Expected Infimum and persistence probabilities of Log-Normal Stationary Brown-Resnick Processes

Abstract

We investigate the asymptotics of the expected infimum of log-normal Brown-Resnick stationary processes, a class of processes that arise naturally in the study of extremes of Gaussian processes and max-stable processes. Specifically, we analyse the functional $$\mathcal{G}_V(T) = \mathbb{E}\left\{\inf_{t \in [0,T]}e^{ \sqrt{2}V(t)-σ^2_V(t)} \right\}, \quad T>0, $$ where $V$ is a centered Gaussian process with stationary increments, continuous sample paths and variance $σ_V^2$, and a closely related problem of the decay rate of the persistence probability $$p_V(T,C)=\mathbb{P}\left\{\inf_{t\in [0,T]} (\sqrt{2} V(t)- σ^2_V(t)) > C\right\}$$ for some constant $C<0$. For both $\mathcal{G}_V(T)$ and $p_V(T,C)$ we derive exact asymptotics as $T \to \infty$ for a broad class of processes $V$, including fractional Brownian motion with Hurst parameter $H \in (1/2,1]$. For the latter, the behavior in the short-range dependence regime $H \in (0, 1/2]$ is markedly different and, in general, more delicate; we find logarithmic asymptotics for a family of processes that includes this case. Our results provide sharp bounds and comparison principles, and highlight the contrast between the behavior of infimum and supremum functionals for such processes. The discrete-time analogues and connections to Pickands constants are also discussed.