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We prove an almost sure central limit theorem on the Poisson space, which is perfectly tailored for stabilizing functionals arising in stochastic geometry. As a consequence, we provide almost sure central limit theorems for (i) the total edge length of the k-nearest neighbors random graph, (ii) the clique count in random geometric graphs, and (iii) the volume of the set approximation via the Poisson–Voronoi tessellation.

We generalize a theorem of Shao [Proc. Amer. Math. Soc.123 (1995) 575–582] on the almost-sure limiting behavior of the maximum of standardized random walk increments to multidimensional arrays of i.i.d. random variables. The main difficulty is the absence of an appropriate strong approximation result in the multidimensional setting. The multiscale statistic under consideration was used recently for the selection of the regularization parameter in a number of statistical algorithms as well as for the multiscale signal detection.

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Almost sure central limit theorems in stochastic geometry

Almost sure functional limit theorem for the product of partial sums

Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays

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3 results

Almost sure central limit theorems in stochastic geometry

Almost sure functional limit theorem for the product of partial sums

Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays