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Some properties of generalized Pickands constants. (English) Zbl 1089.60035

Theory Probab. Appl. 50, No. 2, 290-298 (2006) and Teor. Veroyatn. Primen. 50, No. 2, 396-404 (2005).
Summary: We study properties of generalized Pickands constants \({\mathcal H}_{\eta}\), which appear in the extreme value theory of Gaussian processes and are defined via the limit \[ {\mathcal H}_{\eta}=\lim_{T \to\infty}\frac{{\mathcal H}_{\eta}(T)}{T}, \] where \({\mathcal H}_{\eta}(T)={\mathbf E}\exp(\max_{t \in [0,T]} (\sqrt{2}\, \eta(t)-\text{Var}(\eta(t))))\) and \(\eta(t)\) is a centered Gaussian process with stationary increments. We give estimates of the rate of convergence of \({\mathcal H}_{\eta}(T)/T\) to \({\mathcal H}_{\eta}\) and prove that if \(\eta_{(n)}(t)\) weakly converges in \(C([0,\infty))\) to \(\eta(t)\), then under some weak conditions, \(\lim_{n\to\infty}{\mathcal H}_{\eta_{(n)}}={\mathcal H}_{\eta}\). As an application we prove that \(\Upsilon(\alpha)={\mathcal H}_{B_{\alpha/2}}\) is continuous on \((0,2]\), where \(B_{\alpha/2}(t)\) is a fractional Brownian motion with Hurst parameter \(\alpha/2\).

MSC:

60G70 Extreme value theory; extremal stochastic processes
60G15 Gaussian processes
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