Bibliographic Notes Rather than providing a complete bibliography, our intention is to guide the reader into parts of the literature that we have found relevant for supplementary reading. For well-established results, we usually refer to textbooks or research monographs, whereas original sources are cited for specialized and more recent results. We apolo- gize for the inevitable (and entirely unintentional) omission of important references. Chapter 1 The infinitesimal approach to the Poisson process can be found in C¸inlar[69]or Karlin and Taylor [118]. For an extensive treatment of the Poisson process, also in higher dimensions, we refer to Kingman [120]. Our main references to discrete state-space Markov chains are Asmussen [18], C¸inlar[69], Karlin and Taylor [118], and Bremaud [59]. The material on Markov jump processes is mainly taken from Asmussen [18]andC¸inlar[69]. For a gentle introduction to Markov chains and processes (without measure theory), we suggest Hoel, Port, and Stone [104]. Discrete phase-type distributions were introduced in Neuts [150] and also treated in Neuts [153]. Distributions of jumps and sojourn times in Markov jump processes are treated in Bladt et al. [47]. The embedding problem is treated in general in Kingman [121] and was characterized in three dimensions in Johansen [110]. Uni- formization goes back at least to Jensen [109]. The coupling proof of the conver- gence of transition probabilities for both Markov chains and jump processes is taken from Lindvall [131] and Asmussen [18]. See also Thorisson [188] for further prop- erties of coupling methods and their relationship to regeneration. Properties of Kronecker sums and products can be found in Graham [93]. © Springer Science+Business Media LLC 2017 703 M. Bladt, B.F. Nielsen, Matrix-Exponential Distributions in Applied Probability, Probability Theory and Stochastic Modelling 81, DOI 10.1007/978-1-4939-7049-0 704 Bibliographic Notes Chapter 2 The main references on martingales in discrete time are Neveu [156] and Doob [77]. For continuous-time martingales, we refer to Revuz [173]. The central Martingale central limit theorem of Section 2.2.4 is adapted from Hall and Heyde [97]. The martingale relation to Example 2.2.8 can be found in Li [129] and Pozdnyakov and Kulldorff [166]. For further reading on general Markov processes, we suggest Kallenberg [114], Ethier and Kurtz [81], Revuz [172], Dynkin [79], and Loeve` [133, 134]. For Brownian motion, see, e.g., Kallianpur [115], Schilling [178], or Karatzas and Shreve [117]. Chapter 3 Erlang [65] is usually accredited for the method of stages. However, as was pointed out in [72], the idea occurred earlier, in works by Ellis [80]. The idea of extending the method of stages to absorption times of Markov jump processes is due to Jensen [108]. Neuts [150] started research on phase-type distributions and their application in queueing models; Neuts [153]. See also Latouche and Ramaswami [126], As- mussen [18], Breuer and Baum [63], Asmussen and Albrecher [20], He [99], and Nelson [149]. The denseness of phase-type distributions is due to Schassberger [177]. The proof of Theorem 3.2.9, p. 153, is taken from Johnson [111]. The minimal closure char- acterization (Theorem 3.2.10, p. 153) is from Maier and O’Cinneide [139]. Strassen’s theorem is a deep result in probability theory and was first proved byStrassenin[185] and later also by Dellacherie and Meyer in [75]. Both these proofs are based on functional analysis. Hoffman–Jørgensen proved the result in the preliminaries of [105] using an entirely measure-theoretic approach. For the case of R, a simple proof based on stop-loss transforms has more recently become available in Stoyan and M¨uller [147]. The proof of Theorem 3.3.8, p. 162, is adapted from Asmussen [19] (private communication). The minimality of the coefficient of variation for the Erlang distributions is originally due to Aldous and Shepp [7], who employed a martingale approach. The result was later generalized and proved using Strassen’s theorem and convex ordering in O’Cinneide [163]. For discrete phase-type distributions, a similar result was proved by Telek in [186]. For a more extensive treatment of stochastic ordering, see, e.g., Shaked and Shantikumar [181]. Functional calculus is treated in Haase [96] for general sectorial operators. A nice condensed paper on functions of matrices, defined in terms of the Cauchy integral, is Doolittle [78]. For all other methods of dealing with functions of matrices, we refer to Higham [103]. The application of functional calculus to phase-type distributions in Section 3.4, p. 169, is from Bladt, Campillo, and Nielsen [44]. The material on infinite-dimensional phase-type distributions (NPH) in Sec- tion 3.5, p. 180, is from Bladt, Nielsen, and Samorodnitsky [52]. For results on heavy-tailed distribution, regular variation, and Breiman’s theorem, we refer to Bibliographic Notes 