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

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 $P({\max }_{[0,1{]}^2}\mathrm{\mid }{W}_n-W\mathrm{\mid }>\alpha n^{-\beta })=o(n^{-q})$ for every $\beta <\lambda \mathrm{/}2$, where $\lambda \in (0,\frac{1}{5})$ is the parameter of the construction; this gives rates $n^{-\beta }$ with $\beta <\frac{1}{10}$. In this paper, we show that the same construction, with $\lambda \ge 1$, satisfies $$P\text{\hspace{0.17em}⁣}(\underset{(s,t)\in [0,1{]}^2}{\max }\mathrm{\mid }{W}_n(s,t)-W(s,t)\mathrm{\mid }>\alpha n^{-1\mathrm{/}2}(\log n{)}^3)=o(n^{-q})\text{for every }q>0,$$ which is the one-parameter rate of Gorostiza and Griego~\cite{GG} up to a factor $(\log n{)}^{1\mathrm{/}2}$. In particular the restriction $\lambda <\frac{1}{5}$ is not needed.