The moment index of minima

For a random variable (RV) X its moment index κ ( X ) ≡ sup{ κ : E(| X | κ ) < ∞} 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.