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)

Rio-type inequality Rio [11] proved that Cov(X,Y ) ≤ 2 α(ᏹ,X)/2 0 Q Y (u)Q X (u)du. (1.4) Related results are given in [12, page 9]. These results were generalized by Bradley [2] for a strong-mixing process and by Prakasa Rao [8] for rth-order joint cumulant under rth-order strong mixing. In a recent work, Dedecker and Doukhan [3] proved that (1.5) and obtained an improved version of the above inequality. If X i , 1 ≤ i ≤ n, are positivevalued random variables, it is easy to see that For a proof, see [12, Lemma 2.1, page 35].
We now obtain an improved version of the above inequality following the techniques of Dedecker and Doukhan [3] and Block and Fang [1].

Main result
Let {X i , 1 ≤ i ≤ n} be a sequence of nonnegative random variables defined on a probability space (Ω,Ᏺ,P). Then the random variable X i can be represented in the form where Let H X1,X2,...,Xn (u) = λ x 2 ,...,x n : g X1 x 2 ,...,x n > u , (2.8) where λ is the Lebesgue measure on the space R n−1 + . Hence Observe that from the Fréchet's inequality [4]. Here Q X1 (·) is the generalized inverse of the function (2.13) 10 Rio-type inequality Hence the set is contained in the set In particular, it follows that the Lebesgue measure of the former set is less than or equal to that of the latter. Let denote the Lebesgue measure of the set (2.15). Then We have proved the following inequality. where the functions H,Q * , and G are as defined earlier.

Applications
We now suppose that the random variables {X i , 1 ≤ i ≤ n} are arbitrary but with and define M X1 (·),Q X1 (·),Q * X2,...,Xn , and G X1 accordingly. The following theorem follows by arguments analogous to those given in Section 2. B. L. S. Prakasa Rao 11 Theorem 3.1. Let X i , 1 ≤ i ≤ n, be arbitrary random variables defined on a probability space (Ω,Ᏺ,P). Then where the functions H,Q * , and G are as defined above.
In particular, for n = 2, we have since Q * X = Q X for any univariate random variable X. Furthermore, Therefore, for any two functions Applying Theorem 3.1 for the random variables X 1 − E(X 1 ),X 2 ,...,X n , we get that Observing that G X1 (·) is the inverse of the function M X1 (y) = y 0 Q X1 (t)dt, it follows that Hence we have the following result. 12 Rio-type inequality Theorem 3.2. Let X i , 1 ≤ i ≤ n, be arbitrary random variables defined on a probability space (Ω,Ᏺ,P) with E|X 1 | < ∞ and E|X 1 X 2 ··· X n | < ∞. Then (3.11) holds.
Observe that Q * X = Q X for any univariate random variable X. Let n = 2 in Theorem 3.2. Then Q * X2 = Q X2 and the above result reduces to As a further consequence, we get that we obtain that Then it follows that This inequality is different from the inequality in [12, page 9].
Let f 1 and f 2 be differentiable functions on It is easy to see that B. L. S. Prakasa Rao 13 by the Fubini's theorem.
Observe that and hence Here χ A (·) denotes the indicator function of the set A. Let  An analogous inequality holds by interchanging f 1 (X 1 ) and f 2 (X 2 ):