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Zbl 1130.60079
Khoshnevisan, Davar; Levin, David A.; Méndez-Hernández, Pedro J.
Exceptional times and invariance for dynamical random walks. (English)
[J] Probab. Theory Relat. Fields 134, No. 3, 383-416 (2006). ISSN 0178-8051; ISSN 1432-2064/e

In a dynamical random walk, each increment is carrying an independent Poisson clock, and is independently resampled whenever this clock strikes. A property of the random walk is called dynamically stable, if it holds almost surely, simultaneously for all times. This concept turns out to be extremely interesting, because some properties (like the law of the iterated logarithm) turn out to be dynamically stable, while others (like the integral test for the upper class) are not. In the latter case it is natural to ask for the Hausdorff dimension of the set of exceptional times, at which a given function (which satisfies the integral test) fails to be in the upper class. \par The main result of the present paper solves this problem. The paper also provides some finer characterizations of this set, making use of the Kolmogorov $\epsilon$-entropy. A further interesting result is that, under suitable conditions, the recurrence to the origin of a mean zero random walk on the integers is dynamically stable. To obtain these results, the authors prove a lot of spin-off results, many of which are of independent interest. This interesting paper is well worth a closer look!
[Peter Mörters (Bath)]

MSC 2000:: *60J25 Markov processes with continuous parameter
60J05 Markov processes with discrete parameter
60F05 Weak limit theorems
28A78 Hausdorff measures
28C20 Set functions and measures and integrals in infinite-dim. spaces
60F17 Functional limit theorems

Keywords: dynamical walks; Hausdorff dimension; upper functions; Kolmogorov entropy; Ornstein-Uhlenbeck process

Citations: Zbl 1021.60055

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