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Consider a stationary Poisson process in a $d$-dimensional hyperbolic space.For $R>0$ define the point process $\xi_R^{(k)}$ of exceedance heights over asuitable threshold of the hyperbolic volumes of $k$th nearest neighbour ballscentred around the points of the Poisson process within a hyperbolic ball ofradius $R$ centred at a fixed point. The point process $\xi_R^{(k)}$ iscompared to an inhomogeneous Poisson process on the real line with intensityfunction $e^{-u}$ and point process convergence in the Kantorovich-Rubinsteindistance is shown. From this, a quantitative limit theorem for the hyperbolicmaximum $k$th nearest neighbour ball with a limiting Gumbel distribution isderived.

Large nearest neighbour balls in hyperbolic stochastic geometry