Do empirical processes of Laguerre tessellation generators satisfy functional central limit theorems?

Two processes, one counting and one volume-weighted

The paper studies empirical statistics built only on generators that are actually observable from tessellation data. A generator x = (v, h) is called extreme when its spatial location v lies inside its own Laguerre cell C(x; φ); equivalently, no competing point lies in the region D(x) below the downward paraboloid with apex at x. Generators with empty cells, or whose location is swallowed by another cell, are excluded, because they leave no directly observable contribution to the tessellation.

Two functionals are then defined over a growing convex window W_n = nW. The unweighted statistic G_n(z) counts extreme generators in the window whose marks do not exceed a prescribed level z. The volume-weighted statistic G_V_n(z) sums the Lebesgue volumes |C(X_i; P)| of the cells of those same extreme generators. Letting z vary turns each into an empirical process indexed by mark level, so the two objects answer different questions: how many observable extreme generators appear up to a mark threshold, and how much cell volume they collectively occupy.

The paper motivates the weighted version by observational stability. Under the unweighted count, an extreme point with a very small Laguerre cell contributes exactly as much as one with a large cell, even though small cells are typically more susceptible to geometric perturbations. The volume-weighted functional instead emphasizes geometrically stable extreme points by assigning larger weight to generators with larger cells.

Empty cells make the mark law only partly observable

In a Poisson-Laguerre model the generators form a homogeneous Poisson process on R^d with independent real-valued marks drawn from a common law F, and the mark h acts as a vertical shift in the Laguerre distance L(x, u) = ||v − u||^2 + h, so smaller marks have a competitive advantage. Unlike the Voronoi case, a generator may end up with an empty cell, and even among generators with nonempty cells the generator location need not lie in its own cell.

This is why the paper restricts attention to extreme generators. Earlier nonparametric estimators for F were built from observable generators and cell geometry, and one of them is based precisely on generators contained in their own cells. The present paper asks how empirical statistics based on those observable generators fluctuate as the observation window grows, rather than how to estimate F itself.

The geometric difficulty is that a generator with a sufficiently negative mark can influence the tessellation far from its spatial location. The lower tail of the mark law therefore enters the dependence estimates, and exponential localization cannot be assumed. This is the technical reason the results are stated under moment conditions on H_− = max(0, −H) rather than under light-tail assumptions.

Functional CLTs hold where the mark law has a bounded density

The central result, Theorem 2.4, proves functional central limit theorems on compact intervals where the mark law F has a bounded density. After centering and division by the square root of the window volume |W_n|, each empirical process converges in the Skorokhod topology to a centered Gaussian process. The convergence is driven by the growth of the observation window, not by any change in the mark distribution.

The moment assumptions are stated explicitly and are designed to allow heavy-tailed marks. Existence of the tessellation itself requires E[H_−^{d/2}] < ∞, which guarantees that the Poisson input generates a tessellation. Stronger assumptions are imposed only where quantitative rates are needed.

The proof strategy separates two tasks. Increments over short intervals of mark levels are controlled through covariance and fourth-cumulant bounds, and tightness is then established to complete the functional limit theorem. The paper notes that in this setting the increment estimates must accommodate both the dependence created by the mark distribution and the unbounded weights given by cell volumes.

Covariance kernels carry a surface-order error

Theorem 2.2 identifies the limiting covariance kernels of the Gaussian processes and establishes covariance asymptotics with a surface-order error. The kernels describe the dependence between Gaussian fluctuations at different mark levels, which is what distinguishes a functional limit theorem from a single-level central limit theorem.

The surface-order error term is the quantitative content of this result: the covariance asymptotics are not merely stated to leading order in the window volume, but with an explicit error of surface order. For readers comparing this to fixed-level results, the distinction matters, because a fixed-level CLT says nothing about how fluctuations at two different levels covary.

The volume-weighted case is substantially more involved here. The score function associated with G_V_n(z) depends on the volume of the Laguerre cell, which may be influenced by points outside the local region D(x). To handle this, the paper uses the cover time, which quantifies how far outside D(x) one must look to determine the Laguerre cell of x.

Cover times bound how far a cell reaches

The cover time T(x; φ) is defined as the smallest t at which the growing crystal C_t(x; φ) has reached its final shape C(x; φ), with T(x; φ) = −∞ when the cell is empty. The paper shows that C_t(x; φ) equals C(x; φ) intersected with the ball of radius sqrt(t − h) around v, so the cover time is exactly h plus the squared maximal distance from v to a point of the cell.

