Extremes of Reflecting Gaussian Processes on Discrete Grid

For $\{X(t), t \in G_δ\}$ a centered Gaussian process with stationary increments and a.s. sample paths on a discrete grid $G_δ=\{0,δ,2δ, ...\}$, where $δ>0$, we investigate the stationary reflected process $Q_{δ,X}(t) = \sup\limits_{s\in [t,\infty)\cap G_δ}\big( X(s)-X(t)-c(s-t)\big), \ t \in G_δ$ with $c>0$. We derive the exact asymptotics of $\mathbb P\{\sup\limits_{t\in [0,T]\cap G_δ} Q_{δ,X}(t)>u\} \quad \text{and} \quad \mathbb P\{\inf\limits_{t\in [0,T]\cap G_δ} Q_{δ,X}(t)>u\},$ as $u\to\infty$, with $T>0$. It appears that $φ=\lim_{u\to\infty} \frac{σ^2(u)}{u}$ determines the asymptotics, leading to three qualitatively different scenarios: $φ=0$, $φ\in(0,\infty)$ and $φ=\infty$.

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