705 Bingham et al. [41]. While the distributions of NPH may be genuinely heavy-tailed, other papers have dealt with the approximation of heavy-tailed distributions using phase-type distributions and Markovian arrival processes; see, e.g., Feldmann and Whitt [88] for the former and Andersen and Nielsen [9, 13, 14] for the latter. Chapter 4 Distributions with rational Laplace transforms were discussed in Cox [71]. It is standard in linear systems theory (Farina and Rinaldi [87], Astr˚ om¨ [33]) that matrix- exponential functions are exactly those having a rational Laplace transform. The proof that the dominant eigenvalue is real is taken from Fackrell [84]. A companion matrix approach was given in Asmussen and Bladt [23]. The idea of Lemma 4.2.5, p. 212, is from O’Cinneide [161]. Theorem 4.2.16, p. 216, for matrix-exponential distributions is from Asmussen and Bladt [23]. The ideas were applied to RAPs in Buchholz and Telek [66]. Details on the poles of rational Laplace transforms has been treated partially in O’Cinneide [162], where also the characterization in terms of a finite-dimensional vector space spanned by the residual life operator is given. General algebraic prop- erties and related linear vector spaces has been taken from Asmussen and Bladt [23]. The material on flow interpretation is based on Bladt and Neuts [48] and Bean and Nielsen [38]. The expression of matrix-exponential distributions in terms of their moments using Hankel matrices, and the resulting order determination is based on Bladt and Nielsen [49] and has previously been noted in the unpublished technical report Van de Liefvoort [130].SeealsoHe[101] and Bodrog, Horvath,´ and Telek [58]. The method using permanents is taken from the survey paper by Balakrishnan [36] on order statistics for i.n.i.d., i.e., independent nonidentically distributed, ran- dom variables. Abdelkader [2] obtains Laplace transforms of order statistics for i.n.i.d. phase-type distributed random variables. The distribution of the spread for a phase-type renewal process was discussed in Kao [116] but without providing a specific representation. The latter was derived in Bladt and Nielsen [51], and with a proof based on functional calculus in Bladt, Campillo, and Nielsen [44]. The characterization of phase-type distributions in terms of their densities is orig- inally due to O’Cinneide [162]. The proof leans on convex analysis, for which we refer to Rockafellar [174] and on semigroup theory (see, e.g., Davis [73]). The con- cept of invariant polytopes and the residual life operator were introduced to show that the sufficient conditions ensured that the distribution could be written as a con- vex combination of phase-type distributions and was thus of phase type. A proof based on automata theory was given in Maier [138]. An algebraic proof was given in Horvarth´ and Telek [106]. The latter is somewhat related to unicyclic representa- tions, which can be found in Mocanu and Commault [144] based on the terminology of O’Cinneide [162]. See also Fackrell et al. [83] for related work. 706 Bibliographic Notes Chapter 5 For general renewal theory we refer to Asmussen [18], C¸inlar[69], and Lindvall [131]. Renewal reward processes of Section 5.2.4, p. 320, are from Sigman and Wolff [183]andWolff[192]. Anscombe’s theorem (Theorem 5.2.20, 321) is from Gut [95]. Phase-type renewal theory is now well established and can be found, e.g., in Neuts [153] and Latouche and Ramaswami [126]. Moment distributions of phase- type (Section 5.5, p. 330) are from Bladt and Nielsen [51]. Renewal theory for matrix-exponential distributions is from Asmussen and Bladt [23] and Bladt and Neuts [48], while renewal theory for phase-type distributions is presented in Neuts [151]. Reversible phase-type representations seem to have appeared first in Ramaswami [168]. The concept was used by Andersen, Neuts, and Nielsen [11, 12]. Chapter 6 The main references to random walks are Feller [89] and Asmussen [18]. Random walks with a phase-type component (Section 6.2.1, p. 368) are from Asmussen [15]. For a matrix-exponential component, we refer to Asmussen and Bladt [23]. The occupation measures (6.6) and (6.7), both p. 365, are from Pitman [165]. For application of phase-type and matrix-exponential distributions in queueing theory, we refer to the monographs by Neuts [153, 154], Asmussen [18], [132], Nelson [149], Latouche and Ramaswami [126], Breuer [63], Bini, Latouche and Meini [42], and He [99]. For a classical introduction to queueing theory with an emphasis on rational transform methods, we refer to Cohen [70]. Some important papers and introductions to queues using matrix-analytic methods are Lucantoni [135, 136], Latouche and Ramaswami [127], Gail, Hantler, and Taylor [91], Bright and Taylor [64], Bean, Pollett, and Taylor [37]. Concerning applications of random walks and phase-type distributions in risk theory, we refer to Asmussen [30, 27].