对数正态平稳Brown-Resnick过程的期望下确界与持续概率
Expected Infimum and persistence probabilities of Log-Normal Stationary Brown-Resnick Processes
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中文总结 AI 辅助
本文研究对数正态Brown-Resnick平稳过程的下确界期望与持续概率的渐近行为，对一大类过程（含分数布朗运动）导出精确或对数渐近，提供尖锐界与比较原理，并揭示下确界与上确界行为的差异。
中文摘要 AI 辅助
我们研究了对数正态Brown-Resnick平稳过程的下确界期望的渐近行为，这类过程在Gaussian过程和max-stable过程的极值研究中自然出现。具体而言，我们分析了泛函 $$\mathcal{G}_V(T) = \mathbb{E}\left\{\inf_{t \in [0,T]}e^{ \sqrt{2}V(t)-\sigma^2_V(t)} \right\}, \quad T>0, $$ 其中 $V$ 是具有平稳增量、连续样本路径和方差 $\sigma_V^2$ 的中心化Gaussian过程，以及一个密切相关的问题：持续概率 $$p_V(T,C)=\mathbb{P}\left\{\inf_{t\in [0,T]} (\sqrt{2} V(t)- \sigma^2_V(t)) > C\right\}$$ 对于某个常数 $C<0$ 的衰减速率。对于 $\mathcal{G}_V(T)$ 和 $p_V(T,C)$，我们在 $T \to \infty$ 时，对一大类过程 $V$ 导出了精确渐近，包括Hurst参数 $H \in (1/2,1]$ 的分数布朗运动。对于后者，在短程依赖区间 $H \in (0, 1/2]$ 的行为显著不同，且通常更为精细；我们为包含此情形的一族过程找到了对数渐近。我们的结果提供了尖锐的界和比较原理，并凸显了此类过程的下确界与上确界泛函行为之间的对比。还讨论了离散时间类比以及与Pickands常数的联系。
英文摘要
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.
发表机构
- University of Wrocław（弗罗茨瓦夫大学）
- University of Lausanne（洛桑大学）
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