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

Source 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 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.