1196216 (93k:60123)
- O. Ī. Ponomarenko and Yu. D. Perun, Multidimensional weakly stationary random functions on semigroups, Teor. Ĭmovīr. Mat. Stat. 73 (2005), 134–145; English transl. in Theory Probab. Math. Statist. 73 (2006), 151–162. MR 2213849 (2007b:60092)
- H. Glökner, Positive definite functions on infinite-dimensional convex cones, Memoirs of AMS 166 (2003), no. 789. MR 2008256 (2004m:43008)
- E. L. Lehmann, Testing Statistical Hypotheses, 2nd ed., Springer-Verlag, New York, 1997. MR 1481711 (98m:62004)
- E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R.I., 1974. MR 0423094 (54:11077)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60G10,
60G57
Retrieve articles in all journals
with MSC (2000):
60G10,
60G57
Additional Information
O. I. Ponomarenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Mechanics and Mathematics Faculty, Taras Shevchenko National University, Glushkov Ave., 6, Kyïv 03127, Ukraine
Yu. D. Perun
Affiliation:
Auditorship Department, National Bank of Ukraine, Instituts’ka Street, 9, Kyïv 01601, Ukraine
Email:
perun@bank.gov.ua
Keywords:
Additively stationary random function in a Hilbert space,
convex cone,
convex set,
$\alpha$-boundedness,
spectral expansion
Received by editor(s):
April 13, 2005
Published electronically:
July 5, 2007
Article copyright:
© Copyright 2007
American Mathematical Society