Rates of convergence for bivariate extremes
Let \(\{(X_n,Y_n)\), \(n\geq 1\}\) be independent, identically distributed (i.i.d.) random vectors with common distribution function \(F\). Suppose \(F\) is in the domain of attraction of an extreme value distribution, i.e., there exist constants \(a_n>0\), \(c_n>0\), \(b_n\), and \(d_n\) such that \[ F^n(a_nx+ b_n, c_ny+ d_n)\to G(x,y) \tag{1} \] for \((x,y)\in \mathbb{R}^2\), where \(G\) has non-degenerate marginal distributions, i.e., \[ P\Biggl\{ \frac{\max_{i\leq n}X_i-b_n}{a_n}\leq x, \frac{\max_{i\leq n}y_i-d_n}{c_n}\leq y\Biggr\}\to G(x,y) \qquad (n\to\infty). \] Then \(G(x,\infty)\) and \(G(\infty,x)\) are extreme value distributions. Therefore, without loss of generality we can assume \[ G(x,\infty)= \exp\{-(1+\gamma_1 x)^{- 1/\gamma_1}\}, \qquad G(\infty,y)= \exp\{-(1+\gamma_2 y)^{-1/\gamma_2}\} \] for some \(\gamma_1,\gamma_2\in \mathbb{R}\) and \(1+\gamma_1x>0\), \(1+\gamma_2y>0\). We are interested in rates of convergence in the limit relation. Under a second order regular variation condition, rates of convergence of the distribution of bivariate extreme order statistics to its limit distribution are given both in the total variation metric and in the uniform metric.
scientific article; zbMATH DE number 1040617