L\'evy Processes and L\'evy White Noise as Tempered Distributions

We identify a necessary and sufficient condition for a L'evy white noise to be a tempered distribution. More precisely, we show that if the L'evy measure associated with this noise has a positive absolute moment, then the L'evy white noise almost surely takes values in the space of tempered distributions. If the L'evy measure does not have a positive absolute moment of any order, then the event on which the L'evy white noise is a tempered distribution has probability zero.