2017/10/29 by Borgo, Martina dal, Méliot, Pierre-Loïc, Nikeghbali, Ashkan
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1710.10679
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 a local limit theorem in the framework of mod-phi convergence: one proof based on the notion of zone of control, and one proof based on the notion of mod-ϕ convergence in L1(iR). 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.