Maxima of bivariate random vectors: Between independence and complete dependence
Let \(\{(X_{in}, Z_{in})\), \(1\leq i\leq n\), \(n\geq 1\}\) be a triangular array of bivariate random vectors which are independent for each fixed \(n\). Furthermore, let \(M_{n1}= \max_{1\leq i\leq n} X_{in}\) and \(M_{n2}= \max_{1\leq i\leq n} Z_{in}\). Suppose that there are sequences \(\{a_{ni}\}\), \(\{b_{ni}\}\) such that \(P(a_{ni} (M_{ni}- b_{ni})\leq x)\to G_ i(x)\) for \(i=1,2\). \(G_ i\) is assumed to be continuous. The author gives sufficient conditions such that \[ P(a-{n_ 1} (M_{n1}- b_{n1})\leq x,\;a_{n2} (M_{n2}- b_{n2})\leq z) @>d>> G(x,z). \tag{1} \] Among other things it is proved that if \(P(a_{n2} (Z_{1n}- b_{n2})> z\mid a_{n1} (X_{1n}- b_{n1})= x)\to 0\) \((n\to\infty)\) for every continuity point \((x,z)\), then (1) holds with \(G(x,z)= G_ 1(x) G_ 2(z)\). Moreover, he derives necessary conditions for independence and complete dependence (i.e. \(G(x,z)= \min\{ G_ 1(x), G_ 2(z)\}\) of the limit distributions. As a special model he considers the interesting case \(Z_ i= X_ i+ \rho_ n Y_ i\) with \(\rho_ n>0\). The variables \(\{X_ i\}\) and \(\{Y_ i\}\) are assumed to be i.i.d. and independent of each other. If \(F_ X\) and \(F_ Y\) belong to a domain of attraction of the Fréchet distribution, it is shown that the limit distribution \(G\) is in general not max-stable. In addition, it is described how his results can be extended to triangular arrays of random vectors which are for large \(n\) stationary.
triangular array of bivariate random vectors
domain of attraction of the Fréchet distribution
scientific article; zbMATH DE number 706915