@article{aalen1995phase,
  title={Phase type distributions in survival analysis},
  author={Aalen, Odd O},
  journal={Scandinavian journal of statistics},
  pages={447--463},
  year={1995},
  publisher={JSTOR}
}

@article{acal2019phase,
  title={Phase-type distributions for studying variability in resistive memories},
  author={Acal, Christian and Ruiz-Castro, Juan Eloy and Aguilera, Ana M and Jim{\'e}nez-Molinos, Francisco and Rold{\'a}n, Juan Bautista},
  journal={Journal of Computational and Applied Mathematics},
  volume={345},
  pages={23--32},
  year={2019},
  publisher={Elsevier}
}


@article{faddy1999analysing,
  title={Analysing data on lengths of stay of hospital patients using phase-type distributions},
  author={Faddy, MJ and McClean, SI0966},
  journal={Applied Stochastic Models in Business and Industry},
  volume={15},
  number={4},
  pages={311--317},
  year={1999},
  publisher={Wiley Online Library}
}



@misc{genomedk,
    title = {Genome DK cluster},
    howpublished={\url{https://genome.au.dk}}
}

@book{he2014,
  title={Fundamentals of Matrix-Analytical Methods},
  author={Qi-Ming He},
  volume={},
  year={2014},
  publisher={Springer},
  address={New York}
}

@book{Neuts1981,
  title={Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach},
  author={Marcel F Neuts},
  volume={},
  year={1981},
  publisher={John Hopkins University Press},
  address={Baltimore}
}

@book{Duff2017,
  title={Direct Methods for Sparse Matrices},
  author={Duff, I.S. and Erisman, A.M. and Reid, J.K.},
  volume={},
  year={2017},
  publisher={Oxford University Press}
}

@article{sherlock2021,
  title={Direct statistical inference for finite Markov jump processes via the matrix exponential},
  author={Chris Sherlock},
  journal={Computational Statistics},
  volume={36},
  pages={2863--2887},
  year={2021}
}

@article{SandersonCurtin2016,
  title={Armadillo: a template-based C++ library for linear algebra},
  author={C. Sanderson and R. Curtin},
  journal={Journal of Open Source Software},
  volume={1},
  pages={},
  year={2016}
}

@article{EddelbuettelSanderson2016,
  title={RcppArmadillo: Accelerating R with high-performance C++ linear algebra},
  author={D Eddelbuettel and C Sanderson},
  journal={Computational Statistics & Data Analysis},
  volume={71},
  pages={1054-1063},
  year={2014}
}

@article{MolerVanLoan2003,
  title={Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later},
  author={Cleve Moler and Charles Van Loan},
  journal={SIAM Review},
  volume={45},
  pages={3--49},
  year={2003}
}

@article{hobolth2019phase,
  title={Phase-type distributions in population genetics},
  author={Hobolth, Asger and Siri-Jegousse, Arno and Bladt, Mogens},
  journal={Theoretical population biology},
  volume={127},
  pages={16--32},
  year={2019},
  publisher={Elsevier}
}

@article{hobolth2021sfs,
  title={Multivariate phase-type theory for the site frequency spectrum},
  author={Hobolth, Asger and Bladt, Mogens and Andersen, Lars N},
  journal={Journal of Mathematical Biology},
  volume={},
  pages={},
  year={2021},
  publisher={Springer}
}

@article{kingman1982genealogy,
  title={On the genealogy of large populations},
  author={Kingman, John FC},
  journal={Journal of applied probability},
  volume={19},
  number={A},
  pages={27--43},
  year={1982},
  publisher={Cambridge University Press}
}

@book{wakeley2009coalescent,
  title={Coalescent theory: an introduction},
  author={Wakeley, John},
  year={2009},
  publisher={Roberts   \&   Company,   
Greenwood Village, CO},
ISBN={9780974707754}
}

@book{bladt2017matrix,
  title={Matrix-exponential distributions in applied Probability},
  author={Bladt, Mogens and Nielsen, Bo Friis},
  volume={81},
  year={2017},
  publisher={Springer},
  address={New York}
}

@article{sbeity2008phase,
  title={Phase-type distributions in stochastic automata networks},
  author={Sbeity, Ihab and Brenner, Leonardo and Plateau, Brigitte and Stewart, William J},
  journal={European Journal of Operational Research},
  volume={186},
  number={3},
  pages={1008--1028},
  year={2008},
  publisher={Elsevier}
}

@misc{cormen2009introduction,
  title={Introduction to Algorithms, third edit ed},
  author={Cormen, TH and Leiserson, CE and Rivest, RL and Stein, C},
  year={2009},
  publisher={MIT Press}
}

