We investigate the use of distribution functions to characterise the geophysical signals and noise components embedded in the geodetic Very Long Baseline Interferometry (VLBI) data sets across some of the International VLBI Service (IVS) stations. The rationale of using α-stable distributions as a tool to model the noise components in geodetic observables is due to the existence of impulsive signals/noise bursts (which often take the form of excursions with intermittent occurrences) in the data sets suggesting deviations from Gaussian distribution. A deviation from Gaussian distribution type would therefore suggest that statistical techniques such as least squares analysis, often used for analyzing the geodetic data (which are often based on Gaussian assumptions) could not be robust. In this paper, the properties of a long-range α-stable distribution with long tails and infinite moments in geodetic data are investigated by way of statistically testing their distribution using a family of stable distributions. The choice of stable distributions is based on the ease with which the statistical properties of the non-Gaussian processes are defined. Results indicate that the independent geophysical noise components reconstructed from geodetic VLBI baseline data exhibit distributions that have asymptotic power-law decay (albeit variable power indices) whose underlying process can be modelled as a long-range dependent process with an α-stable distribution (i.e., the stable varieties have small characteristic exponents).