  A time-dependent finite-state Markov chain that uses doubly stochastic
transition matrices, is considered. Entropic quantities that describe the
randomness of the probability vectors, and also the randomness of the discrete
paths, are studied. Universal convex polytopes are introduced which contain all
future probability vectors, and which are based on the Birkhoff-von Neumann
expansion for doubly stochastic matrices. They are universal in the sense that
they depend only on the present probability vector, and are independent of the
doubly stochastic transition matrices that describe time evolution in the
future. It is shown that as the discrete time increases these convex polytopes
shrink, and the minimum entropy of the probability vectors in them increases.
These ideas are applied to a sequence of non-selective measurements (with
different projectors in each step) on a quantum system with $d$-dimensional
Hilbert space. The unitary time evolution in the intervals between the
measurements, is taken into account. The non-selective measurements destroy
stroboscopically the non-diagonal elements in the density matrix. This
`hermaphrodite' system is an interesting combination of a classical
probabilistic system (immediately after the measurements) and a quantum system
(in the intervals between the measurements). Various examples are discussed. In
the ergodic example, the system follows asymptotically all discrete paths with
the same probability. In the example of rapidly repeated non-selective
measurements, we get the well known quantum Zeno effect with `frozen discrete
paths' (presented here as a biproduct of our general methodology based on
Markov chains with doubly stochastic transition matrices).
