The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S n ) n≥0 on $${\mathbb{R}}$$ d there exists a smallest measurable subgroup $${\mathbb{G}}$$ of $${\mathbb{R}}$$ d , called minimal subgroup of (S n ) n≥0, such that P(S n ∈ $${\mathbb{G}}$$ )=1 for all n≥1. $${\mathbb{G}}$$ can be defined as the set of all x∈ $${\mathbb{R}}$$ d for which the difference of the time averages n −1 ∑ n k=1 P(S k ∈ċ) and n −1 ∑ n k=1 P(S k +x∈ċ) converges to 0 in total variation norm as n→∞. The related subgroup $${\mathbb{G}}$$ * consisting of all x∈ $${\mathbb{R}}$$ d for which lim n→∞ ‖P(S n ∈ċ)−P(S n +x∈ċ)‖=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S n ) n≥0. In the final section we consider quasi-invariance and admissible shifts of probability measures on $${\mathbb{R}}$$ d . The main result shows that, up to regular linear transformations, the only subgroups of $${\mathbb{R}}$$ d admitting a quasi-invariant measure are those of the form $${\mathbb{G}}$$ ′1×...× $${\mathbb{G}}$$ ′ k × $${\mathbb{R}}$$ l−k ×{0} d−l , 0≤k≤l≤d, with $${\mathbb{G}}$$ ′1,..., $${\mathbb{G}}$$ ′ k being countable subgroups of $${\mathbb{R}}$$ . The proof is based on a result recently proved by Kharazishvili(3) which states no uncountable proper subgroup of $${\mathbb{R}}$$ admits a quasi-invariant measure.