Asymptotics of dominated Gaussian maxima (Q1424688)

Let \(\{X_{n}, n\geq1\}\) be a sequence of independent Gaussian random vectors in \(R^{d}, d\geq2\), with common distribution function. The author studies the asymptotic behavior of the probability that for given \(n\in N\) some \(d\)-dimensional Gaussian random vector \(Y_{n}\) dominates the sample maximum \(M_{n}:= (\max_{1\leq i\leq n}X_{i1},\ldots, \max_{1\leq i\leq n}X_{id})\), i.e. \(\text{Pr}\{M_{n}\leq Y_{n}\}\) as \(n\to\infty\). The paper deals with the following cases: \(Y_{n}=a_{n}Z+b_{n}\) and \(Y_{n}=a_{n}X_{1}+b_{n}\) with \(a_{n}\in R_{+}^{d}, b_{n}\in R^{d}\) and \(Z\) is another Gaussian random vector in \(R^{d}\), independent of \(\{X_{n}, n\geq1\}\).