The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S n ) n>0 on Rd there exists a smallest measurable subgroup G of Rd , called minimal subgroup of (S n ) n > 0, such that P(S n (element of) G)=1 for all n > 1. G can be defined as the set of all x (element of) Rd for which the difference of the time averages n –1 (sigma)n k=1 P(S k (element of).) and n –1 (sigma)n k=1 P(S k +x(element of).) converges to 0 in total variation norm as n-(infinity). The related subgroup G* consisting of all x(element of) Rd for which lim n-(infinity) |P(S n (element of).)–P(S n +x(element of).)|=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 Rd . The main result shows that, up to regular linear transformations, the only subgroups of Rd admitting a quasi-invariant measure are those of the form G’1×...× G’k × Rl–k ×{0} d–l , 0