  In this paper, we give out a setting of an Diaconis and Freedman's chain in a
multidimensional simplex and consider its asymptotic behavior. By using
techniques in random iterated functions theory and quasi-compact operators
theory, we first give out some sufficient conditions which ensure the existence
and uniqueness of an invariant probability measure. In some particular cases,
we give out explicit formulas of the invariant probability density. Moreover,
we completely classify all behaviors of this chain in dimensional two.
Eventually, some other settings of the chain are discussed.
