Markov chains on finite groups I (commutative groups)

Abstract

We now turn to state spaces which have the additional structure of a group; implicitly we have already met them, e.g., when prescribing rules like “with equal probability go to i + 1 mod N or to i - 1 mod N” (on {0,..., N-1}). In a group it is possible to move from a state i to a new position by composing i with the elements j of the group, where j is chosen in accordance with a certain probability law which is independent of i (in the preceding example +1 and -1 have been chosen each with probability 1/2).