Quasi-infinitely divisible distributions

• Applicant: Professor Dr. Alexander Lindner
• Subject Area: Mathematics
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 419461105

• Final Report Year: 2021

Final Report Abstract

We studied multivariate quasi-infinitely divisible distributions and could address various distributional properties of these probability measures, like moments, convergence or support properties. We also gave methods on how to construct multivariate quasi-infinitely divisible distributions from one-dimensional ones. We could show that a distribution on Zd is quasi-infinitely divisible if and only if its characteristic function is zero-free, from which we could deduce a Cramér-Wold device for infinite divisibility of Zd-valued distributions. This came as a surprise, because a Cramér-Wold device for infinite divisibility does not hold for general Rd-valued distributions. It was known before that the set of one-dimensional quasi-infinitely divisible distributions is dense in the set of probability measures on R with respect to weak convergence, and it was a surprise for us to find out that this does not generalise to higher dimensions. Indeed, it could be shown that the class of Rd-valued quasi-infinitely divisible distributions is dense in the class of Rd-valued probability measures with respect to weak convergence if and only if the dimension d is equal to 1. We further introduced L´evy driven CARMA random fields in a natural way by defining them as generalized random processes that solve a stochastic partial differential equation driven by Lévy noise. This is the first approach to define such processes by appealing directly to the dynamics given by the differential equations of such processes. Finally, central limit theorems for moving average random fields with nonrandom and random sampling were obtained.

Publications

• (2019), Infinitely divisible and related distributions and Lévy driven stochastic partial differential equations, PhD thesis, Ulm University
  Berger, D.
  (See online at https://dx.doi.org/10.18725/OPARU-25286)
• (2020), Lévy driven CARMA generalized processes and stochastic partial differeential equations, Stoch. Process. Appl. 130, 5865–5887
  Berger, D.
  (See online at https://doi.org/10.1016/j.spa.2020.04.009)
• (2021) Central limit theorems for moving average random fields with nonrandom and random sampling, ALEA Lat. Am. J. Probab. Math. Stat.
  Berger, D.
  
• (2021) On Multivariate Quasi-infinitely Divisible Distributions. In: Chaumont L., Kyprianou A.E. (eds) A Lifetime of Excursions Through Random Walks and Lévy Processes. Progress in Probability, vol 78. Birkhäuser, Cham
  Berger, D., Kutlu, M. and Lindner, A.
  (See online at https://doi.org/10.1007/978-3-030-83309-1_6)
• (2021), A Cramér–Wold device for infinite divisibility of Zd - valued distributions
  Berger, D. and Lindner, A.
  
• (2021), On a denseness result for quasi-infinitely divisible distributions, Statistics Probab. Lett. 176, 109139
  Kutlu, M.
  (See online at https://doi.org/10.1016/j.spl.2021.109139)