Limit Theorems for Random Walks with Absorption

Abstract

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. Our main results serve as a toolkit which allows obtaining persistence and scaling limit results for many different examples in this class. Further, our results reveal new connections between results in Kemperman (The passage problem for a stationary Markov chain. Statistical research monographs, The University of Chicago Press, Chicago, 1961) and Vysotsky (Stoch Processes Appl 125(5):1886–1910, 2015).