Tempered positive Linnik processes and their representations.

We study several classes of processes associated with the tempered positive Linnik (TPL) distribution, in both the purely absolutely-continuous and mixed law regimes. We explore four main ramifications. Firstly, we analyze several subordinated representations of TPL L\'evy processes; in particular we establish a stochastic self-similarity property of positive Linnik (PL) L\'evy processes, connecting TPL processes with negative binomial subordination. Secondly, in finite activity regimes we show that the explicit compound Poisson representations gives rise to innovations following novel Mittag-Leffler type laws. Thirdly, we characterize two inhomogeneous TPL processes, namely the Ornstein-Uhlenbeck (OU) L\'evy-driven processes with stationary distribution and the additive process generated by a TPL law. Finally, we propose a multivariate TPL L\'evy process based on a negative binomial mixing methodology of independent interest. Some potential applications of the considered processes are also outlined in the contexts of statistical anti-fraud and financial modelling.