Max-convolution semigroups and extreme values in limit theorems for the free multiplicative convolution

We investigate relations between additive convolution semigroups and max-convolution semigroups through the law of large numbers for the free multiplicative convolution. Based on these relations, we give a formula related with the Belinschi–Nica semigroup and the max-Belinschi–Nica semigroup. Finally, we give several limit theorems for classical, free and Boolean extreme values.