A Markov jump process associated with the matrix-exponential distribution

Abstract

AbstractLet f be the density function associated to a matrix-exponential distribution of parameters $(\boldsymbol{\alpha}, T,\boldsymbol{{s}})$ . By exponentially tilting f, we find a probabilistic interpretation which generalizes the one associated to phase-type distributions. More specifically, we show that for any sufficiently large $\lambda\ge 0$ , the function $x\mapsto \left(\int_0^\infty e^{-\lambda s}f(s)\textrm{d} s\right)^{-1}e^{-\lambda x}f(x)$ can be described in terms of a finite-state Markov jump process whose generator is tied to T. Finally, we show how to revert the exponential tilting in order to assign a probabilistic interpretation to f itself.