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

TL;DR

This paper investigates the asymptotic behavior of the normalized inverse-Gaussian process, establishing limit theorems and an almost sure approximation that facilitates efficient simulation of the process.

Contribution

It introduces new asymptotic results and an almost sure finite sum-representation for the normalized inverse-Gaussian process, enhancing understanding and simulation methods.

Findings

Established an empirical functional central limit theorem.

Proved a strong law of large numbers and Glivenko-Cantelli theorem.

Derived an almost sure finite sum-representation for simulation.

Abstract

In this paper, we present some asymptotic properties of the normalized inverse-Gaussian process. In particular, when the concentration parameter is large, we establish an analogue of the empirical functional central limit theorem, the strong law of large numbers and the Glivenko-Cantelli theorem for the normalized inverse-Gaussian process and its corresponding 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 efficiently the normalized inverse-Gaussian process.