Uniform transport processes and the Brownian sheet:the one-parameter rate of convergence is preserved

Abstract

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 $λ\in(0,\tfrac{1}{5})$ is the parameter of the construction; this gives rates $n^{-β}$ with $β<\tfrac{1}{10}$. In this paper, we show that the same construction, with $λ\geq 1$, satisfies \[ P\!\left(\max_{(s,t)\in[0,1]^2} |W_n(s,t)-W(s,t)| > α\, n^{-1/2}(\log n)^3\right) = o(n^{-q}) \quad\text{for every }q>0, \] which is the one-parameter rate of Gorostiza and Griego~\cite{GG} up to a factor $(\log n)^{1/2}$. In particular the restriction $λ<\tfrac{1}{5}$ is not needed.