Decompositions and antiirreducibles in Max-semigroups of bivariate random variables (Q1600630)

Let \(S\subset R^2\) be a semigroup w.r.t. coordinatewise max, i.e. \(x\vee y\in S\) when \(x\in S\) and \(y\in S\). To \(S\) corresponds the semigroup \(M(S)\), with pointwise multiplication, of distribution functions of \(S\)-valued random vectors: when \(X\) and \(Y\) are independent \(S\)-valued random vectors, so is \(X\vee Y\) and \(P(X\vee Y\leq u)= P(X\leq u)P(Y\leq u)\). It is shown that if \(S\) is closed and has a smallest element \(c\), so that \(\delta(c,.)\) is the unit of \(M(S)\), then \(M(S)\) is a so-called Hun semigroup. This implies that any \(F\in M(S)\) has a decomposition \((*)\) \(F= H\prod_i G_i\) analogous to Khinchin's decomposition for ordinary convolution. Here the \(G_i\) are irreducible and \(H\) is anti-irreducible and therefore infinitely divisible. Examples are given of elements of \(M(R^2)\) and \(M((0,\infty)\times (0,\infty))\) that are nondecomposable and of an anti-irreducible \(F\) that is not infinitely divisible. A class of sets \(S\) where \((*)\) does not hold for all elements is derived.