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

This article is cited in 3 scientific papers (total in 3 papers)

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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Received: 20.08.2002
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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
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\by K.~Debicki
\paper Some properties of generalized Pickands constants
\jour Teor. Veroyatnost. i Primenen.
\yr 2005
\vol 50
\issue 2
\pages 396--404
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\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}
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    Citing articles on Google Scholar: Russian citations, English citations
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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.  crossref  mathscinet  zmath  isi  scopus
    2. Dębicki K., Kisowski P., “A note on upper estimates for Pickands constants”, Statist. Probab. Lett., 78:14 (2008), 2046–2051  crossref  mathscinet  zmath  isi  scopus
    3. Dieker A.B., Yakir B., “On Asymptotic Constants in the Theory of Extremes For Gaussian Processes”, Bernoulli, 20:3 (2014), 1600–1619  crossref  mathscinet  zmath  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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