Abstract
Let ξ 1, ξ2, ξ3 be independent random variables with values in a locally compact Abelian group X with nonvanishing characteristic functions, and aj, bj be continuous endomorphisms of X satisfying some restrictions. Let L1 = a1ξ 1+ a2 ξ2 + a3ξ 3, L2 = b1ξ 1 + b2ξ 2 + b3ξ 3. It was proved that the distribution of the random vector (L1, L2 ) determines the distributions of the random variables ξj up a shift. This result is a group analogue of the well-known C.R. Rao theorem. We also prove an analogue of another C.R. Rao's theorem for independent random variables with values in an a-adic solenoid.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
C. R. Rao, Sankhya Ser. A 33, 265–270 (1971).
I. Kotlarski, Pacific J. Math. 20, 69–76 (1967).
K. R. Parthasarathy, Probability Measures on Metric Spaces (Academic, New York, 1967).
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis (Springer-Verlag, Berlin, 1970).
B. L. S. Prakasa Rao, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 9, 98–100 (1968).
G. M. Feldman, Aequat. Math. 91, 949–967 (2017).
G. M. Feldman, Functional Equations and Characterization Problems on Locally Compact Abelian Groups, EMS Tracts in Mathematics, Vol. 5 (Eur. Math. Soc., Zurich, 2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
The article was translated by the author.
Rights and permissions
About this article
Cite this article
Feldman, G.M. On C.R. Rao’s Theorem for Locally Compact Abelian Groups. Dokl. Math. 99, 48–51 (2019). https://doi.org/10.1134/S1064562419010149
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562419010149