# Seminario di Probabilità: “Large Deviations Principles in Renewal Theory”

## Venerdì 23 Marzo 2018, ore 11:30 - Sala Riunioni VII Piano Aula 1BC45 - Marco Zamparo

ARGOMENTI: Seminari

Venerdì 23 Marzo 2018 alle ore 11:30 in Sala Riunioni VII Piano Aula 1BC45, Marco Zamparo (Politecnico di Torino) terrà un seminario dal titolo “Large Deviations Principles in Renewal Theory”.

Abstract
Cramér's theorem provides a large deviations principle for the empirical mean of independent and identically distributed random variables. In this seminar, I establish a generalization of Cramér's theorem for multivariate renewal-reward processes under optimal hypotheses. The way to attack the problem is the following. A large deviations principle is first obtained for renewal processes constrained by the event that one of the renewals occurs at a predetermined time. Then, results are transferred to standard renewal processes by resorting to conditioning. The constrained setting makes it possible to solve the problem by using an argument based on convexity and sub-additivity, which is inspired by the approach of Bahadur and Zabell to derive the modern form of the Cramér's theorem. Interestingly, with a different interpretation of the time coordinate, constrained renewal processes include important models of statistical mechanics, such as the pinning model of polymers and the Poland-Scheraga model of DNA denaturation. These models are briefly reviewed in the seminar.

M. Zamparo, Large Deviations in Renewal Theory and Renewal Models of Statistical Mechanics, arXiv:1801.09941.