  We study scaling limits of non-increasing Markov chains with values in the
set of non-negative integers, under the assumption that the large jump events
are rare and happen at rates that behave like a negative power of the current
state. We show that the chain starting from $n$ and appropriately rescaled,
converges in distribution, as $n\rightarrow \infty$, to a non-increasing
self-similar Markov process. This convergence holds jointly with that of the
rescaled absorption time to the time at which the self-similar Markov process
reaches first 0. We discuss various applications to the study of random walks
with a barrier, of the number of collisions in $\Lambda$-coalescents that do
not descend from infinity and of non-consistent regenerative compositions.
Further applications to the scaling limits of Markov branching trees are
developed in our paper, Scaling limits of Markov branching trees, with
applications to Galton--Watson and random unordered trees (2010).
