Rio-type inequality for the expectation of products of random variables

In [5], Lehmann gave a simple proof of this identity and used it in his study of some concepts of dependence. This identity was generalized to functions h(X) and g(Y) with E[h2(X)] <∞ and E[g2(Y)] <∞ and with finite derivatives h′(·) and g′(·) by Newman [6]. Multidimensional versions of these results were proved by Block and Fang [1], Yu [13], and more recently by Prakasa Rao [7]. Related covariance identities for exponential and other distributions are given by Prakasa Rao in [9, 10]. Suppose that is a sub-σ-algebra of and Y is measurable with respect to . Let σ(X) be the sub-σ-algebra generated by the random variable X . Define α( ,X) = sup{∣∣P(A∩B)−P(A)P(B)∣∣, A∈ , B ∈ σ(X)}. (1.2)