Let $\{X_i\}_{i\in\mathbb N}$ be a sequence of random variables identically distributed as $X$, possibly dependent, and $f$ be a measurable function.
I'm trying to show that in general $Cov(f(X_i),f(X_j))\neq Cov(f(X_1),f(X_2)),\forall i\neq j$.
Question: How would you argue that $Cov(f(X_i),f(X_j))= Cov(f(X_1),f(X_2)),\forall i\neq j,$ is false?
Comments: $Cov(X_i,X_j)$ can vary depending on the indices $i,j$, and these covariances seem to affect, somehow, their respective $Cov(f(X_i),f(X_j))$, which should also vary according to $i,j$. From this idea, $Cov(f(X_i),f(X_j))= Cov(f(X_1),f(X_2)),\forall i\neq j,$ would be impossible. That is, if all covariaces $Cov(X_i,X_j), i\neq j,$ are not the same, how could all $Cov(f(X_i),f(X_j))$ be the same? Unfortunately, I cannot formalize this argument. Can you help me?