Abstract
The vertices of an infinite locally finite tree T are labelled by a collection of i.i.d. real random variables {Xσ}σ∈T which defines a tree indexed walk\(S_\sigma = \sum\limits_{\theta< \tau \leqslant \sigma } {X_\tau } \). We introduce and study the oscillations of the walk:
where Φ(n) is an increasing sequence of positive numbers. We prove that for each Φ belonging to a certain class of sequences of different orders, there are ξ′s depending on Φ such that 0<OSCΦ(ξ)<∞. Exact Hausdorff dimension of the set of such ξ′s is calculated. An application is given to study the local variation of Brownian motion. A general limsup deviation problem on trees is also studied.
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This paper was presented in the Fractal Satellite Conference of ICM 2002 in Nanjing.
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Aihua, F. Limsup deviations on trees. Anal. Theory Appl. 20, 113–148 (2004). https://doi.org/10.1007/BF02901437
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DOI: https://doi.org/10.1007/BF02901437
Key words
- limsup deviation
- tree-indexed walk
- oscillation
- Hausdorff dimension
- Brownian motion
- Percolation
- random covering
- indexed martingale
- Peyrière measure