Hostname: page-component-7f64f4797f-pffzl Total loading time: 0 Render date: 2025-11-04T13:55:45.215Z Has data issue: false hasContentIssue false
  • English
  • Français

THE SKITOVICH–DARMOIS THEOREM FOR LOCALLY COMPACT ABELIAN GROUPS

Part of: Probability theory on algebraic and topological structures Abstract harmonic analysis

Published online by Cambridge University Press:  11 June 2010

GENNADIY FELDMAN  and
PIOTR GRACZYK
GENNADIY FELDMAN*
Affiliation:
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave, Kharkov 61103, Ukraine LAREMA Laboratoire de Mathématiques, Université d’Angers, 2 bd. Lavoisier, 49045 Angers, France (email: feldman@ilt.kharkov.ua)
PIOTR GRACZYK
Affiliation:
LAREMA Laboratoire de Mathématiques, Université d’Angers, 2 bd. Lavoisier, 49045 Angers, France (email: piotr.graczyk@univ-angers.fr)
*
✉For correspondence; e-mail: feldman@ilt.kharkov.ua
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

According to the Skitovich–Darmois theorem, the independence of two linear forms of n independent random variables implies that the random variables are Gaussian. We consider the case where independent random variables take values in a second countable locally compact abelian group X, and coefficients of the forms are topological automorphisms of X. We describe a wide class of groups X for which a group-theoretic analogue of the Skitovich–Darmois theorem holds true when n=2.

Keywords

locally compact abelian groupSkitovich–Darmois theorem

MSC classification

Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 3 , June 2010 , pp. 339 - 352
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This research was supported in part by the Ukrainian–French grant PICS No 4769 (2009–2011).

References

[1]Chistyakov, G. P. and Götze, F., ‘Independence of linear forms with random coefficients’, Probab. Theory Related Fields 137(1–2) (2007), 1–24.CrossRefGoogle Scholar
[2]Darmois, G., ‘Analyse générale des liaisons stochastiques’, Rev. Inst. Intern. Stat. 21 (1953), 2–8.CrossRefGoogle Scholar
[3]Fel′dman, G. M., ‘On the Skitovich–Darmois theorem on Abelian groups’, Theory Probab. Appl. 37 (1992), 621–631.CrossRefGoogle Scholar
[4]Fel′dman, G. M., Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups, American Mathematical Society Translation of Mathematical Monographs, 116 (American Mathematical Society, Providence, RI, 1993).CrossRefGoogle Scholar
[5]Feldman, G. M., ‘More on the Skitovich–Darmois theorem for finite Abelian groups’, Theory Probab. Appl. 45 (2000), 507–511.CrossRefGoogle Scholar
[6]Feldman, G. M., ‘A characterization of the Gaussian distribution on Abelian groups’, Probab. Theory Related Fields 126 (2003), 91–102.CrossRefGoogle Scholar
[7]Feldman, G. M., Functional Equations and Characterization Problems on Locally Compact Abelian Groups, EMS Tracts in Mathematics, 5 (European Mathematical Society, Zurich, 2008).CrossRefGoogle Scholar
[8]Feldman, G. M., ‘On a theorem of K. Schmidt’, Bull. Lond. Math. Soc. 41, (2009), 103–108.CrossRefGoogle Scholar
[9]Feldman, G. M. and Graczyk, P., ‘On the Skitovich–Darmois theorem on compact Abelian groups’, J. Theoret. Probab. 13 (2000), 859–869.CrossRefGoogle Scholar
[10]Feldman, G. M. and Graczyk, P., ‘On the Skitovich–Darmois theorem for discrete Abelian groups’, Theory Probab. Appl. 49 (2005), 527–531.CrossRefGoogle Scholar
[11]Ghurye, S. G. and Olkin, I., ‘A characterization of the multivariate normal distribution’, Ann. Math. Stat. 33 (1962), 533–541.CrossRefGoogle Scholar
[12]Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis, Vol. 1 (Springer, Berlin, 1963).Google Scholar
[13]Heyer, H. and Rall, Ch., ‘Gausssche Wahrscheinlichkeitsmasse auf Corwischen Gruppen’, Math. Z. 128 (1972), 343–361.CrossRefGoogle Scholar
[14]Kagan, A. M., Linnik, Yu. V. and Rao, C. R., Characterization Problems in Mathematical Statistics (Wiley, New York, 1973).Google Scholar
[15]Kagan, A. and Wesołowski, J., ‘An extension of the Darmois–Skitovitch theorem to a class of dependent random variables’, Statist. Probab. Lett. 47 (2000), 69–73.CrossRefGoogle Scholar
[16]Parthasarathy, K. R., Probability Measures on Metric Spaces (Academic Press, New York, 1967).CrossRefGoogle Scholar
[17]Ruhin, A. L., ‘A certain theorem of S. N. Bernstein’, Math. Notes 6 (1969), 638–641.CrossRefGoogle Scholar
[18]Schmidt, K., ‘On a characterization of certain infinitely divisible positive definite functions and measures’, J. Lond. Math. Soc.(2) 4 (1971/72), 401–407.Google Scholar
[19]Skitovich, V. P., ‘On a property of a normal distribution’, Dokl. Acad. Nauk SSSR 89 (1953), 217–219 (in Russian).Google Scholar