Abstract.
The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus
which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish.
References
Darmois, G.: Analyse generale des liaisons stochastiques. Rev. Inst. Intern. Stat. 21, 2–8 (1953)
Feldman, G.M.: On the decomposition of Gaussian distributions on groups. Theory Probab. Appl. 22, 133–140 (1977)
Feldman, G.M.: Marcinkiewicz and Lukacs theorems on Abelian groups. Theory Probab. Appl. 34, 290-297 (1989).
Feldman, G.M.: Arithmetic of probability distributions and characterization problems on Abelian groups. AMS translation of mathematical monographs 116, Providence, RI, 1993
Feldman, G.M.: More on the Skitovich-Darmois theorem for finite Abelian groups. Theory Probab. Appl. 45, 507–511 (2001)
Feldman, G.M.: A characterization of the Gaussian distribution on Abelian groups. Probab. Theory Relat. Fields. 126, 91–102 (2003)
Feldman, G.M., Graczyk, P.: On the Skitovich-Darmois theorem on compact Abelian groups. J. of Theoretical Probability. 13, 859–869 (2000)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. 1. Springer-Verlag, Berlin Gottingen Heildelberg, 1963
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. 2. Springer-Verlag, Berlin Heildelberg New York, 1970
Heyde, C.C.: Characterization of the normal low by the symmetry of a certain conditional distribution. Sankhya, Ser. A. 31, 115–118 (1969)
Kagan, A.M., Linnik, Ju. V., Rao, C.R.: Characterization problems of mathematical statistics. Wiley, New York 1973
Linnik Ju.V., Ostrovskii I.V.: Decomposition of random variables and vectors. AMS translation of mathematical monographs 48, Providence, RI, 1977
Neuenschwander, D. and Schott, R.: The Bernstein and Skitovic-Darmois characterization theorems for Gaussian distributions on groups, symmetric spaces, and quantum groups. Expo. Math. 15, 289-314 (1997)
Parthasarathy, K.R.: Probability measures on metric spaces. Academic Press, New York London 1967
Skitovich, V.P.: On a property of a normal distribution. Doklady Academii nauk SSSR 89, 217–219 (in Russian) (1953)
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Feldman, G. On a characterization theorem for locally compact abelian groups. Probab. Theory Relat. Fields 133, 345–357 (2005). https://doi.org/10.1007/s00440-005-0429-4
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DOI: https://doi.org/10.1007/s00440-005-0429-4
Keywords
- Heyde theorem
- Locally compact Abelian group
- Characterization of probability distributions
Mathematics Subject Classifications (2000)
- Primary 62E10
- secondary 60B15