  We introduce the space of virtual Markov chains (VMCs) as a projective limit
of the spaces of all finite state space Markov chains (MCs), in the same way
that the space of virtual permutations is the projective limit of the spaces of
all permutations of finite sets. We introduce the notions of virtual initial
distribution (VID) and a virtual transition matrix (VTM), and we show that the
law of any VMC is uniquely characterized by a pair of a VID and VTM which have
to satisfy a certain compatibility condition. Lastly, we study various
properties of compact convex sets associated to the theory of VMCs, including
that the Birkhoff-von Neumann theorem fails in the virtual setting.
