Generalized Gaussian Estimates and Local Limit Theorems for Discrete Convolution Powers of Complex Functions: The $d$-dimensional case

TL;DR

This work extends generalized Gaussian bounds and local limit theorems for convolution powers from one to $d$ dimensions, tying the asymptotics of complex-valued functions on $\mathbb{Z}^d$ to Legendre-Fenchel transforms of positive-homogeneous polynomials and associated heat-kernel attractors. The authors develop a robust framework using holomorphic Fourier analysis, contour deformations, and local homogeneous expansions to derive off-diagonal Gaussian-type bounds and a d-dimensional local limit theorem with Gaussian-type error, capturing both on- and off-diagonal behavior. Key contributions include a finite-support refinement of the bounds, explicit attractor representations $H_{P_k}^t$, and a quantified on-diagonal rate $O(n^{-(\mu+\lambda)})$, with concrete examples such as the two-drifting-packets scenario illustrating sharpness. The results have practical relevance to the stability analysis of numerical difference schemes for PDEs and generalize prior one-dimensional results to higher dimensions. The paper also provides methodological tools (e.g., contour deformations tailored to each $\Omega(\phi)$ point) that pave the way for further multi-dimensional local limit theorems with cumulants.

Abstract

We establish generalized Gaussian bounds and local limit theorems with Gaussian-type error for the convolution powers of certain complex-valued functions on $\mathbb{Z}^d$. These global space-times estimates/error, which are sharp in certain cases, are written in terms of the Legendre-Fenchel transforms of positive-homogeneous polynomials and are mirrored by estimates satisfied by the heat kernels associated to a related class of partial differential operators. The results obtained here enjoy applications to the analysis and stability of numerical difference schemes to partial differential equations. This work extends several recent results, pertaining to one and several dimensions, of P. Diaconis, L. Saloff-Coste, J.-F. Coulombel, G. Faye, L. Coeuret, and the second author.

Figures (6)

• Figure 1: The graphs of $\phi^{(n)}$ for $n=100$ and $n=1,000$ are displayed on the grid $-50\leq x,y\leq 50$.
• Figure 2: On the grid $-50\leq x\leq 50$, $\abs{\phi^{(n)}}$ are illustrated by the solid light brown surfaces (for $n=100$ and $1,000$). The generalized Gaussian bounds, appearing on the right-hand side of \ref{['eq:Ex1Bound']}, are illustrated by the transparent "nets" above.
• Figure 3: The top row illustrates $\phi^{(n)}$ for $n=100$ and $n=1,000$. The second row displays the sum of attractors $A^n$. In the final row, the absolute errors for $n=100$ and $n=1,000$, $\abs{\phi^{(n)}-A^n}$, are illustrated by the solid light brown surfaces and the generalized Gaussian bounds, appearing on the right-hand side of \ref{['eq:Ex1LLT']}, are illustrated by the transparent "nets" above.
• Figure 4: Contour deformation for the coordinate $\widehat{\xi}_d$ (assuming $\sigma_d>0$.)
• Figure 5: For $n=30$ and $n=60$, the graphs of $\operatorname{Re}(\phi^{(n)})$ and $\operatorname{Re}(A^n)$ appear in the first two rows. In the final row, the absolute errors $\abs{\phi^{(n)}-A^n}$ are illustrated by the solid light brown surfaces and the generalized Gaussian bounds for this error are illustrated by the transparent "nets" above.

Theorems & Definitions (40)

• Definition 1.1
• Example 1
• Proposition 1.2
• Proposition 1.3: Proposition 1.2 of BR22
• Definition 1.4
• Proposition 1.5: Proposition 8.15 of RSC17
• Example 2
• Proposition 1.6
• Definition 1.7
• Theorem 1.8