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Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function
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Foster, James and Habermann, Karen (2023) Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function. Combinatorics, Probability and Computing, 32 (3). pp. 370-397. doi:10.1017/S096354832200030X ISSN 0963-5483.
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Official URL: http://dx.doi.org/10.1017/S096354832200030X
Abstract
We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Brownian motion processes, Probabilities, Approximation theory, Functions, Zeta | ||||||||
Journal or Publication Title: | Combinatorics, Probability and Computing | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 0963-5483 | ||||||||
Official Date: | May 2023 | ||||||||
Dates: |
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Volume: | 32 | ||||||||
Number: | 3 | ||||||||
Page Range: | pp. 370-397 | ||||||||
DOI: | 10.1017/S096354832200030X | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 3 November 2022 | ||||||||
Date of first compliant Open Access: | 4 November 2022 | ||||||||
RIOXX Funder/Project Grant: |
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