Uncorrelatedness sets for random variables with given distributions

Abstract

Let ξ 1 \xi _1 and ξ 2 \xi _2 be random variables having finite moments of all orders. The set \[ U ( ξ 1 , ξ 2 ) := { ( j , l ) ∈ N 2 : E ( ξ 1 j ξ 2 l ) = E ( ξ 1 j ) E ( ξ 2 l ) } U(\xi _1,\xi _2):=\left \{(j,l)\in \textbf {N}^2:\textbf {E}\left (\xi _1^j\xi _2^l\right )=\textbf {E}\left (\xi _1^j\right )\textbf {E}\left ( \xi _2^l\right )\right \} \] is said to be an uncorrelatedness set of ξ 1 \xi _1 and ξ 2 . \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.