Local limit theorems and mod-phi convergence

We prove local limit theorems for mod-ϕ convergent sequences of random variables, ϕ being a stable distribution. In particular, we give two new proofs of the local limit theorem stated in Delbaen et al. (2015): one proof based on the notion of zone of control introduced in Féray et al. (2019+a), and one proof based on the notion of mod-ϕ convergence in $\textit{L}^1$$(i\mathbb{R})$. These new approaches allow us to identify the infinitesimal scales at which the stable approximation is valid. We complete our analysis with a large variety of examples to which our results apply, and which stem from random matrix theory, number theory, combinatorics or statistical mechanics