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2 results

  • Phase-type distributions and invariant polytopes

  • Colm Art O'Cinneide
  • Journal:
    Advances in Applied Probability / Volume 23 / Issue 3 / September 1991
    Published online by Cambridge University Press:
    01 July 2016, pp. 515-535
    Print publication:
    September 1991

  • The notion of an invariant polytope played a central role in the proof of the characterization of phase-type distributions. The purpose of this paper is to develop invariant polytope techniques further. We derive lower bounds on the number of states needed to represent a phase-type distribution based on poles of its Laplace–Stieltjes transform. We prove that every phase-type distribution whose transform has only real poles has a bidiagonal representation. We close with three short applications of the invariant polytope idea. Taken together, the results of this paper show that invariant polytopes provide a natural approach to many questions about phase-type distributions.

  • On Rank One Commutators and Triangular Representations

  • Tsoy-Wo Ma
  • Journal:
    Canadian Mathematical Bulletin / Volume 29 / Issue 3 / 01 September 1986
    Published online by Cambridge University Press:
    20 November 2018, pp. 268-273
    Print publication:
    01 September 1986
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  • Starting with the extension of Lomonosov's Lemma by Tychonoff fixed point theorem, a result of Daughtry and Kim — Pearcy-Shields on rank-one commutators is extended to the context of locally convex spaces. Non-zero diagonal coefficients, eigenvalues and simultaneous triangular representations of compact operators on locally convex spaces are studied.