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/

Global: Zendy Tools annual

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

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

This paper studies the joint tail asymptotics of extrema of themulti-dimensional Gaussian process over random intervals defined as  $$ P(u):=\mathbb{P}\left\{\cap_{i=1}^n \left(\sup_{t\in[0,\mathcal{T}_i]} (X_{i}(t) +c_i t )>a_i u \right)\right\}, \ \ \ u\to\infty, $$ where $X_i(t),t\ge0$, $i=1,2,\cdots,n,$ are independent centered Gaussian processes withstationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \cdots,\mathcal{T}_n)$ is a regularly varying random vector with positive components,which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$,$a_i>0$, $i=1,2,\cdots,n$. Our result shows that the structure of theasymptotics of $P(u)$ is determined by the signs of the drifts $c_i$'s. We alsodiscuss a relevant multi-dimensional regenerative model and derive thecorresponding ruin probability.

Extrema of multi-dimensional Gaussian processes over random intervals