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Random Walks With Stochastically Bounded Increments: Foundations and Characterization Results

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We consider random walks S, adapted to a filtration \(\cal F_N\), whose conditional increment distribution functions are bounded from above and/or below by an integrable distribution function. A further stability condition on the conditional increment means is also introduced. Such random walks share a number of properties with those having i.i.d. increments, in particular a uniform law of large numbers. In this paper, which is accompanied by a second one on renewal theory, we derive their basic properties and give equivalent characterizations in terms of certain drift constants which are introduced before and of great importance for a renewal theoretic analysis.

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Alsmeyer, G. Random Walks With Stochastically Bounded Increments: Foundations and Characterization Results. Results. Math. 19, 22–45 (1991). https://doi.org/10.1007/BF03322413

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