@article{trygubenko2006graph,
  title={Graph transformation method for calculating waiting times in Markov chains},
  author={Trygubenko, Semen A and Wales, David J},
  journal={The Journal of chemical physics},
  volume={124},
  number={23},
  pages={234110},
  year={2006},
  publisher={American Institute of Physics}
}
@book{serfozo2009basics,
  title={Basics of applied stochastic processes},
  author={Serfozo, Richard},
  year={2009},
  publisher={Springer Science \& Business Media}
}
@book{wald2004sequential,
  title={Sequential analysis},
  author={Wald, Abraham},
  year={2004},
  publisher={Courier Corporation}
}

@article{o1991phase,
  title={Phase-type distributions and invariant polytopes},
  author={O'Cinneide, Colm Art},
  journal={Advances in Applied Probability},
  volume={23},
  number={3},
  pages={515--535},
  year={1991},
  publisher={Cambridge University Press}
}

@article{bobbio2003acyclic,
  title={Acyclic discrete phase type distributions: Properties and a parameter estimation algorithm},
  author={Bobbio, Andrea and Horv{\'a}th, Andr{\'a}s and Scarpa, Marco and Telek, Mikl{\'o}s},
  journal={Performance evaluation},
  volume={54},
  number={1},
  pages={1--32},
  year={2003},
  publisher={Elsevier}
}
@article{cumani1982canonical,
  title={On the canonical representation of homogeneous Markov processes modelling failure-time distributions},
  author={Cumani, Aldo},
  journal={Microelectronics Reliability},
  volume={22},
  number={3},
  pages={583--602},
  year={1982},
  publisher={Elsevier}
}
@article{murthy2012graphs,
  title={Graphs: Associated Markov Chains},
  author={Murthy, Garimella Rama},
  journal={arXiv preprint arXiv:1209.3387},
  year={2012}
}
@article{jacobsen1974splitting,
  title={Splitting times for Markov processes and a generalised Markov property for diffusions},
  author={Jacobsen, Martin},
  journal={Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und verwandte Gebiete},
  volume={30},
  number={1},
  pages={27--43},
  year={1974},
  publisher={Springer}
}
@article{gillespie1976general,
  title={A general method for numerically simulating the stochastic time evolution of coupled chemical reactions},
  author={Gillespie, Daniel T},
  journal={Journal of computational physics},
  volume={22},
  number={4},
  pages={403--434},
  year={1976},
  publisher={Elsevier Science}
}
@article{pulungan2009reduction,
  title={Reduction of acyclic phase-type representations},
  author={Pulungan, Muhammad Reza},
  year={2009}
}

@article{varmazyar2019classification,
  title={Classification and properties of acyclic discrete phase-type distributions based on geometric and shifted geometric distributions},
  author={Varmazyar, Mohsen and Akhavan-Tabatabaei, Raha and Salmasi, Nasser and Modarres, Mohammad},
  journal={Journal of Industrial Engineering International},
  volume={15},
  number={4},
  pages={651--665},
  year={2019},
  publisher={Springer}
}

@article{rosenberg2002genealogical,
  title={Genealogical trees, coalescent theory and the analysis of genetic polymorphisms},
  author={Rosenberg, Noah A and Nordborg, Magnus},
  journal={Nature Reviews Genetics},
  volume={3},
  number={5},
  pages={380--390},
  year={2002},
  publisher={Nature Publishing Group}
}
@article{polanski2003note,
  title={A note on distributions of times to coalescence, under time-dependent population size},
  author={Polanski, Andrzej and Bobrowski, Adam and Kimmel, Marek},
  journal={Theoretical population biology},
  volume={63},
  number={1},
  pages={33--40},
  year={2003},
  publisher={Elsevier}
}

@book{hahn2019molecular,
  title={Molecular population genetics},
  author={Hahn, Matthew William},
  year={2019},
  publisher={Sinauer Associates New York}
}

@article{waxman2011unified,
  title={A unified treatment of the probability of fixation when population size and the strength of selection change over time},
  author={Waxman, David},
  journal={Genetics},
  volume={188},
  number={4},
  pages={907--913},
  year={2011},
  publisher={Genetics Soc America}
}

@article{sawyer1992population,
  title={Population genetics of polymorphism and divergence.},
  author={Sawyer, Stanley A and Hartl, Daniel L},
  journal={Genetics},
  volume={132},
  number={4},
  pages={1161--1176},
  year={1992},
  publisher={Genetics Soc America}
}


