Uncorrelatedness sets for random variables with given distributions

Author:
Sofiya Ostrovska

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1239-1246

MSC (2000):
Primary 60E05

DOI:
https://doi.org/10.1090/S0002-9939-04-07698-1

Published electronically:
October 18, 2004

MathSciNet review:
2117227

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Abstract | References | Similar Articles | Additional Information

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

Email:
ostrovskasofiya@yahoo.com

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.