  A novel framework for the analysis of observation statistics on time discrete
linear evolutions in Banach space is presented. The model differs from
traditional models for stochastic processes and, in particular, clearly
distinguishes between the deterministic evolution of a system and the
stochastic nature of observations on the evolving system. General Markov chains
are defined in this context and it is shown how typical traditional models of
classical or quantum random walks and Markov processes fit into the framework
and how a theory of quantum statistics ({\it sensu} Barndorff-Nielsen, Gill and
Jupp) may be developed from it. The framework permits a general theory of joint
observability of two or more observation variables which may be viewed as an
extension of the Heisenberg uncertainty principle and, in particular, offers a
novel mathematical perspective on the violation of Bell's inequalities in
quantum models. Main results include a general sampling theorem relative to
Riesz evolution operators in the spirit of von Neumann's mean ergodic theorem
for normal operators in Hilbert space.
