Strict limit types for monotone convolution

We introduce the notion of strict domain of attraction in the context of monotone probability theory and prove that a law has a nonempty strict domain of attraction if and only if it is strictly stable. We also show that the monotonic weak law of large numbers is equivalent to the classical and the free weak laws of large numbers. Finally, a necessary condition for a measure μ to belong to the strict domain of attraction of a nondegenerate strictly stable law is given in terms of the absolute moments of μ.