The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S_n)_n0 on ^d there exists a smallest measurable subgroup of ^d, called minimal subgroup of (S_n)_n0, such that P(S_n)=1 for all n1. can be defined as the set of all x^d for which the difference of the time averages n^{–1 n}_k=1P(S_k) and n^{–1 n}_k=1P(S_k+x) converges to 0 in total variation norm as n. The related subgroup * consisting of all x^d for which lim_n P(S_n)–P(S_n+x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S_n)_n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on ^d. The main result shows that, up to regular linear transformations, the only subgroups of ^d admitting a quasi-invariant measure are those of the form ₁×...×_k×^l–k×{0}^d–l, 0kld, with ₁,...,_k being countable subgroups of . The proof is based on a result recently proved by Kharazishvili⁽³⁾ which states no uncountable proper subgroup of admits a quasi-invariant measure.