This paper investigates random-shift representations of $\alpha$-homogeneous shift-invariant classes of random fields (rf's) $ K_{\alpha}[ Z]$, which were introduced in \cite{hashorva2021shiftinvariant}. Here $ Z(t),t\in T$ is a stochastically continuous $\mathbb{R}^d$-valued rf with $ T=\mathbb{R}^l$ or $T=\mathbb{Z}^l$. We show that that random-shift representations of interest are obtained by constructing cluster rf's, which play a crucial role in the study of extremes of stationary regularly varying rf's. An important implication of those representations is their close relationship with Rosi\'nski (or mixed moving maxima) representations of max-stable rf's. We show that for a given $ K_{\alpha}[ Z]$ different cluster rf's can be constructed, which is useful for the derivation of new representations of extremal functional indices, Rosi\'nski representations of max-stable rf's as well as for random-shift representations of shift-invariant tail measures.
翻译:本文对随机字段( rf's) $K ⁇ alpha} [ Z] 的随机临时变化类别表示进行了调查, 这些随机变化类别在 \ cite{hashorva2021 shiftinevarient} 中引入了 。 Z( t), t\ in T$ 是一个连续连续的 $mathbb{R ⁇ d$- 价值为 rf 的随机变化类别, 以 $ T ⁇ mathbb{R ⁇ l$ 或$T ⁇ mathbb ⁇ l$ 。 我们表明, 随机变化组合表示有兴趣的表示可以通过建立 rof's' 来获得。 这些表示的一个重要含义是它们与 Rosi\ nski ( 或混合移动最大值) 的 最大代表关系。 我们显示, 对于给定的 $ K ⁇ alpha} [ Z] 和 不同的组 rf' 。 能够构建一个随机变化表示兴趣的组合, 有助于将 新的功能变化表示作为 最上位 的 的 状态 的 代表 。