Equivalence and Separation for Multivariate Matrix-Exponential and Phase-Type Distribution Classes
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| Format: | Preprint |
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2026
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| _version_ | 1866911686189907968 |
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| author | Peralta, Oscar |
| author_facet | Peralta, Oscar |
| contents | We resolve two questions left open by Bladt and Nielsen (2010) concerning multivariate families of matrix-exponential and phase-type distributions. First, in the matrix-exponential case, the projection-defined class MVME coincides with Kulkarni's algebraic class MME*. Our proof combines a multivariate state-space realization theorem with elementary augmentations that put the realization into Kulkarni's form. Thus every proper rational multivariate Laplace transform has a finite-dimensional Kulkarni-type representation once Markovian sign constraints are removed. Second, in the phase-type setting, the inclusion of MPH* in MVPH is strict from the trivariate case onward. The separation is obtained through a factorization condition for MPH* that appears not to have been previously identified in the PH literature. A Wishart trace distribution belongs to MVPH but fails this condition, hence providing the required example outside MPH*. The example also shows that projection-based multivariate phase-type laws may have density and support geometry that are absent from the usual univariate theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_15395 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivalence and Separation for Multivariate Matrix-Exponential and Phase-Type Distribution Classes Peralta, Oscar Probability We resolve two questions left open by Bladt and Nielsen (2010) concerning multivariate families of matrix-exponential and phase-type distributions. First, in the matrix-exponential case, the projection-defined class MVME coincides with Kulkarni's algebraic class MME*. Our proof combines a multivariate state-space realization theorem with elementary augmentations that put the realization into Kulkarni's form. Thus every proper rational multivariate Laplace transform has a finite-dimensional Kulkarni-type representation once Markovian sign constraints are removed. Second, in the phase-type setting, the inclusion of MPH* in MVPH is strict from the trivariate case onward. The separation is obtained through a factorization condition for MPH* that appears not to have been previously identified in the PH literature. A Wishart trace distribution belongs to MVPH but fails this condition, hence providing the required example outside MPH*. The example also shows that projection-based multivariate phase-type laws may have density and support geometry that are absent from the usual univariate theory. |
| title | Equivalence and Separation for Multivariate Matrix-Exponential and Phase-Type Distribution Classes |
| topic | Probability |
| url | https://arxiv.org/abs/2605.15395 |