Details

Autor(en) / Beteiligte

• KHOSHNEVISAN, Davar
• LEVIN, David A
• MENDEZ-HERNANDEZ, Pedro J

Titel

Exceptional times and invariance for dynamical random walks

Ist Teil von

• Probability theory and related fields, 2006-03, Vol.134 (3), p.383-416

Ort / Verlag

Heidelberg: Springer

Erscheinungsjahr

2006

Quelle

EBSCOhost Business Source Ultimate

Beschreibungen/Notizen

• Benjamini et al. (2003) introduced the dynamical walk S n (t)=X 1(t)+···+X n (t), and proved among other things that the LIL holds for n[rightwards arrow from bar]S n (t) for all t. In other words, the LIL is dynamically stable. Subsequently (2004b), we showed that in the case that the X i (0)'s are standard normal, the classical integral test is not dynamically stable. Presently, we study the set of times t when n[rightwards arrow from bar]S n (t) exceeds a given envelope infinitely often. Our analysis is made possible thanks to a connection to the Kolmogorov [varepsilon]-entropy. When used in conjunction with the invariance principle of this paper, this connection has other interesting by-products some of which we relate. In addition, we extend a result of Benjamini et al. (2003) by proving that if the X i (0)'s are lattice, mean-zero variance-one, and possess (2+[varepsilon]) finite absolute moments for some [varepsilon]>0, then the recurrence of the origin is dynamically stable. To prove this we derive a gambler's ruin estimate that is valid for all lattice random walks that have mean zero and finite variance. We believe the latter may be of independent interest. [PUBLICATION ABSTRACT]