  Given a power-bounded linear operator T in a Banach space and a probability F
on the non-negative integers, one can form a `subordinated' operator S = \sum_k
F(k) T^k. We obtain asymptotic properties of the subordinated discrete
semigroup (S^n: n=1,2,...) under certain conditions on F. In particular, we
study probabilities F with the property that S satisfies the Ritt resolvent
condition whenever T is power-bounded. Examples and counterexamples of this
property are discussed. The hypothesis of power-boundedness of T can sometimes
be replaced by the weaker Kreiss resolvent condition.