@article{eldon2015can,
  title={Can the site-frequency spectrum distinguish exponential population growth from multiple-merger coalescents?},
  author={Eldon, Bjarki and Birkner, Matthias and Blath, Jochen and Freund, Fabian},
  journal={Genetics},
  volume={199},
  number={3},
  pages={841--856},
  year={2015},
  publisher={Genetics Soc America}
}

@book{buchholz2014input,
  title={Input modeling with phase-type distributions and Markov models: theory and applications},
  author={Buchholz, Peter and Kriege, Jan and Felko, Iryna},
  year={2014},
  publisher={Springer}
}

@article{korneliussen2013calculation,
  title={Calculation of Tajima’s D and other neutrality test statistics from low depth next-generation sequencing data},
  author={Korneliussen, Thorfinn Sand and Moltke, Ida and Albrechtsen, Anders and Nielsen, Rasmus},
  journal={BMC bioinformatics},
  volume={14},
  number={1},
  pages={289},
  year={2013},
  publisher={BioMed Central}
}
@article{ren2017cmph,
  title={CMPH: a multivariate phase-type aggregate loss distribution},
  author={Ren, Jiandong and Zitikis, Ricardas},
  journal={Dependence Modeling},
  volume={5},
  number={1},
  pages={304--315},
  year={2017},
  publisher={De Gruyter Open}
}
@article{bobbio2004scale,
  title={The scale factor: a new degree of freedom in phase-type approximation},
  author={Bobbio, Andrea and Horv{\'a}th, Andr{\'a}s and Telek, Mikl{\'o}s},
  journal={Performance Evaluation},
  volume={56},
  number={1-4},
  pages={121--144},
  year={2004},
  publisher={Elsevier}
}


@article{eisele2006recursions,
  title={Recursions for compound phase distributions},
  author={Eisele, Karl-Theodor},
  journal={Insurance: Mathematics and Economics},
  volume={38},
  number={1},
  pages={149--156},
  year={2006},
  publisher={Elsevier}
}



@article{frydenberg1990chain,
  title={The chain graph {M}arkov property},
  author={Frydenberg, Morten},
  journal={Scandinavian Journal of Statistics},
  pages={333--353},
  year={1990},
  publisher={JSTOR}
}
@inproceedings{younes2004solving,
  title={Solving generalized semi-{M}arkov decision processes using continuous phase-type distributions},
  author={Younes, H{\aa}kan LS and Simmons, Reid G},
  booktitle={AAAI},
  volume={4},
  pages={742},
  year={2004}
}


@article{fu1993maximum,
  title={Maximum likelihood estimation of population parameters.},
  author={Fu, Yun-Xin and Li, Wen-Hsiung},
  journal={Genetics},
  volume={134},
  number={4},
  pages={1261--1270},
  year={1993},
  publisher={Genetics Soc America}
}

@article{ramirez2009correcting,
  title={Correcting estimators of $\theta$ and tajima's D for ascertainment biases caused by the single-nucleotide polymorphism discovery process},
  author={Ram{\'\i}rez-Soriano, Anna and Nielsen, Rasmus},
  journal={Genetics},
  volume={181},
  number={2},
  pages={701--710},
  year={2009},
  publisher={Genetics Soc America}
}

@article{abdellaoui2013population,
  title={Population structure, migration, and diversifying selection in the Netherlands},
  author={Abdellaoui, Abdel and Hottenga, Jouke-Jan and De Knijff, Peter and Nivard, Michel G and Xiao, Xiangjun and Scheet, Paul and Brooks, Andrew and Ehli, Erik A and Hu, Yueshan and Davies, Gareth E and others},
  journal={European journal of human genetics},
  volume={21},
  number={11},
  pages={1277--1285},
  year={2013},
  publisher={Nature Publishing Group}
}



@article{knudsen2009accurate,
  title={Accurate and fast methods to estimate the population mutation rate from error prone sequences},
  author={Knudsen, Bjarne and Miyamoto, Michael M},
  journal={BMC bioinformatics},
  volume={10},
  number={1},
  pages={247},
  year={2009},
  publisher={Springer}
}

@article{jorde2007unbiased,
  title={Unbiased estimator for genetic drift and effective population size},
  author={Jorde, Per Erik and Ryman, Nils},
  journal={Genetics},
  volume={177},
  number={2},
  pages={927--935},
  year={2007},
  publisher={Genetics Soc America}
}

@unpublished{hobolth2019unpublished,
author  = {Hobolth, Asger},
title   = {Course Phase-type distributions in population genetics, Handout 6},
year    = {2019},
howpublished = "\url{https://blackboard.au.dk/bbcswebdav/pid-2416009-dt-content-rid-7536861_1/xid-7536861_1}",
note    = {unpublished}
}

