Local Central Limit Theorem

• Local Central Limit Theorem is a refinement of the classical CLT that provides precise, local Gaussian approximations for small-interval probabilities of normalized sums.
• It uses characteristic function analysis and spatial partitioning to control error terms and ensure that nearly independent contributions dominate the local behavior.
• The theorem underpins applications in stochastic geometry, percolation, and random graphs by transferring fine local approximations from regular to global functional behaviors.

The local central limit theorem (LCLT) refines the classical central limit theorem by providing precise asymptotics for pointwise or small-interval probabilities of sums of many random variables or local functionals, showing that—after appropriate normalization—these probabilities are uniformly approximated by the density of a Gaussian distribution. The LCLT is particularly fundamental in modern probability, stochastic geometry, random matrix theory, statistical mechanics, combinatorics, and geometric probability, as it enables transfer of fine local approximations from independent or weakly dependent contributions to quantities defined on complex geometric or combinatorial structures.

1. Formal Statement and General Structure

A classical local CLT states that for a sum $W_n$ of independent (or suitably weakly dependent) random variables, the probability that $W_n$ falls in a small interval (or at a lattice point) is locally approximated by a rescaled Gaussian density: $$\lim_{n \to \infty} \sup_u \left| c_n\, P(W_n \in [u, u+b)) - \varphi\left( \frac{u - \mathbb{E}W_n}{c_n} \right) \right| = 0$$ with scaling $c_n \sim \sqrt{n}$ for variance $\sigma^2$, fixed $b>0$, and $$\varphi(x) = (1/\sigma\sqrt{2\pi}) e^{-x^2/(2\sigma^2)}$$ the standard normal density.

The central contribution in "Local central limit theorems in stochastic geometry" (Penrose et al., 2010) is an extension of this principle to random variables and functionals expressible as

$Z_n = Y_n + S_n$

where $S_n$ is a "smoothing" component (often a sum of contributions from nearly independent, regular parts of a complex structure), and $Y_n$ is a residual or error term (for example, arising from the boundary or exceptions), which is negligible at the correct scaling. The general LCLT (Theorem 2.1) states that if:

• $n^{-1/2}(Z_n - \mathbb{E}Z_n) \to^d N(0, \sigma^2)$ (global CLT),
• $S_n$ satisfies a classical local CLT at the same order,
• $Y_n$ is "small"—its scaled distribution does not affect local probabilities, then

$$\sup_{u \in \mathbb{R}} \left| c_n\, P\left[ Z_n \in [u, u+b) \right] - \varphi\left( \frac{u - \mathbb{E}Z_n}{c_n} \right) \right| \to 0$$

where $c_n \sim \sqrt{n}$ and the interpretation adapts in the lattice case.

The technical innovation is to show that, under weak dependence and negligible error from $Y_n$, the local smoothing properties of $S_n$ are inherited by $Z_n$.

2. Methodological Foundation and Proof Sketch

The proof of the general LCLT for such decomposed functionals relies on:

• Classical LCLT for $S_n$,
• Characteristic function analysis,
• Bounding the error from coupling $Y_n$ and $S_n$, ensuring that after normalization the residual $Y_n$ does not affect the main term,
• Partitioning the underlying geometric, combinatorial, or spatial domain into "good" (regular, i.i.d.-like) and "bad" (error) regions.

Schematically, the method involves:

1. Verifying a local CLT for the main sum $S_n$ using known LCLT methods (for instance, partitioning space into independent or stabilized boxes/cells and applying the classical LCLT to their sum).
2. Proving that $Y_n$, after centering and $\sqrt{n}$-scaling, adds only "global smoothing": its contribution to the distribution is "flat" and does not affect the shape of the local limiting law.
3. Using characteristic functions, the impact of $Y_n$ can be isolated and controlled, while classical smoothing arguments for $S_n$ establish local convergence to the normal density.
4. In the lattice case, interval $b$ is adjusted to the appropriate span to account for discreteness; in the non-lattice case, uniformity over $u$ holds for fixed (Lebesgue) intervals.

