In this paper, we investigate multivariate generalized gamma convolutions, commonly referred to as the Thorin class, which forms a distinguished subclass of the class of infinitely divisible distributions. The Thorin class contains all elementary gamma distributions and is closed under convolution and weak convergence; moreover, it is the smallest class of probability measures on \(\mathbb {R}^{p}\) enjoying these properties. By developing multivariate Markov–Krein transformations, we establish new and significant connections between multivariate generalized gamma convolutions and multivariate generalized spline distributions. In particular, using the Markov–Krein formula, we show that the asymptotic behavior of multivariate generalized splines naturally leads to Thorin measures. Conversely, we prove that Thorin measures can be characterized through the asymptotics of generalized spline distributions. Finally, within this framework, we derive a representation of the Fourier–Laplace transform of Thorin measures, obtained via the asymptotic properties of multivariate generalized splines and the Markov–Krein transform.
