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

Abstract

We give a complete expansion, at any accuracy order, for the iterated convolution of a complex valued integrable sequence in one space dimension. 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.

Figures (3)

• Figure 1: The integration contour in the case $x_k \ge 0$ (in blue). The bullets correspond to the endpoints of the three segments that define the new contour. The initial contour is depicted in black. Each new integral appears in red.
• Figure 2: Illustration of the scaling factor in the generalized asymptotic expansion provided by Theorem \ref{['thm1']} in the case of the $O3$ scheme. We plot $\log_{10}\|\mathcal{R}^n\|_{\ell^\infty}$ (blue circles) and $\log_{10}\|\mathcal{R}^n\|_{\ell^1}$ (orange circles) as a function of $\log_{10}(n)$ together with a best linear fit for each norm for $n$ ranging from $1$ to $10^3$. For the $\ell^\infty$ norm we find a slope of $-1.2707$ while for the $\ell^1$ norm we find a slope of $-0.9887$ which compare both well with the predicted $-5/4$ and $-1$ scaling factors of Theorem \ref{['thm1']}.
• Figure 3: Illustration of the rescaled remainder term $n^{5/4}\left|\mathcal{R}^n_\ell\right|$ (colored circles) at different time iterations of the $O3$ scheme compared with a fixed generalized Gaussian profile centered at $\ell=\lambda n$ (solid lines) with $\lambda=1/2$. The fixed generalized Gaussian profile is given by the sequence $\ell \mapsto C \exp\left(- c \, \left( \dfrac{|\ell-n/2|}{n^{1/4}} \right)^{4/3} \right)$ with constants $C=0.09$ and $c=0.225$.

Theorems & Definitions (9)

• Lemma 1: Lemma A.1 in CF1
• Theorem 1
• Corollary 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof