The local limit theorem for complex valued sequences: the parabolic case (Q6644108)

For any two complex valued sequences \(\mathbf{a} = (a_l)_{l\in \mathbb{Z}}\) and \(\mathbf{b} = (b_l)_{l\in \mathbb{Z}}\) such that the quantity below makes sense, the convolution \(\mathbf{a}\star\mathbf{b}\) of \(\mathbf{a}\) and \(\mathbf{b}\) is defined as:\N\[\N\mathbf{a}\star\mathbf{b}=\left(\sum_{l'\in \mathbb{Z}}a_{l-l'}b_{l'}\right)_{l\in \mathbb{Z}}.\N\]\NFor \(\mathbf{a}\) and \(\mathbf{b}\) in the space of complex valued integrable sequences \(l^{1}(\mathbb{Z};\mathbb{C})\), the convolution \(\mathbf{a}\star\mathbf{b}\) is well defined and also belongs to \(l^{1}(\mathbb{Z};\mathbb{C})\), which endows this space with a Banach algebra structure. The goal of this article is to study some geometric sequences in this algebra. The authors consider a fixed complex valued sequence \(\mathbf{a} = (a_l)_{l\in \mathbb{Z}}\) and make the following assumption.\N\N\textbf{Assumption 1.} The sequence \(\mathbf{a} = (a_l)_{l\in \mathbb{Z}}\) belongs to \(l^{1}(\mathbb{Z};\mathbb{C})\) and its associated Fourier series\N\[\NF_{\mathbf{a}}: \zeta\in \mathbb{C} \to \sum_{l\in \mathbb{Z}}a_{l}\zeta^l\N\]\Ndefines a holomorphic function on an annulus \(\{\zeta\in \mathbb{C}:1-\varepsilon < |\zeta| < 1+\varepsilon\}\) for some \(\varepsilon>0\). Furthermore, the following holds:\N\[\N\sup_{\substack{\kappa\in \mathbb{S}^1}}|F_{\mathbf{a}}(\kappa)|=1.\N\]\NThe authors give a complete expansion, at any accuracy order, for the iterated convolutions \(\mathbf{a}\star\cdots\star \mathbf{a}\) when \(\mathbf{a}\) satisfies \textbf{Assumption 1} and also satisfies some additional condition related to \(F_{\mathbf{a}}\). The remainders are estimated sharply with generalized Gaussian bounds. The result applies in probability theory for random walks as well as in numerical analysis for studying the large time behavior of numerical schemes.