On multivariate extremal processes
Let \({\mathbf X}_{n,N}=(X^{(1)}_{n,N},\dots,X^{(m)}_{n,N})\), \(1 \leq n \leq N\), \(N \geq 1\), be a triangular array of \(m\)-dimensional random vectors. For each \(N\) assume that \({\mathbf X}_{1,N},\dots,{\mathbf X}_{N,N}\) are exchangeable and arrive at time points \(1/N\), \(2/N,\dots,1\), respectively. Now, for a Borel set \(B\) in \((0,1]\), let \(\xi_ N^{(l,k)}(B)=\#\{n:n/N \in B\) and \({\mathbf X}_{n,N}\) has a \(k\)-record in its \(l\)-th component\}. For a fixed \(K>1\), the paper obtains necessary and sufficient conditions for the convergence of the \(mK\)-variate process \(\{\xi_ N^{(l,k)}:1\leq l \leq m;\;1 \leq k \leq K\}\) to a Poisson process with independent components. These conditions are related to the one which states that the probability that the records are simultaneously broken at different components prior to time \(n/N\) approaches to 0 as \(n \to \infty\).
scientific article; zbMATH DE number 445126