Abstract
This paper analyzes various classes of processes associated with the tempered positive Linnik (TPL) distribution. We provide several subordinated representations of TPL Lévy processes and in particular establish a stochastic self-similarity property with respect to negative binomial subordination. In finite activity regimes we show that the explicit compound Poisson representations give raise to innovations following Mittag-Leffler type laws which are apparently new. We characterize two time-inhomogeneous TPL processes, namely the Ornstein-Uhlenbeck (OU) Lévy-driven processes with stationary distribution and the additive process determined by a TPL law. We finally illustrate how the properties studied come together in a multivariate TPL Lévy framework based on a novel negative binomial mixing methodology. Some potential applications are outlined in the contexts of statistical anti-fraud and financial modelling.
Funding Statement
This research has been financially supported by the programme “FIL-Quota Incentivante” of the University of Parma and co-sponsored by Fondazione Cariparma.
Acknowledgments
The authors thank Peter Carr and Luca Pratelli for the helpful discussions on a previous draft. They are also grateful to Domenico Perrotta and Francesca Torti of the Joint Research Centre of the European Commission for inspiring the anti-fraud applications of the tempered processes described in this work.
Citation
Lorenzo Torricelli. Lucio Barabesi. Andrea Cerioli. "Tempered positive Linnik processes and their representations." Electron. J. Statist. 16 (2) 6313 - 6347, 2022. https://doi.org/10.1214/22-EJS2090
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