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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 2, Pages 396–404 (Mi tvp118)

Short Communications

Some properties of generalized Pickands constants

K. Debicki

Wroclaw University

Abstract: 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 $\mathcal{H}_{\eta}(T)=\mathbf{E}\exp(\max_{t \in[0,T]}(\sqrt{2} \eta(t)-\mathrm{Var}(\eta(t))))$ 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 $\alpha/2$.

Keywords: exact asymptotics, extremes, fractional Brownian motion, Gaussian process, generalized Pickands constants.

DOI: https://doi.org/10.4213/tvp118

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English version:
Theory of Probability and its Applications, 2006, 50:2, 290–298

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Citation: K. Debicki, “Some properties of generalized Pickands constants”, Teor. Veroyatnost. i Primenen., 50:2 (2005), 396–404; Theory Probab. Appl., 50:2 (2006), 290–298

Citation in format AMSBIB
\Bibitem{Deb05} \by K.~Debicki \paper Some properties of generalized Pickands constants \jour Teor. Veroyatnost. i Primenen. \yr 2005 \vol 50 \issue 2 \pages 396--404 \mathnet{http://mi.mathnet.ru/tvp118} \crossref{https://doi.org/10.4213/tvp118} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2222683} \zmath{https://zbmath.org/?q=an:1089.60035} \elib{http://elibrary.ru/item.asp?id=9153133} \transl \jour Theory Probab. Appl. \yr 2006 \vol 50 \issue 2 \pages 290--298 \crossref{https://doi.org/10.1137/S0040585X97981755} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000238760000009} 

• http://mi.mathnet.ru/eng/tvp118
• https://doi.org/10.4213/tvp118
• http://mi.mathnet.ru/eng/tvp/v50/i2/p396

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This publication is cited in the following articles:
1. Wu Dongsheng, “Generalized pickands constants”, J. Math. Phys., 48:5 (2007), 053513, 9 pp.
2. Dębicki K., Kisowski P., “A note on upper estimates for Pickands constants”, Statist. Probab. Lett., 78:14 (2008), 2046–2051
3. Dieker A.B., Yakir B., “On Asymptotic Constants in the Theory of Extremes For Gaussian Processes”, Bernoulli, 20:3 (2014), 1600–1619
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