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Limsup deviations on trees

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Analysis in Theory and Applications

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:

$$OSC_\Phi (\xi ) = \overline {lim} _{\sigma \to \xi \in \partial T} \frac{{X_\sigma }}{{\Phi \left( {\left| \sigma \right|} \right)}},$$

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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Authors and Affiliations

  1. Départment de Mathématiques, Picardie University, France

    Fan Aihua

  2. Department of Mathematics, Wuhan University, 430072, Wuhan, P. R. China

    Fan Aihua

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  1. Fan Aihua
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Correspondence to Fan Aihua.

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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

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