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The general class of probability measures on <math>\mathbb{R}</math> with (proper) rational characteristic functions are densities corresponding to mixtures of the pointmass at zero ("[[delta distribution]]") and 2-EPT densities. Unlike [[Phase-type_distribution|phase-type]] and matrix geometric<ref>Neuts, M. "Probability Distributions of Phase Type", Liber Amicorum Prof. Emeritus H. Florin pages 173-206, Department of Mathematics, University of Louvain, Belgium 1975</ref> distributions, the 2-EPT probability density functions are defined on the whole real line. It has been shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations have been illustrated for the two-sided framework in Sexton and Hanzon.<ref>Sexton, C. and Hanzon,B.,"State Space Calculations for two-sided EPT Densities with Financial Modelling Applications", ''www.2-ept.com''</ref> The most involved operation is the [[convolution]] of 2-EPT densities using state space techniques. Much of the work centers on the ability to decompose the rational characteristic function into the sum of two rational functions with poles located in either the open left or open right half plane. The [[variance-gamma distribution]] density has been shown to be a 2-EPT density under a parameter restriction and the [[variance gamma process]]<ref>Madan, D., Carr, P., Chang, E. (1998) "The Variance Gamma Process and Option Pricing", ''European Finance Review'' 2: 79–105</ref> can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes.
It can be shown using [[Parseval's theorem]] and an isometry that approximating the discrete time rational transform is equivalent to approximating the 2-EPT density
==Applications==
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