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We establish precise bounds on cumulants for a rather general class ofnon-linear geometric functionals satisfying the stabilization property under asimple, stationary (marked) point process admitting fast decay of itscorrelation functions and thereby conclude a Berry-Esseen bound, aconcentration inequality, a moderate deviation principle and aMarcinkiewicz-Zygmund-type strong law of large numbers. The result is appliedto the germ-grain model as well as to random sequential absorption for${\alpha}$-determinantal point processes having fast decaying kernels andcertain Gibbsian point processes. The proof relies on cumulant expansions usinga clustering result as well as factorial moment expansions for point processes.

Asymptotic results for stabilizing functionals of point processes having fast decay of correlations