@article{akkouchi2008convolution,
  title={On the convolution of exponential distributions},
  author={Akkouchi, Mohamed},
  journal={J. Chungcheong Math. Soc},
  volume={21},
  number={4},
  pages={501--510},
  year={2008}
}
@article{knuth1971optimum,
  title={Optimum binary search trees},
  author={Knuth, Donald E.},
  journal={Acta informatica},
  volume={1},
  number={1},
  pages={14--25},
  year={1971}
}

@inproceedings{adel1962algorithm,
  title={An algorithm for organization of information},
  author={Adelson-Velskii, George M and Landis, Evgenii Mikhailovich},
  booktitle={Doklady Akademii Nauk},
  volume={146},
  number={2},
  pages={263--266},
  year={1962},
  organization={Russian Academy of Sciences}
}

@article{hardy1918asymptotic,
  title={Asymptotic formula{\ae} in combinatory analysis},
  author={Hardy, Godfrey H and Ramanujan, Srinivasa},
  journal={Proceedings of the London Mathematical Society},
  volume={2},
  number={1},
  pages={75--115},
  year={1918},
  publisher={Wiley Online Library}
}

@Article{partitionsR,
    title = {Additive integer partitions in R},
    author = {R. K. S. Hankin},
    journal = {Journal of Statistical Software, Code Snippets},
    year = {2006},
    month = {May},
    volume = {16},
    issue = {1},
  }
  
  @article{junges2019parameter,
  title={Parameter synthesis for Markov models},
  author={Junges, Sebastian and Abraham, Erika and Hensel, Christian and Jansen, Nils and Katoen, Joost-Pieter and Quatmann, Tim and Volk, Matthias},
  journal={arXiv preprint arXiv:1903.07993},
  year={2019}
}

@article{goyal1987evaluation,
  title={Evaluation of performability for degradable computer systems},
  author={Goyal, Ambuj and Tantawi, Asser N.},
  journal={IEEE Transactions on Computers},
  number={6},
  pages={738--744},
  year={1987},
  publisher={IEEE}
}

@article{meyer1982closed,
  title={Closed-form solutions of performability},
  author={Meyer, John F.},
  journal={IEEE Transactions on Computers},
  number={7},
  pages={648--657},
  year={1982},
  publisher={IEEE}
}

@book{denardo2011linear,
  title={Linear programming and generalizations: A problem-based introduction with spreadsheets},
  author={Denardo, Eric V},
  volume={149},
  year={2011},
  publisher={Springer Science \& Business Media}
}

@article{lawrence1991polytope,
  title={Polytope volume computation},
  author={Lawrence, Jim},
  journal={mathematics of computation},
  volume={57},
  number={195},
  pages={259--271},
  year={1991}
}

@article{gravin2018moments,
  title={On moments of a polytope},
  author={Gravin, Nick and Pasechnik, Dmitrii V and Shapiro, Boris and Shapiro, Michael},
  journal={Analysis and Mathematical Physics},
  volume={8},
  number={2},
  pages={255--287},
  year={2018},
  publisher={Springer}
}

@article{huff2010mobile,
  title={Mobile elements reveal small population size in the ancient ancestors of Homo sapiens},
  author={Huff, Chad D and Xing, Jinchuan and Rogers, Alan R and Witherspoon, David and Jorde, Lynn B},
  journal={Proceedings of the National Academy of Sciences},
  volume={107},
  number={5},
  pages={2147--2152},
  year={2010},
  publisher={National Acad Sciences}
}

@article{lipson2015calibrating,
  title={Calibrating the human mutation rate via ancestral recombination density in diploid genomes},
  author={Lipson, Mark and Loh, Po-Ru and Sankararaman, Sriram and Patterson, Nick and Berger, Bonnie and Reich, David},
  journal={PLoS genetics},
  volume={11},
  number={11},
  year={2015},
  publisher={Public Library of Science}
}


@inproceedings{papp2013canonical,
  title={Canonical representation of discrete phase type distributions of order 2 and 3},
  author={Papp, J{\'a}nos and Telek, Mikl{\'o}s},
  booktitle={Proc. of UK Performance Evaluation Workshop, UKPEW},
  volume={2013},
  year={2013}
}

@book{Slatkin2013AnIT,
  title={An Introduction to Population Genetics: Theory and Applications},
  author={Montgomery Slatkin and Rasmus Nielsen},
  year={2013},
  ISBN={1605351539},
  publisher={sinauer}
}

