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

Let $X(t),t\in R^d$ be a centered Gaussian random field with continuoustrajectories and set $\xi_u(t)= X(f(u)t),t\in R^d$ with $f$ some positivefunction. Classical results establish the tail asymptotics of $P\{\Gamma(\xi_u) > u\}$ as $u\to \infty$ with $\Gamma(\xi_u)= \sup_{t \in [0,T]^d} \xi_u(t),T>0$ by requiring that $f(u) \to 0$ with speed controlled by thelocal behaviour of the correlation function of $X$. Recent research shows thatfor applications more general continuous functionals than supremum should beconsidered and the Gaussian field can depend also on some additional parameter$\tau_u \in K$, say $\xi_{u,\tau_u}(t),t\in R^d$. In this contribution wederive uniform approximations of $P\{ \Gamma(\xi_{u,\tau_u})> u\}$ with respectto $\tau_u$ in some index set $K_u$, as $u\to\infty$. Our main result haveimportant theoretical implications; two applications are already included in[10,11]. In this paper we present three additional ones, namely i) we deriveuniform upper bounds for the probability of double-maxima, ii) we extendPiterbarg-Prisyazhnyuk theorem to some large classes of homogeneous functionalsof centered Gaussian fields $\xi_{u}$, and iii) we show the finiteness ofgeneralized Piterbarg constants.

Uniform tail approximation of homogenous functionals of Gaussian fields