Uniform Window Discrepancy Bound for a Counting Process with Poisson Increments on the Integer Grid

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $n$ and $m$ be natural numbers, and let $\mathsf{P}=(\mathsf{P}_u)_{u\ge0}$ be a stochastic process on $(\Omega,\mathcal{F},P)$, all of whose paths $u\mapsto\mathsf{P}_u(\omega)$ are counting paths, such that for all integers $i,i'$ with $0\le i<i'\le n$ the increment $\mathsf{P}_{i'}-\mathsf{P}_{i}$ has the Poisson distribution with parameter $i'-i$ (in the sense of Distribution and Cumulative Distribution Function of a Random Variable). For instance, a homogeneous Poisson process with rate $1$ all of whose paths are counting paths has this property by condition 3 of Stochastic Process, Independent Increments, and Inhomogeneous Poisson Process, its mean function being $\Lambda(t)=t$. 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

$$P(\Omega\setminus G)\ \le\ 2\,(n+1)(m+3)\,\exp\bigl(-\varpi_{m+2}(x)\bigr),$$

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

$$\bigl|\mathsf{P}_{u'}(\omega)-\mathsf{P}_{u}(\omega)-(u'-u)\bigr|\ \le\ x+2 .$$