Convergence of random measures in geometric probability

Abstract

Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure ${\nu }_n={\sum }_i{\xi }_i$, where ${\xi }_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near $X_i$. Technically, this means here that ${\xi }_i$ 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 ${\nu }_n(f)$. The latter implies weak convergence of ${\nu }_n(\cdot )$, 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.

51 pages

References in corpus (1)

• Gaussian limits for random measures in geometric probability