  In this paper we study the states of Poisson type and infinitely divisible
states on compact quantum groups. Each state of Poisson type is infinitely
divisible, i.e., it admits $n$-th root for all $n\geq1$. The main result is
that on finite quantum groups infinitely divisible states must be of Poisson
type. This generalizes B\"oge's theorem concerning infinitely divisible
measures (commutative case) and Parthasarathy's result on infinitely divisible
positive definite functions (cocommutative case). Two proofs are given.
