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Uncorrelatedness sets for random variables with given distributions


Author: Sofiya Ostrovska
Journal: Proc. Amer. Math. Soc. 133 (2005), 1239-1246
MSC (2000): Primary 60E05
Published electronically: October 18, 2004
MathSciNet review: 2117227
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Abstract: Let $\xi_1$ and $\xi_2$ be random variables having finite moments of all orders. The set

\begin{displaymath}U(\xi_1,\xi_2):=\left\{(j,l)\in {\bf N}^2:{\bf E}\left(\xi_1^... ...{\bf E}\left(\xi_1^j\right){\bf E}\left( \xi_2^l\right)\right\}\end{displaymath}

is said to be an uncorrelatedness set of $\xi_1$ and $\xi_2.$ It is known that in general, an uncorrelatedness set can be arbitrary. Simple examples show that this is not true for random variables with given distributions. In this paper we present a wide class of probability distributions such that there exist random variables with given distributions from the class having a prescribed uncorrelatedness set. Besides, we discuss the sharpness of the obtained result.


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  • 1. Viktor Beneš and Josef Štěpán (eds.), Distributions with given marginals and moment problems, Kluwer Academic Publishers, Dordrecht, 1997. MR 1614650
  • 2. E. W. Cheney, Introduction to approximation theory, AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1982) edition. MR 1656150
  • 3. C. M. Cuadras, Theoretical, experimental foundations and new models of factor analysis, Investigación Pesquera, 39 (1972), pp. 163-169 (in Spanish)
  • 4. C. M. Cuadras, First principal component characterization of a continuous random variable, Universitat de Barcelona, Institut de Matemàtica, Mathematics Preprint Series, No 327 (2003)
  • 5. G. Dall’Aglio, S. Kotz, and G. Salinetti (eds.), Advances in probability distributions with given marginals, Mathematics and its Applications, vol. 67, Kluwer Academic Publishers Group, Dordrecht, 1991. Beyond the copulas; Papers from the Symposium on Distributions with Given Marginals held in Rome, April 1990. MR 1215942
  • 6. W. Feller, An Introduction to Probability Theory and Its Applications. Wiley, New-York, 1986
  • 7. Jacek Jakubowski and Stanisław Kwapień, On multiplicative systems of functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 9, 689–694 (English, with Russian summary). MR 600722
  • 8. J. Komlós, A central limit theorem for multiplicative systems, Canad. Math. Bull. 16 (1973), 67–73. MR 0324753
  • 9. Paul Koosis, The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, vol. 12, Cambridge University Press, Cambridge, 1988. MR 961844
  • 10. Ju. V. Linnik and Ĭ. V. Ostrovs′kiĭ, Decomposition of random variables and vectors, American Mathematical Society, Providence, R. I., 1977. Translated from the Russian; Translations of Mathematical Monographs, Vol. 48. MR 0428382
  • 11. Sofiya Ostrovska, Uncorrelatedness and correlatedness of powers of random variables, Arch. Math. (Basel) 79 (2002), no. 2, 141–146. MR 1925381, 10.1007/s00013-002-8296-z
  • 12. Sofia Ostrovska, A scale of degrees of independence of random variables, Indian J. Pure Appl. Math. 29 (1998), no. 5, 461–471. MR 1627847
  • 13. Ludger Rüschendorf, Berthold Schweizer, and Michael D. Taylor (eds.), Distributions with fixed marginals and related topics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 28, Institute of Mathematical Statistics, Hayward, CA, 1996. MR 1485518
  • 14. Jordan Stoyanov, Dependency measure for sets of random events or random variables, Statist. Probab. Lett. 23 (1995), no. 1, 13–20. MR 1333372, 10.1016/0167-7152(94)00089-Q
  • 15. Jordan M. Stoyanov, Counterexamples in probability, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1987. MR 930671
  • 16. Jordan Stoyanov, Krein condition in probabilistic moment problems, Bernoulli 6 (2000), no. 5, 939–949. MR 1791909, 10.2307/3318763
  • 17. Jordan Stoyanov, Sets of binary random variables with a prescribed independence/dependence structure, Math. Sci. 28 (2003), no. 1, 19–27. MR 1995161

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Additional Information

Sofiya Ostrovska
Affiliation: Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey
Email: ostrovskasofiya@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-04-07698-1
Keywords: Uncorrelatedness, independence, uncorrelatedness set, quasianalytic class, characteristic function
Received by editor(s): September 22, 2003
Received by editor(s) in revised form: December 22, 2003
Published electronically: October 18, 2004
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.