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In this paper we define the closure under weak convergence of the class ofp-tempered {\alpha}-stable distributions. We give necessary and sufficientconditions for convergence of sequences in this class. Moreover, we show thatany element in this class can be approximated by the distribution of a linearcombination of elementary p-tempered {\alpha}-stable random variables.

Limit Theorems For Sequences of Tempered Stable and Related Distributions