On asymptotic properties and almost sure approximation of the normalized inverse-gaussian process

In this paper, similar to the frequentist asymptotic theory, we present large sample theory for the normalized inverse-Gaussian process and its corre-sponding 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 representa-tion 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.