On conditional extreme values of random vectors with polar representation (Q488091)

Let \((X,Y)\) denote a random vector of the form \((X,Y)=R\times (u(T),v(T))\), where \(R\) and \(T\) are independent random variables and \(u(t)\) and \(v(t)\) are functions. In the paper, the author assumes that \(R\) is in the max-domain of attraction of the Gumbel distribution and \(T\) has a density \(g(t)\) which is regularly varying at \(t=0\). The functions \(u(t),v(t)\) are of the form \( u(t)=1-L(t)\), \(v(t)=(t+\rho )\times u(t)\), where \(\rho \) is a real parameter and \(L(.)\) is slowly varying at \(t=0\). Under some technical assumptions about \(L^{\prime }(t)\) and \(g(t)\), the author obtains the precise asymptotic behaviour (as \(x\rightarrow \infty \)) of \(\operatorname{P}(X>x)\), of \(\operatorname{P}(Y\leq \rho x+xA(x)v\mid X>x)\) and of \(\operatorname{P}(X\leq x+B(x)u,Y\leq \rho x+xA(x)v\mid X>x)\).\ Here \(A(x),B(x)\) are known functions related to \(L(x)\) and \(\operatorname{P}(R>x)\). The results generalize the results of \textit{A.-L. Fougères} and \textit{P. Soulier} [Stoch. Models 26, No. 1, 54--77 (2010; Zbl 1195.60025)].