[Paper Review] Convergence of random measures in geometric probability

Mathew D. Penrose|ArXiv.org|Aug 24, 2005

Stochastic processes and statistical mechanics18 references6 citations

TL;DR

This paper establishes a law of large numbers and central limit theorem for random measures generated by marked point processes in $\mathbb{R}^d$, under power-law stabilization conditions—extending prior results on exponential stabilization. The key contribution is weak convergence of rescaled and centered random measures to a Gaussian field, applicable to geometric objects like surface measures in germ-grain models with unbounded grains.

ABSTRACT

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

Motivation & Objective

• To generalize existing limit theorems for random measures from exponential to power-law stabilization, broadening applicability to models with long-range dependence.
• To establish a law of large numbers and central limit theorem for random measures defined via local contributions from marked point processes, even when the measure is not concentrated at points.
• To allow for indicator functions as test functions, extending prior results restricted to continuous test functions.
• To provide a framework for analyzing surface and volume measures in germ-grain models with unbounded grain sizes, which were previously outside the scope of stabilization-based limit theorems.
• To de-Poissonize central limit theorems, extending results from Poisson to binomial point processes without relying on external stabilization concepts.

Proposed method

• Define a random measure $\nu_n = \sum_i \xi_i$, where $\xi_i$ is a measure derived from the local configuration around each point $X_i$, stabilized with power-law tail decay in the radius of stabilization.
• Use a refined version of the objective method to compute second moments and covariance structures for the limiting Gaussian field under Poissonian settings.
• Apply Stein's method for normal approximation to prove the limiting field is Gaussian, avoiding the method of moments used in prior work.
• Perform second moment calculations via the objective method to de-Poissonize results, extending central limit theorems from Poisson to binomial point processes.
• Introduce translation invariance and moment conditions on the grain size distribution to ensure integrability and moment bounds for the measure components.
• Verify moment conditions on both positive and negative parts of signed measures (e.g., surface measures) to satisfy the required integrability for the central limit theorem.

Experimental results

Research questions

• RQ1 Under what conditions does the sum of locally defined random measures converge weakly to a Gaussian field in geometric probability?
• RQ2 Can the central limit theorem for random measures be extended beyond exponential stabilization to power-law stabilization?
• RQ3 Can the theory accommodate non-continuous test functions, such as indicator functions of Borel sets?
• RQ4 How can the limit theorems be extended from Poisson to binomial point processes without introducing external stabilization?
• RQ5 Can the framework handle surface and volume measures in germ-grain models with unbounded grain sizes?

Key findings

• A law of large numbers holds for $\nu_n(f)$ under power-law stabilization, with convergence of the normalized measure to a deterministic limit for bounded test functions.
• A central limit theorem is established for $\nu_n(f)$, showing weak convergence of the rescaled and centered measure to a Gaussian field indexed by bounded test functions.
• The limiting Gaussian field arises from a covariance structure derived via refined second moment calculations using the objective method.
• The results apply to surface measures in germ-grain models with unbounded grains, provided $\mathbb{E}[|T|^{dp}]\mathbb{E}[|\partial T|^p] < \infty$ for some $p > 1$.
• For the surface measure case, the condition $P[|T| > r] \leq C e^{-r/C}$ or $P[|T| > r] \leq C r^{-q}$ with $q > d(150 + 6/p)$ ensures the moment conditions for the CLT.
• The framework successfully handles signed measures such as the difference between positive and negative surface contributions, ensuring integrability and convergence.