Bilateral Matrix-Exponential Distributions

Abstract

In this article we define the classes of bilateral and multivariate bilateral matrix-exponential distributions. These distributions have support on the entire real space and have rational moment-generating functions. The distributions extend the class of bilateral phase-type distributions of Ahn and Ramaswami [Stoch. Models 21, 239–259 (2005)] and the class of multivariate matrix-exponential distributions of Bladt and Nielsen [Stoch. Models 26, 1–26 (2010)]. We prove a characterization theorem stating that a random variable has a bilateral multivariate distribution if and only if all linear combinations of the coordinates have a univariate bilateral matrix-exponential distribution. As an application we demonstrate that certain multivariate diffusions, which are governed by the underlying Markov jump process generating a phase-type distribution, have a bilateral matrix-exponential distribution at the time of absorption.

Appendix

Existence of B ₊ and B ₋

In what follows, we will give an analysis of the existence of B ₊ and B ₋ assuming that we do not have an atom at zero.

Suppose that the polynomial A(s) can be written as $A(s) ={ \prod \nolimits }_{j=1}^{r}{(s - {\lambda }_{j})}^{{\nu }_{j}}$ for some r such as ∑ _j = 1 ^rν _j  = deg(A) and whose poles are given by λ _j . Then for

$${A}_{k}(s) ={ \prod \nolimits }_{j\neq k}{(s - {\lambda }_{j})}^{{\nu }_{j} } = \frac{A(s)} {{(s - {\lambda }_{k})}^{{\nu }_{k}}},\quad k = 1,\ldots, r,$$

we obtain that

$$\frac{B(s)} {A(s)} ={ \sum \limits _{j=1}^{r}} \frac{{C}_{j}(s)} {{(s - {\lambda }_{j})}^{{\nu }_{j}}},$$

(3.20)

where the polynomial C _j (s) is the Taylor polynomial of $\frac{B(s)} {{A}_{j}(s)}$ of order ν _j  − 1 at the point λ _j , i.e.,

$${C}_{j}(s) :={ \sum \limits _{k=0}^{{\nu }_{j}-1}} \frac{1} {k!}{\left ( \frac{B(s)} {{A}_{j}(s)}\right )}^{k}{\lambda }_{ j}{(s - {\lambda }_{j})}^{k}.$$

Taylor’s theorem (in the real or complex case) provides a proof of the existence and uniqueness of the partial fraction decomposition and a characterization of the coefficients. If we define

$${A}_{+}(s) :={ \prod \limits _{j=1}^{r}}{(s - {\lambda }_{j})}^{{\nu }_{j} }{1}_{\{{\lambda }_{j}>0\}},\quad {A}_{-}(s) :={\prod \limits _{j=1}^{r}}{(s - {\lambda }_{ j})}^{{\nu }_{j}}{1}_{\{{\lambda }_{j}<0\}},$$

then from (3.20) we obtain

$$\begin{array}{rcl} \frac{B(s)} {A(s)}& =& {\sum \nolimits }_{j=1}^{r} \frac{{C}_{j}(s)} {{(s - {\lambda }_{j})}^{{\nu }_{j}}}{1}_{\{{\lambda }_{j}>0\}} +{ \sum \nolimits }_{j=1}^{r} \frac{{C}_{j}(s)} {{(s - {\lambda }_{j})}^{{\nu }_{j}}}{1}_{\{{\lambda }_{j}<0\}} \\ & =& \frac{{B}_{+}(s)} {{A}_{+}(s)} + \frac{{B}_{-}(s)} {{A}_{-}(s)}, \\ \end{array}$$

where

$$\begin{array}{rcl} {B}_{+}(s)& :=& {\sum \limits_{j=1}^{r}}{C}_{ j}(s){1}_{\{{\lambda }_{j}>0\}}{ \prod \nolimits}_{k\neq j}^{r}{(s - {\lambda }_{ k})}^{{\nu }_{k} }{1}_{\{{\lambda}_{k}>0\}}, \\ {B}_{-}(s)& :=& {\sum \limits _{j=1}^{r}}{C}_{j}(s){1}_{\{{\lambda }_{j}<0\}}{ \prod \nolimits }_{k\neq j}^{r}{(s- {\lambda }_{ k})}^{{\nu }_{k} }{1}_{\{{\lambda}_{k}<0\}}.\end{array}$$