Proposition 3.3 gives exponential tail bounds for the cover time, uniformly over the event that the point is extreme. For t > h the bound takes the form of a polynomial prefactor times exp(−2^{−d} Q((h + t)/2)), where Q(h) = κ_d E[(h − H)_+^{d/2}] is the Q-measure of the region D(v, h). Choosing the net parameter proportional to sqrt(t − h) balances the polynomial prefactor against the exponential decay.

Corollary 3.5 converts these tails into moment bounds for the excess cover time, and Lemma 3.6 gives E[1{x extreme} |C(x; P)|^{4+p}] ≤ C(1 + (z − h)_+^{(4+p)d/2}) for h ≤ z. These estimates are what allow the volume-weighted contributions to be localized, and they are the key input needed later for the weighted process.

Univariate bounds decay as the inverse square root of window volume

Theorem 2.1 gives non-asymptotic univariate normal approximation bounds. For the unweighted statistic, assuming E[H_−^{3d}] < ∞, the distance d_* = max(d_W, d_K) between the standardized statistic and a standard normal is O(|W_n|^{−1/2}). For the volume-weighted statistic, the assumption is E[H_−^{m_0}] < ∞ for some m_0 > 11d/2, together with the requirement that F is not a Dirac measure, and the same O(|W_n|^{−1/2}) rate holds.

The two distances measure different things and should not be conflated. The Wasserstein distance d_W uses the class of Lipschitz-1 functions, while the Kolmogorov distance d_K uses indicator functions of half-lines. The paper reports the maximum of the two, so the stated rate holds simultaneously in both senses.

Theorem 2.3 extends this to multivariate normal approximation, using the d_3 distance defined over C^3 functions whose second and third derivatives are bounded by 1. The paper notes that the same geometric estimates enter the multivariate Poisson normal-approximation bounds, and that the univariate bounds decay as the inverse square root of the window volume in both Wasserstein and Kolmogorov distance under the stated nondegeneracy assumptions.

Bounded density and dimension two and above

The functional central limit theorems are proved on compact intervals where the mark law has a bounded density. This is a genuine restriction on the level sets covered, and it sits alongside the moment assumptions rather than replacing them. The covariance asymptotics and the quantitative approximation results carry their own stated assumptions, including nondegeneracy conditions and, for the weighted univariate bound, the requirement that F is not a Dirac measure.

The paper assumes d ≥ 2 throughout. The case d = 1 could also be considered using the same approach, but some case distinction would be needed for the rates, and the authors state that d = 1 is less interesting from the application perspective, so they do not treat it. Readers working in one dimension should therefore not read the stated rates as directly transferable.

The results are asymptotic in the growing convex window W_n = nW, with W a compact convex body containing the origin in its interior. The constants in the O-notations may depend on fixed parameters such as d, F, z, and I, but not on n. The paper also notes that the closest prior Gaussian-approximation results for Laguerre tessellations concerned counts of generators with nonempty cells in the β-, β′-, and Gaussian-Voronoi models, whereas here the observables select generators contained in their own cells and retain their marks as an indexing parameter.

Choose the weighted process when cell size matters

The paper's own comparison points to a practical priority. If the question is how many observable extreme generators fall below a mark level, the unweighted process G_n(z) is the relevant object, and its univariate Gaussian approximation requires only E[H_−^{3d}] < ∞. If the question is how much tessellation volume those generators account for, the volume-weighted process G_V_n(z) is the relevant object, but it demands a stronger moment condition, E[H_−^{m_0}] < ∞ for some m_0 > 11d/2, plus the requirement that F is not a Dirac measure.

That gap in assumptions is the paper's most decision-relevant distinction. The weighted functional is the one that downweights geometrically fragile small cells, which is exactly why it needs additional control: a generator's contribution depends on its entire cell, so the analysis must localize contributions through cover-time tail and moment estimates. The unweighted count avoids that machinery but remains sensitive to observational noise from small cells.

For readers deciding which fluctuation result to invoke, the threshold is therefore not the window size but the mark law's tail and the weighting choice. Heavy-tailed marks are permitted in both cases, but the weighted process pays for that generality with a higher moment requirement and a nondegeneracy condition on F. Where the mark law has a bounded density on the interval of interest and the relevant moment condition holds, both processes converge to centered Gaussian processes with identified covariance kernels and a surface-order covariance error.

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