Upper bound on expectation value of the product of two random variables Hello,
I am trying to find an upper bound on the expectation value of the product of two random variables.
So suppose x, y are two non-independent random variables, given that I know the distribution of x p(x) and the distribution of y q(y), how can I find an upper bound on E[|x * y |] that is a function of p and q?
I know that Holder's inequality gives an upper bound to my problem in terms of moments of x and y, but this is a poor bound for the problem that I am considering. 
Thank you!
Best
Michele
 A: I'll assume that $X$ and $Y$ are non-negative random variables. Let $F_X$ be the cumulative distribution function of $X$ (that is $F_X(t)=\mathbb P(X\le t)$) and $F_Y$ be the cumulative distribution function of $Y$. 
In your notation, probably $F_X(t)=\int_0^t p(s)\,ds$ and $F_Y(y)=\int_0^t q(t)\,dt$.
Now define two functions on $[0,1]$: $g_X(x)=\sup\lbrace t\colon \mathbb P(X\le t)\le x\rbrace $
and similarly $g_Y(x)=\sup\lbrace t\colon \mathbb P(Y\le t)\le x\rbrace$. These functions are the increasing rearrangements of $X$ and $Y$. That is these are non-decreasing functions with the property that $m\lbrace x\colon g_X(x)\le t\rbrace =\mathbb P(X\le t)$ and $m\lbrace x\colon g_Y(x)\le t\rbrace = \mathbb P(Y\le t)$. 
Now the largest possible value of $\mathbb E XY$ given the distributions is $\int_0^1 g_X(t)g_Y(t)\ dt$. Intuitively the reason for this is that the largest value for the expectation is obtained when the largest values of $X$ are multiplied by the largest values of $Y$. Slightly more precisely imagine you've arranged the $X$ values from largest to smallest. Think of these as "weights" for the $Y$ values. Obviously you get the biggest integral if you weight the big $Y$ values with the biggest weights. 
A: I would try yo apply Hoeffding's Lemma, who used his result to identify the bivariate cdfs with given marginal cdfs that minimize or maximize correlation. Let $(X,Y)$ be a random vector with bivariate cdf $H$, let $F$ and $G$ be their marginal cdfs, respectively. It is well known that a sharp upper bound for $H(x,y)$ is $\min(F(x),G(y))$. By Hoeffding's Lemma we get that 
$$E(XY)\leq E(X)E(Y)+\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\big[\min(F(x),G(y))-F(x)G(y)\big]dxdy$$
