Minima and maxima of elliptical arrays and spherical processes (Q358134)

Let \(X_n^{(i)}=_D RA_nU\), \(1\leq i\leq n\), \(n\in\mathbb N\), be independent \(k\)-dimensional random vectors, where the random vector \(U\) is uniformly distributed on the unit sphere of \(\mathbb R^k\), the radial random variable \(R\) is positive, \(R\) and \(U\) are independent, and the sequence of square matrices \(A_n\) satisfies \(\lim_{n\to\infty} c_n((1)-A_nA_n^\intercal)=\Gamma\) for some sequence of constants \(c_n\) tending to infinity, among others.NEWLINENEWLINEThis paper discusses the asymptotic distribution of the vector of componentwise minima of absolute values \(L_n=(L_{n1},\dots,L_{nk})\) and maxima \(M_n=(M_{n1},\dots,M_{nk})\), where \(L_{nj}=\min_{1\leq i\leq n}\left|X_{nj}^{(i)}\right|\) and \(M_{nj}=\max_{1\leq i\leq n}X_{nj}^{(i)}\), \(1\leq j\leq k\), \(n\in\mathbb N\). An application concerns the asymptotics of the maximum and and the minimum of independent spherical processes.