  This paper studies the light-tailed asymptotics of the stationary tail
probability vectors of a Markov chain of M/G/1 type. Almost all related studies
have focused on the typical case, where the transition block matrices in the
non-boundary levels have a dominant impact on the decay rate of the stationary
tail probability vectors and their decay is aperiodic. In this paper, we study
not only the typical case but also atypical cases such that the stationary tail
probability vectors decay periodically and/or their decay rate is determined by
the tail distribution of jump sizes from the boundary level. We derive
light-tailed asymptotic formulae for the stationary tail probability vectors by
locating the dominant poles of the generating function of the sequence of those
vectors. Further we discuss the positivity of the dominant terms of the
obtained asymptotic formulae.
