Multi-excited random walks on integers

Abstract

Abstract. We introduce a class of nearest-neighbor integer random walks in random and non-random media, which includes excited random walks considered in the literature. At each site the random walker has a drift to the right, the strength of which depends on the environment at that site and on how often the walker has visited that site before. We give exact criteria for recurrence and transience and consider the speed of the walk.

Positively and negatively excited random walks on integers, with branching processes

“…see the last line on p. 113 and the first line on p. 114 of [Zer05]. Similarly, by symmetry, if inf n≥0 X n = −∞ a.s. then lim inf n→∞ X n n ≥ −1 u − a.s., where (42)…”

“…In all papers mentioned above a (possible) bias introduced by the consumption of a cookie was assumed to be only in one direction, say, positive. The recurrence and transience, strong law of large numbers [Zer05], conditions for positive linear speed [MPV06], [BS08a], and the rates of escape to infinity for transient walks with zero speed [BS08b] are now well understood. Yet some of the methods and facts used in the proofs (for example, comparison with simple symmetric random walks, submartingale property) depend significantly on this "positive bias" assumption.…”

Trading cookies in a gambler’s ruin scenario

“…It is natural to study this type of problem within a game-theoretic framework, where exact features of the reinforcing mechanism are determined through the interaction between the walker and a supplier. This is in contrast to the usual excited or cookie random walk [Antal and Redner 2005;Zerner 2005] (see [Menshikov et al 2012] for an up-to-date review and references), where the walker, as a pricetaker in a large market, has no effect on determining the parameters of the cookie environment.…”

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“…It is natural to study this type of problem within a game-theoretic framework, where exact features of the reinforcing mechanism are determined through the interaction between the walker and a supplier. This is in contrast to the usual excited or cookie random walk [Antal and Redner 2005;Zerner 2005] (see [Menshikov et al 2012] for an up-to-date review and references), where the walker, as a pricetaker in a large market, has no effect on determining the parameters of the cookie environment.…”