We give a setting of the Diaconis–Freedman chain in a multi-dimensional simplex and consider its asymptotic behavior. By using techniques from random iterated function theory and quasi-compact operator theory, we first give some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure and, in particular cases, explicit formulas for the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimension two. Some other settings of the chain are also discussed.
