Large nearest neighbour balls in hyperbolic stochastic geometry

Abstract Consider a stationary Poisson process in d -dimensional hyperbolic space. For $$R>0$$ R > 0 define the point $$\xi _R^{(k)}$$ ξ ( k ) of exceedance heights over suitable threshold volumes k th nearest neighbour balls centred around points within ball radius R at fixed point. The is compared to an inhomogeneous on real line with intensity function $$e^{-u}$$ e - u and convergence Kantorovich-Rubinstein distance shown. From this, quantitative limit theorem for maximum limiting Gumbel distribution derived.