Densities and Local Probabilities

Abstract

This chapter is devoted to local long-tailedness and to local subexponentiality. First we consider densities with respect to either Lebesgue measure on $\mathbb{R}$ or counting measure on $\mathbb{Z}$. Next we study the asymptotic behaviour of the probabilities to belong to an interval of a fixed length. We give the analogues of the basic properties of the tail probabilities including two analogues of Kesten’s estimate, and provide sufficient conditions for probability distributions to have these local properties.