Local limit theorem for complex valued sequences

TL;DR

This work extends the local limit theorem from real-valued to complex-valued, finitely supported sequences on the integer lattice by employing a spatial-dynamics framework. The authors derive a full asymptotic expansion of the iterated convolution coefficients with explicit attractor terms and a generalized Gaussian remainder, valid to any order, through a careful analysis of the spatial Green's function and its resolvent. The main contribution is Theorem thPrinc, which expresses the asymptotics as a sum over tangency points $oldsymbol{ar extkappa}_k$ with polynomials $oldsymbol{\mathscr{P}}^k_ extsigma$ acting on generalized Gaussians $H_{2 extmu_k}^{eta_k}$, together with sharp bounds on the remainder. They further extend the results to drift-vanishing cases via a relaxed hypothesis H2 bis, provide detailed computations of the correction polynomials, and illustrate the results with numerical examples, including real nonnegative probabilistic sequences and an O3 transport scheme. The methodology and results have implications for stability and accuracy in finite-difference approximations of evolution equations and offer a precise higher-order generalization of the local limit theorem for complex-valued discrete systems.

Abstract

In this article, we study the pointwise asymptotic behavior of iterated convolutions on the one dimensional lattice Z. We generalize the so-called local limit theorem in probability theory to complex valued sequences. A sharp rate of convergence towards an explicitly computable attractor is proved together with a generalized Gaussian bound for the asymptotic expansion up to any order of the iterated convolution.

Figures (9)

• Figure 1: An example of spectrum $\sigma(\mathscr{L}_a)$. The spectrum $\sigma(\mathscr{L}_a)$ (in red) is inside the closed disk $\bar{\mathbb{D}}$ and touches the boundary $\mathbb{S}^1$ in finitely many points. In gray, we have $\mathcal{O}$ the intersection of the unbounded connected component of $\mathbb{C}\backslash\sigma(\mathscr{L}_a)$ and $\left\lbrace z\in\mathbb{C}, |z|>\exp(-\underline{\eta})\right\rbrace$.
• Figure 2: An illustration of the sectors $\mathcal{D}_k$. Here, we have $\alpha_1=-2$, $\alpha_2=0.5$ and $\alpha_3=4$. The rays labeled $\alpha_k$ (resp. $\underline{\delta}_k$, $\overline{\delta}_k$) correspond to the ray $j=n\alpha_k$ (resp. $j=n\underline{\delta}_k$, $j=n\overline{\delta}_k$). We observe that, because $\underline{\delta}_k$, $\alpha_k$ and $\overline{\delta}_k$ have the same sign, $j$ and $\alpha_k$ have the same sign for $(n,j)\in\mathcal{D}_k$. Also, the sectors $\mathcal{D}_k$ do not intersect each other.
• Figure 3: A representation of the path $\Gamma_k$ for $\underline{\tau}_k=0$. It is composed of $\Gamma_{k,out}$ (in red), $\Gamma_{k,res}$ (in green) and $\Gamma_{k,p}$ (in blue). The section of $\Gamma_k$ which lies inside the ball $B_\varepsilon(\underline{\tau}_k)$ (i.e. the reunion of $\Gamma_{k,res}$ and $\Gamma_{k,p}$) is notated $\Gamma_{k,in}$.
• Figure 4: This is a representation of $\Gamma_k$ where we decompose $\Gamma_{k,out}$. The red path corresponds to $\widehat{\Gamma}_0$ the part of $\Gamma_{k,out}$ which lies outside the balls $B_\varepsilon(\hat{\tau}_l)$. The green path corresponds to $\widehat{\Gamma}_l$ the part of $\Gamma_{k,out}$ which lies inside the ball $B_\varepsilon(\hat{\tau}_l)$. The dashed green path corresponds to the deformation we use in the proof of the estimate for $E_1$.
• Figure 5: A representation of the path $\Gamma_{k,in}$, $\Gamma_{k,in}^0$ and $\Gamma^\pm_{comp}$ for $\underline{\tau}_k=0$ used in Lemma \ref{['lemDécal']}.

Theorems & Definitions (35)

• Theorem 1
• Lemma 1
• Corollary 1
• Lemma 2
• Lemma 3: Spectral Splitting
• Lemma 4: Bounds far from the tangency points Cou-Faye
• Lemma 5: Bounds close to the tangency points : cases I and II
• Lemma 6: Bounds close to the tangency points : case III
• Lemma 7
• Lemma 8