@article{munch2016selective,
  title={Selective sweeps across twenty millions years of primate evolution},
  author={Munch, Kasper and Nam, Kiwoong and Schierup, Mikkel Heide and Mailund, Thomas},
  journal={Molecular biology and evolution},
  volume={33},
  number={12},
  pages={3065--3074},
  year={2016},
  publisher={Oxford University Press}
}

@article{yi2010sequencing,
  title={Sequencing of 50 human exomes reveals adaptation to high altitude},
  author={Yi, Xin and Liang, Yu and Huerta-Sanchez, Emilia and Jin, Xin and Cuo, Zha Xi Ping and Pool, John E and Xu, Xun and Jiang, Hui and Vinckenbosch, Nicolas and Korneliussen, Thorfinn Sand and others},
  journal={Science},
  volume={329},
  number={5987},
  pages={75--78},
  year={2010},
  publisher={American Association for the Advancement of Science}
}

@article{kern2017exact,
  title={Exact calculation of the joint allele frequency spectrum for isolation with migration models},
  author={Kern, Andrew D and Hey, Jody},
  journal={Genetics},
  volume={207},
  number={1},
  pages={241--253},
  year={2017},
  publisher={Genetics Soc America}
}

@article{zhou2017importance,
  title={Importance of incomplete lineage sorting and introgression in the origin of shared genetic variation between two closely related pines with overlapping distributions},
  author={Zhou, Y and Duvaux, L and Ren, G and Zhang, L and Savolainen, O and Liu, J},
  journal={Heredity},
  volume={118},
  number={3},
  pages={211--220},
  year={2017},
  publisher={Nature Publishing Group}
}

@article{wakeley1997estimating,
  title={Estimating ancestral population parameters},
  author={Wakeley, John and Hey, Jody},
  journal={Genetics},
  volume={145},
  number={3},
  pages={847--855},
  year={1997},
  publisher={Genetics Soc America}
}

@article{gutenkunst2009inferring,
  title={Inferring the joint demographic history of multiple populations from multidimensional SNP frequency data},
  author={Gutenkunst, Ryan N and Hernandez, Ryan D and Williamson, Scott H and Bustamante, Carlos D},
  journal={PLoS genetics},
  volume={5},
  number={10},
  year={2009},
  publisher={Public Library of Science}
}

@article{wang2010estimating,
  title={Estimating divergence parameters with small samples from a large number of loci},
  author={Wang, Yong and Hey, Jody},
  journal={Genetics},
  volume={184},
  number={2},
  pages={363--379},
  year={2010},
  publisher={Genetics Soc America}
}

@article{mailund2011estimating,
  title={Estimating divergence time and ancestral effective population size of Bornean and Sumatran orangutan subspecies using a coalescent hidden Markov model},
  author={Mailund, Thomas and Dutheil, Julien Y and Hobolth, Asger and Lunter, Gerton and Schierup, Mikkel H},
  journal={PLoS genetics},
  volume={7},
  number={3},
  year={2011},
  publisher={Public Library of Science}
}

@book{hall2015lie,
  title={Lie groups, Lie algebras, and representations: an elementary introduction},
  author={Hall, Brian},
  volume={222},
  year={2015},
  publisher={Springer}
}


@article{he2007matrix,
  title={On matrix exponential distributions},
  author={He, Qi-Ming and Zhang, Hanqin},
  journal={Advances in Applied Probability},
  volume={39},
  number={1},
  pages={271--292},
  year={2007},
  publisher={Cambridge University Press}
}
@article{farzan2008notes,
  title={Transitive Closure vis-à-vis Matrix Multiplication},
  author={Farzan, Arash},
  journal={URL https://www.cs.bgu.ac.il/~dinitz/Course/SS-12/transitiveClosure.pdf},
  year={2008}
}

@inproceedings{cohn2005group,
  title={Group-theoretic algorithms for matrix multiplication},
  author={Cohn, Henry and Kleinberg, Robert and Szegedy, Balazs and Umans, Christopher},
  booktitle={46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)},
  pages={379--388},
  year={2005},
  organization={IEEE}
}


@article{haber2018notes,
  title={Notes on the matrix exponential and logarithm},
  author={Haber, Howard E},
  journal={URL http://scipp. ucsc. edu/\~{} haber/webpage/MatrixExpLog. pdf},
  year={2018}
}
@book{horn2012matrix,
  title={Matrix analysis},
  author={Horn, Roger A and Johnson, Charles R},
  year={2012},
  publisher={Cambridge university press}
}

@book{mckaya2007their,
  title={Their footprints remain: biomedical beginnings across the Indo-Tibetan frontier},
  author={McKaya, Alex},
  year={2007},
  publisher={Amsterdam University Press}
}