A general formula derived is: $$\sup_{u} \left| c_n\, P\left[ Z_n \in [u, u+b) \right] - \varphi\left( \frac{u - \mathbb{E}Z_n}{c_n} \right) \right| \to 0$$ where $Z_n = Y_n + S_n$, $S_n$ satisfies the LCLT, and $Y_n$ is negligible.

3. Applications to Stochastic Geometry: Decomposition and Examples

Functionals in stochastic geometry (and related fields) often possess local dependence, enabling geometric decomposition into (a) sums over "good" (i.i.d.-like) regions, and (b) negligible error regions:

A. Percolation on a Box

• For site percolation on $\mathbb{Z}^d$ in a finite box $B_n$, functionals like the number of connected components $A(B_n)$ and the largest cluster size $L(B_n)$ can be partitioned into contributions from disjoint sub-boxes (with good "local" statistical independence/moderate dependence), plus a boundary or residual part.
• Each sub-box contribution, after conditioning on the configuration outside, is (nearly) independent and satisfies a classical LCLT. The global statistic inherits a local CLT by Theorem 2.1.
• This approach is used to prove local limit theorems for the number of components and the size of the largest component (Theorems 3.1–3.2).

B. Random Geometric Graphs

• The number of subgraphs (edges, triangles, isolated vertices), or connected components, can be represented as a sum over spatial partitions (small cubes), where each partition's contribution has weak dependence and classical LCLT applies.
• Using prior global CLTs and the partition decomposition approach, local CLTs are established (Theorem 4.1).

C. Germ-Grain Coverage Models

• For the total covered volume in germ–grain processes with grains of uniformly bounded support, the sum of contributions from finite-range interactions satisfies the finite-range stabilization criterion.
• A global CLT holds (as in known literature), and the general LCLT theorem yields local central limit behavior for the covered volume.

4. Characterization of Error Terms and Lattice Considerations

• In both lattice and non-lattice settings, the error from the residual $Y_n$ is shown to be negligible compared to the smoothing provided by $S_n$; this is essential for "lifting" the local limit property from $S_n$ to $Z_n$.
• When $S_n$ is lattice-valued, the interval $b$ must be adjusted to be an integer multiple of the lattice span; the result then provides uniform convergence modulo the lattice structure.
• The technical heart of the proof involves Fourier-analytic estimates to show that the characteristic function of the error term does not materially affect the main Gaussian limiting behavior at the correct scale.

5. Extensions to High-Dimensional and Complex Geometric Functionals

The paper extends the general LCLT via stabilization and finite-range interaction criteria (Sections 10–11), covering more complex functionals including:

• Functionals on marked point processes in $\mathbb{R}^d$,
• Independence numbers, nearest-neighbor distance sums, or coverage functionals,
• General stabilized functionals where a positive fraction of the sum comes from almost i.i.d. contributions,
• The use of binomial exponential stabilization to handle settings where independence is only approximate.

The method is to partition the total sum into regularly-behaving and error parts, apply the general LCLT to the regular portion, and show that the error part is negligible at the local scale.

6. Unifying Framework and Implications

The general local central limit theorem as established provides a systematic way to obtain pointwise (local) normal approximations for a broad class of sums and functionals in stochastic geometry and related random spatial models. The main consequences and implications include:

• Transferring robust local Gaussian approximations from independent or stabilized components to global functionals of interest,
• Yielding sharp probability estimates for small (local) deviations, essential for large deviations analyses, precise percolation thresholds, and sharp combinatorial asymptotics,
• Providing a powerful tool for analyzing fluctuations of complex geometric or combinatorial structures in high-dimensional regimes,
• Enabling extension of classical probabilistic limit theorems into the setting of random spatial structures, with applications to percolation, coverage, random graphs, germ–grain models, and beyond.

The systematic decomposition approach, management of weak dependencies and error terms, and connection to classical local CLT methods demonstrate that local Gaussian behavior is prevalent in a wide class of high-dimensional, dependent settings, provided sufficient stabilization or localization is achieved in the decomposition of the functional.