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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 \[ U(\xi _1,\xi _2):=\left \{(j,l)\in \textbf {N}^2:\textbf {E}\left (\xi _1^j\xi _2^l\right )=\textbf {E}\left (\xi _1^j\right )\textbf {E}\left ( \xi _2^l\right )\right \}\] 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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Additional Information

Sofiya Ostrovska
Affiliation: Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey
MR Author ID: 329775

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.