@article{sigman2009continuous,
  title={Continuous-time Markov chains},
  author={Sigman, Karl},
  journal={Notes for},
  volume={1},
  year={2009}
}

@article{salamati2020lyapunov,
  title={A Lyapunov Approach for Time-Bounded Reachability of CTMCs and CTMDPs},
  author={Salamati, Mahmoud and Soudjani, Sadegh and Majumdar, Rupak},
  journal={ACM Transactions on Modeling and Performance Evaluation of Computing Systems (TOMPECS)},
  volume={5},
  number={1},
  pages={1--29},
  year={2020},
  publisher={ACM New York, NY, USA}
}

@article{tolver2016introduction,
  title={An introduction to Markov chains},
  author={Tolver, Anders},
  journal={Recuperado el},
  volume={15},
  year={2016}
}

@inproceedings{nielsen2010model,
  title={Model checking multivariate state rewards},
  author={Nielsen, Bo Friis and Nielson, Flemming and Nielson, Hanne Riis},
  booktitle={2010 Seventh International Conference on the Quantitative Evaluation of Systems},
  pages={7--16},
  year={2010},
  organization={IEEE}
}

@article{williams2014multiplying,
  title={Multiplying matrices in O (n2. 373) time},
  author={Williams, Virginia Vassilevska},
  journal={preprint},
  year={2014},
  publisher={Citeseer}
}


@inproceedings{coppersmith1987matrix,
  title={Matrix multiplication via arithmetic progressions},
  author={Coppersmith, Don and Winograd, Shmuel},
  booktitle={Proceedings of the nineteenth annual ACM symposium on Theory of computing},
  pages={1--6},
  year={1987}
}


@article{moler2003nineteen,
  title={Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later},
  author={Moler, Cleve and Van Loan, Charles},
  journal={SIAM review},
  volume={45},
  number={1},
  pages={3--49},
  year={2003},
  publisher={SIAM}
}

@article{navarro2019order,
  title={Order statistics and multivariate discrete phase-type distributions.},
  author={Navarro, Azucena Campillo},
  year={2019},
  publisher={DTU Compute}
}

@inproceedings{rahmann2003dynamic,
  title={Dynamic programming algorithms for two statistical problems in computational biology},
  author={Rahmann, Sven},
  booktitle={International Workshop on Algorithms in Bioinformatics},
  pages={151--164},
  year={2003},
  organization={Springer}
}

@article{ruschendorf1985wasserstein,
  title={The Wasserstein distance and approximation theorems},
  author={R{\"u}schendorf, Ludger},
  journal={Probability Theory and Related Fields},
  volume={70},
  number={1},
  pages={117--129},
  year={1985},
  publisher={Springer}
}

@article{satta2019two,
  title={Two-dimensional site frequency spectrum for detecting, classifying and dating incomplete selective sweeps},
  author={Satta, Yoko and Zheng, Wanjing and Nishiyama, Kumiko V and Iwasaki, Risa L and Hayakawa, Toshiyuki and Fujito, Naoko T and Takahata, Naoyuki},
  journal={Genes \& Genetic Systems},
  pages={19--00012},
  year={2019},
  publisher={The Genetics Society of Japan}
}

@article{wiuf2006consistency,
  title={Consistency of estimators of population scaled parameters using composite likelihood},
  author={Wiuf, Carsten},
  journal={Journal of mathematical biology},
  volume={53},
  number={5},
  pages={821--841},
  year={2006},
  publisher={Springer}
}

@article{ccinlar1972markov,
  title={Markov additive processes. I},
  author={{\c{C}}inlar, Erhan},
  journal={Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und verwandte Gebiete},
  volume={24},
  number={2},
  pages={85--93},
  year={1972},
  publisher={Springer}
}

@phdthesis{mcmahon2008time,
  title={Time-dependence in Markovian decision processes.},
  author={McMahon, Jeremy James},
  year={2008}
}


@book{lozovanu2015optimization,
  title={Optimization of Stochastic Discrete Systems and Control on: Complex Networks},
  author={Lozovanu, Dmitrii and Pickl, Stefan},
  year={2015},
  publisher={Springer}
}

@book{martin1991introduction,
  title={Introduction to Languages and the Theory of Computation},
  author={Martin, John C},
  volume={4},
  year={1991},
  publisher={McGraw-Hill New York}
}


@inproceedings{horvath2007properties,
  title={On the properties of acyclic bilateral phase type distributions},
  author={Horv{\'a}th, Andr{\'a}s and Telek, Mikl{\'o}s},
  booktitle={Proceedings of the 2nd international conference on Performance evaluation methodologies and tools},
  pages={1--8},
  year={2007}
}

