  We are interested in the asymptotic behavior of Markov chains on the set of
positive integers for which, loosely speaking, large jumps are rare and occur
at a rate that behaves like a negative power of the current state, and such
that small positive and negative steps of the chain roughly compensate each
other. If $X_{n}$ is such a Markov chain started at $n$, we establish a limit
theorem for $\frac{1}{n}X_{n}$ appropriately scaled in time, where the scaling
limit is given by a nonnegative self-similar Markov process. We also study the
asymptotic behavior of the time needed by $X_{n}$ to reach some fixed finite
set. We identify three different regimes (roughly speaking the transient, the
recurrent and the positive-recurrent regimes) in which $X_{n}$ exhibits
different behavior. The present results extend those of Haas & Miermont who
focused on the case of non-increasing Markov chains. We further present a
number of applications to the study of Markov chains with asymptotically zero
drifts such as Bessel-type random walks, nonnegative self-similar Markov
processes, invariance principles for random walks conditioned to stay positive,
and exchangeable coalescence-fragmentation processes.
