Convergence of sequences of ordered selections

In this paper, we introduce a convergence notion for ordered selections. Our convergence notion is based on subpermutation densities and convergences of the marginal distributions. A particular case of this convergence is the well-known convergence of permutation sequences. We also introduce a family of probability measures called generalized permutons. We show that in the family of generalized permutons several convergence notions are equivalent. We embed the set of ordered selections to the set of generalized permutons. We prove that any convergent sequence of ordered selections has a limit which is a generalized permuton. Moreover, any generalized permuton is the limit of a sequence of ordered selections. Our results are generalizations of well-known theorems on convergence of permutation sequences to permutons.