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.