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1. arXiv:2409.09909 (Published 2024-09-16)

   On Approximations of Subordinators in $L^p$ and the Simulation of Tempered Stable Distributions

   Michael Grabchak, Sina Saba
   Categories: math.PR

   Keywords: tempered stable distributions, approximations, randomly stopped levy processes, wasserstein metrics, subordinator belongs
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   Subordinators are infinitely divisible distributions on the positive half-line. They are often used as mixing distributions in Poisson mixtures. We show that appropriately scaled Poisson mixtures can approximate the mixing subordinator and we derive a rate of convergence in $L^p$ for each $p\in[1,\infty]$. This includes the Kolmogorov and Wasserstein metrics as special cases. As an application, we develop an approach for approximate simulation of the underlying subordinator. In the interest of generality, we present our results in the context of more general mixtures, specifically those that can be represented as differences of randomly stopped L\'evy processes. Particular focus is given to the case where the subordinator belongs to the class of tempered stable distributions.

2. arXiv:1205.3690 (Published 2012-05-16, updated 2012-10-22)

   Speed of convergence to equilibrium in Wasserstein metrics for Kac-s like kinetic equations

   Federico Bassetti, Eleonora Perversi
   Categories: math.PR, math-ph, math.MP

   Subjects: 60F05, 82C40

   Keywords: wasserstein metrics, convergence, equilibrium, kantorovich-wasserstein distance, initial datum belongs
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   This work deals with a class of one-dimensional measure-valued kinetic equations, which constitute extensions of the Kac caricature. It is known that if the initial datum belongs to the domain of normal attraction of an \alpha-stable law, the solution of the equation converges weakly to a suitable scale mixture of centered \alpha-stable laws. In this paper we present explicit exponential rates for the convergence to equilibrium in Kantorovich-Wasserstein distances of order p>\alpha, under the natural assumption that the distance between the initial datum and the limit distribution is finite. For \alpha=2 this assumption reduces to the finiteness of the absolute moment of order p of the initial datum. On the contrary, when \alpha<2, the situation is more problematic due to the fact that both the limit distribution and the initial datum have infinite absolute moment of any order p >\alpha. For this case, we provide sufficient conditions for the finiteness of the Kantorovich-Wasserstein distance.