  We suggest a method for constructing positive harmonic functions for a wide
class of transition kernels on $Z^+$. We also find natural conditions under
which these functions have positive finite limits at infinity. Further, we
apply our results on harmonic functions to asymptotically homogeneous Markov
chains on $Z^+$ with asymptotically negative drift. More precisely, assuming
that Markov chain satisfy Cram\'er's condition, we study the tail asymptotics
of the stationary distribution. In particular, we clarify the influence of the
rate of convergence of jumps of the chain towards the limiting distribution.
