Tempered operator stabile Verteilungen

Tempered operator stable laws are operator stable laws without normal component, for which we modify their Levy measure to reduce the expected number of large jumps. We will introduce a characterisation of the obtained Levy measure. Tempered operator stable distributions may have all moments finite. We prove short and long time behavior of the tempered operator stable Levy process: In a short time frame it is close to an operator stable process while in a long time frame it approximates a Brownian motion.

Then we construct a random walk, which converges in distribution to a random vector with a tempered operator stable distribution under a triangular array scheme. We show that the random walk process converges to the Levy process generated by the tempered operator stable distribution in the sense of finite-dimensional distributions.

We find probabilistic representations of tempered operator stable Levy process. Such representation can be used for simulation.