5
$\begingroup$

Suppose $f$ and $g$ are monotonically increasing, and bounded, and let $X$ be a random variable. I want to show $f$, $g$ have positive covariance. I tried to compute it directly but I am not getting anything useful

2 Answers 2

7

I think you can consider $\mathbb{E}[(f(X)-f(Y))(g(X)-g(Y))]$ where $X$ and $Y$ are i.i.d.

3

"It is generally taken for granted that the covariance of two increasing functions of a random variable is positive. The present paper contains an elementary proof of this fact."

  • 2
    The proof presented in this paper is ingenious because it avoids introducing an independent copy of $X$. But the author vastly overstates his case asserting that *The inequality holds indeed, but a proof is difficult to find in the literature; an exception is the book by Schürger [1998; Aufgabe 4.22] who suggests a proof based on the independent product of two probability spaces.* In fact, the proof is arch classical, see for example the very first pages of Hermann Thorisson's paper *Coupling Methods in Probability Theory* in the *Scandinavian Journal of Statistics* dating back from 1995.2011-10-21