  A rigorous and largely self-contained account of (a) the bread-and-butter
concepts and techniques in Markov chain theory and (b) the long-term behaviour
of chains. As much as possible, the treatment is probabilistic instead of
analytical (I stay away from semigroup theory). Personally, I tend to find that
the intuition lies with the former and not the latter. This manuscript is
geared towards those interested in the use of Markov chains as models of
real-life phenomena. For this reason, I focus on the type of chains most
commonly encountered in practice (time-homogeneous, minimal, and
right-continuous in the discrete topology) and choose a starting point very
familiar to this audience: the (Kendall-) Gillespie Algorithm commonly used to
simulate these chains. In order to keep the prerequisite knowledge and
technical complications to a minimum, I take a 'bare-bones' approach that keeps
the focus on chains (instead of more general processes) to an almost
pathological degree: I use the 'jump and hold' structure of chains extensively;
almost no martingale theory; minimal coupling; no stochastic calculus; and,
even though regeneration and renewal (of course!) feature in the manuscript,
they do so exclusively in the context of chains. I have also taken some extra
steps to avoid imposing certain assumptions encountered in other texts that
sometimes prove to be stumbling blocks in practice (e.g., irreducibility of the
state space, boundedness of test functions and of stopping times). For more
details, see the preface.
