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Conditional Limit Results for Type I Polar Distributions

Let (S_1,S_2)=(R \cos(Θ), R \sin (Θ)) be a bivariate random vector with associated random radius R which has distribution function $F$ being further independent of the random angle Θ. In this paper we investigate the asymptotic behaviour of the conditional survivor probability Ψ_{ρ,u}(y):=\pk{ρS_1+ \sqrt{1- ρ^2} S_2> y \lvert S_1> u}, ρ\in (-1,1),\in R when u approaches the upper endpoint of F. On the density function of Θwe require a certain local asymptotic behaviour at 0, whereas for F we require that it belongs to the Gumbel max-domain of attraction. The main result of this contribution is an asymptotic expansion of Ψ_{ρ,u}, which is then utilised to construct two estimators for the conditional distribution function 1- Ψ_{ρ,u}. Further, we allow Θto depend on u.