Central limit theorem for a birth-growth model with Poisson arrivals and random growth speed

Abstract

We consider Gaussian approximation in a variant of the classical Johnson-Mehl birth-growth model with random growth speed. Seeds appear randomly in R d at random times and start growing instantaneously in all directions with a random speed. The location, birth time and growth speed of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed and a weight function h : R d → [0, ∞), we provide sufficient conditions for a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.

Gaussian approximation for sums of region-stabilizing scores

“…Hence, following the proof of Theorem 1 in [1], there exist finite positive constants C 1 and C 2 such that In view of the Mecke formula and the Poisson empty space formula, c 1,s (y) is the expected number of minimal points in P s that dominate y ∈ X. Also note that g s (y) and h s (y) from (2.6) is equal to c ζ,s (y) with ζ = p/(40 + 10p) = 1/50, so that G s (y) ≤ 3 + 2c ζ,s (y) 5 .…”

“…We also allow for multiple points and for a non-uniform bound on the (4 + p)-th moment of the score functions, which is particularly important in examples involving infinite intensity measures, like stationary Poisson processes. Apart from examples presented in the current paper, further applications of our method has been elaborated in [5], where a quantitative central limit theorem is obtained for functionals of growth processes that result in generalized Johnson-Mehl tessellations, and in [4], where such a result is obtained in the context of minimal directed spanning trees in dimensions three and higher, respectively.…”