Consider real, nonnegative random variables $A$, $B$, and $X$, and define $Z = \exp(-A X)$ and $W = \exp(-B X)$, and also $U = \exp(-X - A)$ and $V = \exp(-X -B)$.

What sorts of minimal sufficient conditions on the dependency structure of $(A, B, X)$ will guarantee that $\sigma_{ZW} = \langle Z W \rangle - \langle Z\rangle \langle W \rangle \ge 0$, or that $\sigma_{UV} = \langle U V \rangle - \langle U\rangle \langle V \rangle \ge 0$?

These will hold for example if $A$ and $B$ are both constant, with zero variance, or both equal, with perfect correlation. They also hold if $X$ is constant, with zero variance, and $A$ and $B$ are positively quadrant dependent (PQD) in the sense of Lehmann. They also hold if the joint probability density for (A,B,X) satisfies the Fortuin-Kasteleyn-Ginibre (FKG) conditions.

Are there weaker conditions under which the non-negativity of either or both of these covariances can be guaranteed—in particular, if $A$ and $B$ are sufficiently positively correlated in some sense?

Consider real, nonnegative random variables $A$, $B$, and $X$, and define $Z = \exp(-A X)$ and $W = \exp(-B X)$, and also $U = \exp(-X - A)$ and $V = \exp(-X -B)$.

What sorts of minimal sufficient conditions on the dependency structure of $(A, B, X)$ will guarantee that $\sigma_{ZW} = \langle Z W \rangle - \langle Z\rangle \langle W \rangle \ge 0$, or that $\sigma_{UV} = \langle U V \rangle - \langle U\rangle \langle V \rangle \ge 0$?

These will hold for example if $A$ and $B$ are both constant, with zero variance, or both equal, with perfect correlation. They also hold if $X$ is constant, with zero variance, and $A$ and $B$ are positively quadrant dependent (PQD) in the sense of Lehmann. They also hold if the joint probability density for $(A,B,X)$ satisfies the Fortuin-Kasteleyn-Ginibre (FKG) conditions.

Are there weaker conditions under which the non-negativity of either or both of these covariances can be guaranteed—in particular, if $A$ and $B$ are sufficiently positively correlated in some sense?