  This paper analyzes various classes of processes associated with the tempered
positive Linnik (TPL) distribution. We provide several subordinated
representations of TPL L\'evy 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 gives rise to innovations following Mittag-Leffler type
laws which are apparently new. We characterize two time-inhomogeneous TPL
processes, namely the Ornstein-Uhlenbeck (OU) L\'evy-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 L\'evy framework based on a novel negative binomial mixing methodology.
Some potential applications are outlined in the contexts of statistical
anti-fraud and financial modelling.

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