SPEAKER:
Marco Zamparo
AFFILIATION:
Politecnico di Torino
TITLE:
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.