Uniform Window Discrepancy Bound for the Homogeneous Poisson Process

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $\mathsf{P}=(\mathsf{P}_u)_{u\ge0}$ be a homogeneous Poisson process with rate $1$ on it, all of whose paths $u\mapsto\mathsf{P}_u(\omega)$ are counting paths. Let $n$ and $m$ be natural numbers, let $x>0$ be a real number, write $\exp$ for the exponential function, and let $\varpi_{m+2}(x)=\min\bigl(\tfrac{x^{2}}{4(m+2)},\tfrac{x}{2}\bigr)$ be the exponent of Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution with parameter $m+2$.

Then there is an event $G$ belonging to the $\sigma$-algebra generated by the random variables $\mathsf{P}_0,\mathsf{P}_1,\dots,\mathsf{P}_n$, with

such that for every $\omega\in G$ and all real numbers $u,u'$ with $0\le u\le u'\le n$ and $u'-u\le m$,