Stopped Markov chains with stationary occupation times

Let E be a finite set equipped with a group G of bijective transformations and suppose that X is an irreducible Markov chain on E that is equivariant under the action of G. In particular, if E=G with the corresponding transformations being left or right multiplication, then X is a random walk on G. We show that when X is started at a fixed point there is a stopping time U such that the distribution of the random vector of pre-U occupation times is invariant under the action of G. When G acts transitively (that is, E is a homogeneous space), any non-zero, finite expectation stopping time with this property can occur no earlier than the time S of the first return to the starting point after all states have been visited. We obtain an expression for the joint Laplace transform of the pre-S occupation times for an arbitrary finite chain and show that even for random walk on the group of integers mod r the pre-S occupation times do not generally have a group invariant distribution. This appears to contrast with the Brownian analog, as there is considerable support for the conjecture that the field of local times for Brownian motion on the circle prior to the counterpart of S is stationary under circular shifts.