Maxima of normal random vectors: Between independence and complete dependence
Let \(F_{\rho}\) be the distribution function of an \(R^ 2\)-valued Gaussian random vector \((X_ 1,X_ 2)\), where \(X_ 1\) and \(X_ 2\) are standard normal and have correlation \(\rho\). Let \((X_{n1},X_{n2})\) be a sequence of independent random vectors having the common distribution function \(F_{\rho}\). Then the sample maximum \((M_{n1},M_{n2})\), where \(M_{nj}=\max (X_{1j},...,X_{nj}),\) \(j=1,2\), has the distribution function \(F^ n_{\rho}.\) The authors consider the asymptotic behavior of \((M_{n1},M_{n2})\) when \(\rho\) varies with n, and prove that if \((1-\rho (n))\log n\to \lambda^ 2\in [0,\infty]\) as \(n\to \infty\), then \[ F^ n_{\rho (n)}(b_ n+x/b_ n,b_ n+y/b_ n) \to H_{\lambda}(x,y), \] \[ where\quad H_{\lambda}(x,y) = \exp [-\Phi (\lambda +(x- y)/(2\lambda))e^{-y}-\Phi (\lambda +(y-x)/(2\lambda))e^{-x}], \] \(b_ n\) is determined by \(b_ n=n\phi (b_ n)\), \(\phi\) and \(\Phi\) are the density function and the distribution function of N(0,1), respectively. The limiting distribution function \(H_{\lambda}\) is max- stable. The result is extended to \(R^ d\)-valued random variables.
scientific article; zbMATH DE number 4113765