Convergence of random measures in geometric probability

arXivmath/0508464MaRDI QIDQ6475933

Mathew D. Penrose

Publication date: 24 August 2005

Abstract: Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $u_n=su{m}_{i}x{i}_i$, where $x{i}_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that $x{i}_i$ stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions $f$ on $R^d$, we give a law of large numbers and central limit theorem for $u_n(f)$. The latter implies weak convergence of $u_n(cdot)$, suitably scaled and centred, to a Gaussian field acting on bounded test functions. The general result is illustrated with applications including the volume and surface measure of germ-grain models with unbounded grain sizes.

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) Central limit and other weak theorems (60F05) Random measures (60G57) Random convex sets and integral geometry (aspects of convex geometry) (52A22) (L^p)-limit theorems (60F25)

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