  We introduce a class of absorption mechanisms and study the behavior of
real-valued centered random walks with finite variance that do not get
absorbed. In particular, we prove persistence and scaling limit results, which,
in many cases of interests, reduce the analysis of the considered situation to
well understood classical persistence and scaling limit questions. Our results
cover results in Kemperman (1961) and Vysotsky (2015) and can be applied for
many more examples.
