Let X be a finite Abelian group, let ξ i , i = 1, 2,…, n, n ≥ 2, be independent random variables with values in X and distributions μ i , and let α ij , i, j = 1, 2,…, n, be automorphisms of X. We prove that the independence of n linear forms \( {L_j} = \sum\nolimits_{i = 1}^n {{\alpha_{ij}}} {\xi_i} \) implies that all μ i are shifts of the Haar distributions of a certain subgroup of the group X. This theorem is an analog of the Skitovich–Darmois theorem for finite Abelian groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. P. Skitovich, “On a property of the normal distribution,” Dokl. Akad. Nauk SSSR, 89, 217–219 (1953).
G. Darmois, “Analyse generale des liasions stochastiques. Etude particuliere de l‘analyse factorielle lineaire,” Rev. Inst. Int. Statist., 21, 2–8 (1953).
A. M. Kagan, Yu. V. Linnik, and C. R. Rao, Characterization Problems in Mathematical Statistics, Wiley, New York (1973).
S. G. Ghurye and I. Olkin, “A characterization of the multivariate normal distribution,” Ann. Math. Statist., 33, 533–541 (1962).
G. M. Feldman, “On the Skitovich–Darmois theorem for finite Abelian groups,” Theory Probab. Appl., 37, 621–631 (1992).
G. M. Feldman, “On the Skitovich–Darmois theorem on compact groups,” Theory Probab. Appl., 41, 768–773 (1996).
G. M. Feldman, “The Skitovich–Darmois theorem for discrete periodic Abelian groups,” Theory Probab. Appl., 42, 611–617 (1997).
G. M. Feldman, “More on the Skitovich–Darmois theorem for finite Abelian groups,” Theory Probab. Appl., 45, 507–511 (2001).
G. M. Feldman and P. Graczyk, “On the Skitovich–Darmois theorem on compact Abelian groups,” J. Theor. Probab., 13, 859–869 (2000).
G. M. Feldman and P. Graczyk, “On the Skitovich–Darmois theorem for discrete Abelian groups,” Theory Probab. Appl., 49, 527–531 (2005).
G. M. Feldman and P. Graczyk, “The Skitovich–Darmois theorem for locally compact Abelian groups,” J. Austral. Math. Soc., 88, 339–352 (2010).
P. Graczyk and G. M. Feldman, “Independent linear statistics on finite Abelian groups,” Ukr. Mat. Zh., 53, No. 4, 441–448 (2001); English translation: Ukr. Math. J., 53, No. 4, 499–506 (2001).
W. Krakowiak, “The theorem of Darmois–Skitovich for Banach-valued random variables,” Ann. Inst. H. Poincaré B, 11, No. 4, 397–404 (1975).
M. V. Myronyuk, “On the Skitovich–Darmois and Heyde theorem in a Banach space,” Ukr. Mat. Zh. 60, No. 9, 1234–1242 (2008); English translation: Ukr. Math. J., 60, No. 9, 1437–1447 (2008).
G. Feldman, Functional Equations and Characterizations Problems on Locally Compact Abelian Groups, European Mathematical Society, Zurich (2008).
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer, Berlin (1963).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 11, pp. 1512–1523, November, 2011.
Rights and permissions
About this article
Cite this article
Mazur, I.P. Skitovich–Darmois theorem for finite Abelian groups. Ukr Math J 63, 1719–1732 (2012). https://doi.org/10.1007/s11253-012-0608-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0608-2