On C.R. RAO’s theorem for locally compact abelian groups

Cover Page

Abstract


Let x1, x2, x3 be independent random variables with values in a locally compact Abelian group X with nonvanish- ing characteristic functions, and aj, bj be continuous endomorphisms of X satisfying some restrictions. Let L1 = a1x1 + a2x2 + a3x3, L2 = b1x1 + b2x2 + b3x3. It was proved that the distribution of the random vector (L1; L2) determines the distributions of the random variables xj 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.


G. M. Feldman

Физико-технический институт им. Б.И. Веркина Национальной академии наук Украины

Author for correspondence.
Email: feldman@ilt.kharkov.ua

Ukraine, Харьков

  1. Rao C.R. // Sankhya. 1971. V. 33. Ser. A. P. 265–270.
  2. Kotlarski I. // Pacific J. Math. 1967. V. 20. P. 69–76.
  3. Parthasarathy K.R. Probability Measures on Metric Spaces. N.Y.; L.: Acad. Press, 1967. 276 p.
  4. Хьюитт Э., Росс К. Абстрактный гармонический анализ. М.: Наука, 1975. 656 с.
  5. Prakasa Rao B.L.S. // Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 1968. V. 9. P. 98–100.
  6. Feldman G.M. // Aequat. Math. 2017. V. 91. P. 949– 967.
  7. Feldman G.M. Functional Equations and Characteriza- tion Problems on Locally Compact Abelian Groups. EMS Tracts in Mathematics. V. Zurich: Eur. Math. Soc., 2008. 268 p.

Views

Abstract - 94

PDF (Russian) - 83

PlumX


Copyright (c) 2019 Российская академия наук