Intermediate  dimension of images of sequences under fractional Brownian motion

Kenneth John Falconer

doi:10.1016/j.spl.2021.109300

Intermediate dimension of images of sequences under fractional Brownian motion

Kenneth John Falconer^*

^*Corresponding author for this work

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that the almost sure θ-intermediate dimension of the image of the set F_p ={0, 1,1/2^p,1/3^p,...} under index-h fractional Brownian motion is θ/(ph+θ), a value that is smaller than that given by directly applying the Hölder bound for fractional Brownian motion. In particular this establishes the box-counting dimension of these images.

Original language	English
Article number	109300
Number of pages	6
Journal	Statistics and Probability Letters
Volume	182
Early online date	5 Nov 2021
DOIs	https://doi.org/10.1016/j.spl.2021.109300
Publication status	Published - Mar 2022

Keywords

Fractional Brownian motion
Fractal
Intermediate dimension
Hausdorff dimension
Box-counting dimension

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10.1016/j.spl.2021.109300

Falconer-Brownian-images-SPL-AAM
Copyright © 2021 Elsevier B.V. All rights reserved. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.spl.2021.109300.
Accepted author manuscript, 225 KBLicence: CC BY-NC-ND

Cite this

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title = "Intermediate dimension of images of sequences under fractional Brownian motion",

abstract = "We show that the almost sure θ-intermediate dimension of the image of the set Fp ={0, 1,1/2p,1/3p,...} under index-h fractional Brownian motion is θ/(ph+θ), a value that is smaller than that given by directly applying the H{\"o}lder bound for fractional Brownian motion. In particular this establishes the box-counting dimension of these images.",

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AU - Falconer, Kenneth John

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N2 - We show that the almost sure θ-intermediate dimension of the image of the set Fp ={0, 1,1/2p,1/3p,...} under index-h fractional Brownian motion is θ/(ph+θ), a value that is smaller than that given by directly applying the Hölder bound for fractional Brownian motion. In particular this establishes the box-counting dimension of these images.

AB - We show that the almost sure θ-intermediate dimension of the image of the set Fp ={0, 1,1/2p,1/3p,...} under index-h fractional Brownian motion is θ/(ph+θ), a value that is smaller than that given by directly applying the Hölder bound for fractional Brownian motion. In particular this establishes the box-counting dimension of these images.

KW - Fractional Brownian motion

KW - Fractal

KW - Intermediate dimension

KW - Hausdorff dimension

KW - Box-counting dimension

UR - https://arxiv.org/abs/2108.12306

U2 - 10.1016/j.spl.2021.109300

DO - 10.1016/j.spl.2021.109300

M3 - Article

SN - 0167-7152

VL - 182

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

M1 - 109300

ER -

Intermediate dimension of images of sequences under fractional Brownian motion

Abstract

Keywords

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