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A distribution with rational Laplace-Stieltjes transform is of phase type if and only if it is either the point mass at zero, or it has a continuous positive density on the positive reals and its Laplace-Stieltjes transform has a unique pole of maximal real part (which is therefore real). This result is proved, and the corresponding characterization of discrete phase-type distributions is stated and proved. Our methods are based on a geometric property of the set of phase-type distributions associated with a Markov chain.
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Colm Art O’Cinneide (1990) studied this question.
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