On Asymptotic Properties and Almost Sure Approximation of the Normalized Inverse-Gaussian Process

Abstract

In this paper, similar to the frequentist asymptotic theory, we present large sample theory for the normalized inverse-Gaussian process and its corresponding quantile process. In particular, when the concentration parameter is large, we establish the functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem for the normalized inverse-Gaussian process and its related quantile process. We also derive a finite sum representation that converges almost surely to the Ferguson and Klass representation of the normalized inverse-Gaussian process. This almost sure approximation can be used to simulate the normalized inverse-Gaussian process.