  We study quasi-stationary distributions and quasi-limiting behavior of Markov
chains in general reducible state spaces with absorption. We propose a set of
assumptions dealing with particular situations where the state space can be
decomposed into three subsets between which communication is only possible in a
single direction. These assumptions allow us to characterize the exponential
order of magnitude and the exact polynomial correction, called polynomial
convergence parameter, for the leading order term of the semigroup for large
time. They also provide explicit convergence speeds to this leading order term.
We apply these results to general Markov chains with finitely or denumerably
many communication classes using a specific induction over the communication
classes of the chain. We are able to explicitely characterize the polynomial
convergence parameter, to determine the complete set of quasistationary
distributions and to provide explicit estimates for the speed of convergence to
quasi-limiting distributions in the case of finitely many communication
classes. We conclude with an application of these results to the case of
denumerable state spaces, where we are able to prove that, in general, there is
existence of a quasi-stationary distribution without assuming irreducibility
before absorption. This actually holds true assuming only aperiodicity, the
existence of a Lyapunov function and the existence of a point in the state
space from which the return time is finite with positive probability.
