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 Xr. We introduce and study theoscillations of the walk:where ...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 Xr. We introduce and study theoscillations of the walk:where Φ(n) is an increasing sequence of positive numbers. We prove that for each $ 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.展开更多
文摘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 Xr. We introduce and study theoscillations of the walk:where Φ(n) is an increasing sequence of positive numbers. We prove that for each $ 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.
