English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Pickands' constants $H_{\alpha}$ appear in various classical limit resultsabout tail probabilities of suprema of Gaussian processes. It is an oftenquoted conjecture that perhaps $H_{\alpha} = 1/\Gamma(1/\alpha)$ for all $0 <\alpha \leq 2$, but it is also frequently observed that this doesn't seemcompatible with evidence coming from simulations.  We prove the conjecture is false for small $\alpha$, and in fact that$H_{\alpha} \geq (1.1527)^{1/\alpha}/\Gamma(1/\alpha)$ for all sufficientlysmall $\alpha$. The proof is a refinement of the "conditioning and comparison"approach to lower bounds for upper tail probabilities, developed in a previouspaper of the author. Some calculations of hitting probabilities for Brownianmotion are also involved.

Pickands' constant $H_{\alpha}$ does not equal $1/\Gamma(1/\alpha)$, for small $\alpha$