Overview
This mathematical study establishes an improved rate of convergence for a sequence of stochastic processes $W_n$ approximating the two-parameter Brownian sheet $W$ on the unit square $[0,1]^2$. The processes $W_n$ are constructed from a family of independent standard Poisson processes. The authors demonstrate that under this construction, the convergence rate matches the classical one-parameter convergence rate established by Gorostiza and Griego, up to a factor of $(\log n)^{1/2}$, while removing earlier parameter constraints.
Research Context
In earlier work referenced as [BFR], a family of processes $W_n$ was constructed using independent standard Poisson processes to achieve almost sure convergence to the Brownian sheet uniformly across $[0,1]^2$. A subsequent study referenced as [Rov] analyzed the tail probabilities of the approximation error, proving that:
$P\left(\max_{[0,1]^2}|W_n - W| > \alpha n^{-\beta}\right) = o(n^{-q})$
for every $\beta < \lambda/2$, where $\lambda \in (0, 1/5)$ denoted the parameter governing the construction. That constraint bounded the admissible rate to $n^{-\beta}$ with $\beta < 1/10$.
Approach
The authors analyze the exact same Poisson-based construction of $W_n$ but evaluate its asymptotic behavior when the parameter satisfies $\lambda \geq 1$, eliminating the prior restriction that required $\lambda < 1/5$.
Findings
For $\lambda \geq 1$, the maximum deviation between the approximating transport process $W_n(s,t)$ and the Brownian sheet $W(s,t)$ over $(s,t) \in [0,1]^2$ satisfies:
$P\left(\max_{(s,t)\in[0,1]^2} |W_n(s,t) - W(s,t)| > \alpha n^{-1/2}(\log n)^3\right) = o(n^{-q})$
for every $q > 0$ and constant $\alpha$.
This result shows that:
- The rate corresponds to the one-parameter rate identified by Gorostiza and Griego [GG] up to a factor of $(\log n)^{1/2}$.
- The previous technical restriction requiring $\lambda < 1/5$ is unnecessary for establishing uniform convergence rates.