@article{horvath2009canonical,
  title={On the canonical representation of phase type distributions},
  author={Horv{\'a}th, G{\'a}bor and Telek, Mikl{\'o}s},
  journal={Performance Evaluation},
  volume={66},
  number={8},
  pages={396--409},
  year={2009},
  publisher={Elsevier}
}

@article{albrecher2019inhomogeneous,
  title={Inhomogeneous phase-type distributions and heavy tails},
  author={Albrecher, Hansj{\"o}rg and Bladt, Mogens},
  journal={Journal of Applied Probability},
  volume={56},
  number={4},
  pages={1044--1064},
  year={2019},
  publisher={Cambridge University Press}
}

@misc{goff2001multivariate,
  title={Multivariate discrete phase-type distributions with their applications},
  author={Goff, Matt},
  year={2001},
  publisher={Academic Press}
}

@article{tavare1984line,
  title={Line-of-descent and genealogical processes, and their applications in population genetics models},
  author={Tavar{\'e}, Simon},
  journal={Theoretical population biology},
  volume={26},
  number={2},
  pages={119--164},
  year={1984},
  publisher={Academic Press}
}


@article{nyholt2004simple,
  title={A simple correction for multiple testing for single-nucleotide polymorphisms in linkage disequilibrium with each other},
  author={Nyholt, Dale R},
  journal={The American Journal of Human Genetics},
  volume={74},
  number={4},
  pages={765--769},
  year={2004},
  publisher={Elsevier}
}

@article{rasmussen2014genome,
  title={Genome-wide inference of ancestral recombination graphs},
  author={Rasmussen, Matthew D and Hubisz, Melissa J and Gronau, Ilan and Siepel, Adam},
  journal={PLoS genetics},
  volume={10},
  number={5},
  year={2014},
  publisher={Public Library of Science}
}

@article{speidel2019method,
  title={A method for genome-wide genealogy estimation for thousands of samples},
  author={Speidel, Leo and Forest, Marie and Shi, Sinan and Myers, Simon R},
  journal={Nature genetics},
  volume={51},
  number={9},
  pages={1321--1329},
  year={2019},
  publisher={Nature Publishing Group}
}

@article{shchur2019distribution,
  title={On the distribution of tract lengths during adaptive introgression},
  author={Shchur, Vladimir and Svedberg, Jesper and Medina, Paloma and Corbett-Detig, Russ and Nielsen, RASMUS},
  journal={bioRxiv},
  pages={724815},
  year={2019},
  publisher={Cold Spring Harbor Laboratory}
}

@article{klein1999some,
  title={Some statistical improvements for estimating population size and mutation rate from segregating sites in DNA sequences},
  author={Klein, Etienne K and Austerlitz, Fr{\'e}d{\'e}ric and Lar{\'e}do, Catherine},
  journal={Theoretical population biology},
  volume={55},
  number={3},
  pages={235--247},
  year={1999},
  publisher={Elsevier}
}

@article{nasri2001optimal,
  title={Optimal parallelization of a recursive algorithm for triangular matrix inversion on MIMD computers},
  author={Nasri, Wahid and Mahjoub, Zaher},
  journal={Parallel Computing},
  volume={27},
  number={13},
  pages={1767--1782},
  year={2001},
  publisher={Elsevier}
}

@book{li2009fast,
  title={Fast algorithms for sparse matrix inverse computations},
  author={Li, Song},
  year={2009},
  publisher={Citeseer}
}

@article{yuster2005fast,
  title={Fast sparse matrix multiplication},
  author={Yuster, Raphael and Zwick, Uri},
  journal={ACM Transactions On Algorithms (TALG)},
  volume={1},
  number={1},
  pages={2--13},
  year={2005},
  publisher={ACM New York, NY, USA}
}


@article{duff1988sparsity,
  title={Sparsity structure and Gaussian elimination},
  author={Duff, Iain S and Erisman, Albert M and Gear, C William and Reid, John K},
  journal={ACM Signum newsletter},
  volume={23},
  number={2},
  pages={2--8},
  year={1988},
  publisher={ACM New York, NY, USA}
}


@article{amestoy2012computing,
  title={On computing inverse entries of a sparse matrix in an out-of-core environment},
  author={Amestoy, Patrick R and Duff, Iain S and L'Excellent, Jean-Yves and Robert, Yves and Rouet, Fran{\c{c}}ois-Henry and U{\c{c}}ar, Bora},
  journal={SIAM journal on Scientific Computing},
  volume={34},
  number={4},
  pages={A1975--A1999},
  year={2012},
  publisher={SIAM}
}

