Dependence

A number of probabilistic approaches to the concept of dependence in stochastic sequences are contrasted. The fundamental idea is a shift transformation. Invariant and remote events and the idea of ergodicity are introduced and the ergodic theorem is proved. The final sections introduce the contrasting idea of dependence on sub-σ‎-fields with the notions of regularity and strong and uniform mixing defined among other variants.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.