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On range and local time of many-dimensional submartingales

Menshikov, M.V.; Popov, S.Yu


S.Yu Popov


We consider a discrete-time process adapted to some filtration which lives on a (typically countable) subset of ℝ d , d≥2. For this process, we assume that it has uniformly bounded jumps, and is uniformly elliptic (can advance by at least some fixed amount with respect to any direction, with uniformly positive probability). Also, we assume that the projection of this process on some fixed vector is a submartingale, and that a stronger additional condition on the direction of the drift holds (this condition does not exclude that the drift could be equal to 0 or be arbitrarily small). The main result is that with very high probability the number of visits to any fixed site by time n is less than n 1 2 −δ for some δ>0. This in its turn implies that the number of different sites visited by the process by time n should be at least n 1 2 +δ .


Menshikov, M., & Popov, S. (2014). On range and local time of many-dimensional submartingales. Journal of Theoretical Probability, 27(2), 601-617.

Journal Article Type Article
Online Publication Date Jul 3, 2012
Publication Date Jun 1, 2014
Deposit Date May 13, 2014
Publicly Available Date May 15, 2014
Journal Journal of Theoretical Probability
Print ISSN 0894-9840
Electronic ISSN 1572-9230
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 27
Issue 2
Pages 601-617
Keywords Strongly directed submartingale, Lyapunov function, Exit probabilities, 60G42, 60J10.


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