The moment index of minima

For a random variable (RV) X its moment index κ(X) ≡ sup(k: E(X^k) < ∞)\ lies in 0 ≤ κ(X) ≤ ∞; it is a critical quantity and finite for heavy-tailed RVs. The paper shows that κ(min(X, Y)) ≥ κ(X) + κ(Y) for independent non-negative RVs X and Y. For independent non-negative ‘excess’ RVs X_s and Y_s whose distributions are the integrated tails of X and Y, κ(X) + κ(Y) ≤ κ(min(X_s, Y_s))+ 2 ≤ κ(min(X, Y)). An example shows that the inequalities can be strict, though not if the tail of the distribution of either X or Y is a regularly varying function.