PreprintCC BY 4.0arXiv
Uniform transport processes and the Brownian sheet:the one-parameter rate of convergence is preserved
Abstract
In~\cite{BFR} we constructed processes Wn, built from a family of independent standard Poisson processes, that converge almost surely to the Brownian sheet uniformly on [0,1]2, and in~\cite{Rov} we proved that P(max[0,1]2|Wn−W|>αn−β)=o(n−q) for every β<λ/2, where λ∈(0,1/5) is the parameter of the construction; this gives rates n−β with β<1/10. In this paper, we show that the same construction, with λ≥ 1, satisfies P (max(s,t)∈[0,1]2 |Wn(s,t)−W(s,t)| > α n−1/2(log n)3) = o(n−q) for every q>0, which is the one-parameter rate of Gorostiza and Griego~\cite{GG} up to a factor (log n)1/2. In particular the restriction λ<1/5 is not needed.
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BibTeX
@misc{rovira2026uniform,
title = {{Uniform transport processes and the Brownian sheet:the one-parameter rate of convergence is preserved}},
author = {Carles Rovira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23347},
eprint = {2610.23347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23347},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform transport processes and the Brownian sheet:the one-parameter rate of convergence is preserved
AU - Carles Rovira
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23347
UR - https://arxiv.org/abs/2610.23347
AB - In~\cite{BFR} we constructed processes W_n, built from a family of independent standard Poisson processes, that converge almost surely to the Brownian sheet uniformly on [0,1]^2, and in~\cite{Rov} we proved that P(max_[0,1]^2|W_n−W|>αn^−β)=o(n^−q) for every β<λ/2, where λ∈(0,1/5) is the parameter of the construction; this gives rates n^−β with β<1/10. In this paper, we show that the same construction, with λ≥ 1, satisfies P (max_(s,t)∈[0,1]^2 |W_n(s,t)−W(s,t)| > α n^−1/2(log n)^3) = o(n^−q) for every q>0, which is the one-parameter rate of Gorostiza and Griego~\cite{GG} up to a factor (log n)^1/2. In particular the restriction λ<1/5 is not needed.
ER -