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 𝒢V(T) = 𝔼{inft ∈ [0,T]e √2V(t)−σ2V(t) }, T>0, where V is a centered Gaussian process with stationary increments, continuous sample paths and variance σV2, and a closely related problem of the decay rate of the persistence probability pV(T,C)=ℙ{inft∈ [0,T] (√2 V(t)− σ2V(t)) > C} for some constant C<0. For both 𝒢V(T) and pV(T,C) we derive exact asymptotics as T → ∞ for a broad class of processes V, including fractional Brownian motion with Hurst parameter H ∈ (1/2,1]. For the latter, the behavior in the short-range dependence regime H ∈ (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.
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BibTeX
@misc{dȩbicki2026expected,
title = {{Expected Infimum and persistence probabilities of Log-Normal Stationary Brown-Resnick Processes}},
author = {Krzysztof Dȩbicki and Enkelejd Hashorva and Svyatoslav Novikov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12321},
eprint = {2609.12321},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12321},
}
RIS
TY - UNPB
TI - Expected Infimum and persistence probabilities of Log-Normal Stationary Brown-Resnick Processes
AU - Krzysztof Dȩbicki
AU - Enkelejd Hashorva
AU - Svyatoslav Novikov
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12321
UR - https://arxiv.org/abs/2609.12321
AB - 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 𝒢_V(T) = 𝔼{inf_t ∈ [0,T]e^ √2V(t)−σ^2_V(t) }, 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)=ℙ{inf_t∈ [0,T] (√2 V(t)− σ^2_V(t)) > C} for some constant C<0. For both 𝒢_V(T) and p_V(T,C) we derive exact asymptotics as T → ∞ for a broad class of processes V, including fractional Brownian motion with Hurst parameter H ∈ (1/2,1]. For the latter, the behavior in the short-range dependence regime H ∈ (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.
ER -