Consider the sequences of random variables $\{X_i\}_{i=1}^n$ and $\{Y_i\}_{i=1}^n$, as well as the corresponding sequence of products, $\{X_i Y_i\}_{i=1}^n$. All $X_i$ share the same mean value, $\mathrm{E}[X_i]= \mathrm{E}[X]$, and likewise, $\mathrm{E}[Y_i]= \mathrm{E}[Y]$. The random variables $X_i$ and $X_j$ are dependent with covariance $$ \mathrm{cov}(X_i, X_j) = \mathrm{E}[X_iX_j]-\mathrm{E}[X]^2, $$ and the same is true for $Y_i$ and $Y_j$, with corresponding covariance $$ \mathrm{cov}(Y_i, Y_j) = \mathrm{E}[Y_iY_j]-\mathrm{E}[Y]^2. $$ $X_i$ and $Y_j$, however, are independent, so that $\mathrm{cov}(X_i, Y_j) = 0$. Assume now that $$ \sum_{i\neq j} \mathrm{cov}(X_i, X_j) \leq 0 $$ and $$ \sum_{i\neq j} \mathrm{cov}(Y_i, Y_j) \leq 0 $$

The question then---given the above information---is the following always true: $$ \sum_{i\neq j} \mathrm{cov}(X_i Y_i, X_j Y_j) \leq 0? $$

This inequality can be reformulated. First, we write the covariance of $X_i Y_i$ and $X_j Y_j$ as $$ \mathrm{cov}(X_i Y_i, X_j Y_j) = \mathrm{E}[X_i X_j Y_i Y_j ]- E[X_i Y_i ] E[X_j Y_j ] $$ Because of independency, $$ \mathrm{E}[X_i X_j Y_i Y_j ] = \mathrm{E}[X_i X_j ] \mathrm{E}[Y_i Y_j ] $$ and therefore $$ \mathrm{cov}(X_i Y_i, X_j Y_j) =\mathrm{E}[X_i X_j ] \mathrm{E}[Y_i Y_j ] - E[X]^2 E[Y]^2 $$ Alternatively, we can use that $$ \mathrm{E}[X_iX_j] = \mathrm{cov}(X_i, X_j) + \mathrm{E}[X]^2 $$
and $$ \mathrm{E}[Y_iY_j] = \mathrm{cov}(Y_i, Y_j) + \mathrm{E}[Y]^2 $$
to write $$ \mathrm{cov}(X_i Y_i, X_j Y_j) = \mathrm{cov}(X_i,X_j) \mathrm{cov}(Y_i,Y_j) + \mathrm{cov}(X_i,X_j) \mathrm{E}[Y]^2 + \mathrm{cov}(Y_i,Y_j) \mathrm{E}[X]^2 $$ The above inequality can therefore be rewritten as

$$ \sum_{i\neq j} \mathrm{E}[X_i X_j ] \mathrm{E}[Y_i Y_j ] \leq n (n-1) E[X]^2 E[Y]^2 $$ or $$ \sum_{i\neq j} \mathrm{cov}(X_i,X_j) \mathrm{cov}(Y_i,Y_j) \leq \mathrm{E}[Y]^2 \sum_{i\neq j}|\mathrm{cov}(X_i,X_j)| + \mathrm{E}[X]^2 \sum_{i\neq j}|\mathrm{cov}(Y_i,Y_j)| $$

Do our starting assumptions provide enough information to guarantee these inequalities, or do we need to add further constraints on the properties of $X_i$ and $Y_i$?

Thanks for any help!