Regular variation and free regular infinitely divisible laws

In this article the relation between the tail behaviours of a free regular infinitely divisible (positively supported) probability measure and its L'evy measure is studied. An important example of such a measure is the compound free Poisson distribution, which often occurs as a limiting spectral distribution of certain sequences of random matrices. We also describe a connection between an analogous classical result of Embrechts et al. [1979] and our result using the Bercovici-Pata bijection.