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Local limit theorems and mod-phi convergence


Dal Borgo, Martina; Méliot, Pierre-Loïc; Nikeghbali, Ashkan (2019). Local limit theorems and mod-phi convergence. ALEA: Latin American Journal of Probability and Mathematical Statistics, 16(1):817-853.

Abstract

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.

Abstract

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Statistics and Probability
Language:English
Date:1 January 2019
Deposited On:31 Oct 2019 08:24
Last Modified:31 Oct 2019 22:12
Publisher:Instituto nacional de matemática pura e aplicada
ISSN:1980-0436
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.30757/alea.v16-30

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