In this thesis, we introduce a new class of set-valued random processes called random set-walk, which is an extension of the classical random walk that takes into account both the nonhomogeneity of the walk's environment, and the additional factor of nondeterminism in the choices of such environments. We also lay down the basic framework for studying random set-walks. We define the notion of a characteristic tuple as a 4-tuple of first-exit probabilities which characterizes the behaviour of a random walk in a nonhomogeneous environment, and a characteristic tuple set as its analogue for a random set-walk. We prove several properties of random set-walks and characteristic tuples, from which we derive our main result: the long-run behaviour of a sequence of random set-walks, relative to the endpoints of the walks, converges as the length of the walks tend to infinity.
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