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Uncorrelatedness sets for random variables with given distributions

Author(s): Sofiya Ostrovska
Journal: Proc. Amer. Math. Soc. 133 (2005), 1239-1246.
MSC (2000): Primary 60E05
Posted: October 18, 2004
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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.

References:

1.
V. Benes, J. Stepan, (eds.) (1997) Distributions with Given Marginals and Moment Problems. Kluwer Academic Publishers MR 1614650 (98k:60003)

2.
E.W. Cheney (1982) Introduction to Approximation Theory. $2^{\rm nd}$ Ed. Chelsea Publishig Company, N.Y. MR 1656150 (99f:41001)

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, G. Salinetti, (eds.) (1991) Advances in Probability Distributions with Given Marginals. Kluwer Academic Publishers. MR 1215942 (93k:60004)

6.
W. Feller, An Introduction to Probability Theory and Its Applications. Wiley, New-York, 1986

7.
A. Jakubowski, S. Kwapien, On multiplicative systems of functions, Bull. Acad. Polon. Sci. Sr. Sci. Math. 27 (1979), (9), pp. 689-694. MR 0600722 (82c:60014)

8.
J. Komlòs, A central limit theorem for multiplicative systems. Canad. Math. Bull., 16 (1973), pp. 67-73. MR 0324753 (48:3102)

9.
P. Koosis, The Logarithmic Integral, Vol. I, Cambridge: Cambridge University Press, 1988. MR 0961844 (90a:30097)

10.
Ju. V. Linnik, I. V. Ostrovskii, Decomposition of Random Variables and Vectors, AMS, Providence, R. I., 1977. MR 0428382 (55:1404)

11.
S. Ostrovska, Uncorrelatedness and correlatedness of powers of random variables, Arch. der Math. 79 (2002), pp. 141-146. MR 1925381 (2003g:60026)

12.
S. Ostrovska, A Scale of Degrees of Independence of Random Variables. Indian J. of Pure and Applied Math., 29 (1998), (5), pp. 461-471. MR 1627847 (99i:60029)

13.
L. Rüschendorf, B. Schweitzer, M. D. Taylor (eds.) Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics. Lecture Notes - Monograph Series, Hayward, California, 1996. MR 1485518 (98g:60004)

14.
J. Stoyanov, Dependency Measure for Sets of Random Events or Random Variables. Statist. & Prob. Letters (1995) 23, pp. 13-20. MR 1333372 (96c:60003)

15.
J. Stoyanov, Counterexamples in Probability. 2nd Ed. Wiley, Chichester - New York, 1997 MR 0930671 (89f:60001)

16.
J. Stoyanov, Krein condition in probabilistic moment problems, Bernoulli 6, No.5(2000), 939-949. MR 1791909 (2001i:44014)

17.
J. Stoyanov, Sets of Binary Random Variables with a Prescribed Independence/Dependence Structure. The Mathematical Scientist, vol. 28, no. 1(2003), 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: 10.1090/S0002-9939-04-07698-1
PII: S 0002-9939(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
Posted: October 18, 2004
Communicated by: Richard C. Bradley
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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