@book{blaesild2018statistics,
  title={Statistics with applications in biology and geology},
  author={Bl{\ae}sild, Preben and Granfeldt, Jorgen},
  year={2018},
  publisher={CRC Press}
}

@article{do2008more,
  title={More on multivariate gaussians},
  author={Do, Chuong B},
  journal={URL http://cs229. stanford. edu/section/more\_on\_gaussians. pdf.[Online},
  year={2008}
}

@article{weiss1976normal,
  title={The normal approximation to the multinomial with an increasing number of classes},
  author={Weiss, Lionel},
  journal={Naval Research Logistics Quarterly},
  volume={23},
  number={1},
  pages={139--149},
  year={1976},
  publisher={Wiley Online Library}
}

@article{spence2016site,
  title={The site frequency spectrum for general coalescents},
  author={Spence, Jeffrey P and Kamm, John A and Song, Yun S},
  journal={Genetics},
  volume={202},
  number={4},
  pages={1549--1561},
  year={2016},
  publisher={Genetics Soc America}
}

@article{tellier2014coalescence,
  title={Coalescence 2.0: a multiple branching of recent theoretical developments and their applications},
  author={Tellier, Aurelien and Lemaire, Christophe},
  journal={Molecular ecology},
  volume={23},
  number={11},
  pages={2637--2652},
  year={2014},
  publisher={Wiley Online Library}
}

@article{birkner2013statistical,
  title={Statistical properties of the site-frequency spectrum associated with Lambda-coalescents},
  author={Birkner, Matthias and Blath, Jochen and Eldon, Bjarki},
  journal={Genetics},
  volume={195},
  number={3},
  pages={1037--1053},
  year={2013},
  publisher={Genetics Soc America}
}

@article{fu1995statistical,
  title={Statistical properties of segregating sites},
  author={Fu, Yun-Xin},
  journal={Theoretical population biology},
  volume={48},
  number={2},
  pages={172--197},
  year={1995},
  publisher={Elsevier}
}

@article{commault2003phase,
  title={Phase-type distributions and representations: some results and open problems for system theory},
  author={Commault, Christian and Mocanu, St{\'e}phane},
  journal={International Journal of Control},
  volume={76},
  number={6},
  pages={566--580},
  year={2003},
  publisher={Taylor \& Francis}
}

@phdthesis{navarro2019discrete,
title = "Order statistics and multivariate discrete phase-type distributions.",
abstract = "In this thesis, we present three different topics of research which are related to the theory of phase-type distributions. Those topics are explained next. The ﬁrst research work is on order statistics from matrix-geometric distributions in the case of a sample of independent and non-identically distributed random variables. We prove that order statistics from matrix-geometric distributions are matrix-geometric distributed and we provide representations for their distributions. The second research work is a study of the discrete version of multivariate phasetype distributions introduced by V. G. Kulkarni. We give an expression for the joint probability-generating function in the similar way than in the continuous time case and under this base we make an analysis of this class of distributions and present examples that are commonly found in the literature. The third research work presented came out with the aim of relating the last two topics. That is, we found a problem which relates the concept of order statistics and multivariate phase-type distributions introduced by V. G. Kulkarni, the last in the case of continuous time. Thus, we present a research on concomitants of phase-type distributions. We provide a procedure to calculate the density function of concomitants of phase-type distributions and we prove that concomitants of phase-type distributions are phase-type distributed.",
author = "{Campillo Navarro}, Azucena",
school = "Technical University of Denmark (Copenhagen, Denmark). Department of Applied Mathematics and Computer Science.",
year = "2018",
language = "English",
series = "DTU Compute PhD-2018",
publisher = "DTU Compute",
note = {ISSN 0909-3192}
}


@article{HBA2021,
title={Multivariate phase-type theory for the site frequency spectrum},
author={Hobolth, Asger and Bladt, Mogens and Andersen, Lars N{\o}rvang},
 journal={Journal of Mathematical Biology},
 volume={83},
 number={6},
 pages={1--28},
 year={2021},
 publisher={Springer},
 DOI={https://doi.org/10.1007/s00285-021-01689-w}
}

@article{PhaseTypeR,
title={PhaseTypeR: phase-type distributions in R with reward transformations and a view towards population genetics},
author={Rivas-González, Iker and Andersen, Lars N{\o}rvang and Hobolth, Asger},
 journal={BioRxiv},
 volume={},
 number={},
 pages={},
 year={2022},
 publisher={BioRxiv},
 DOI={doi: https://doi.org/10.1101/2022.06.16.496381}
}

@article{kingman_1982, title={The coalescent}, volume={13}, DOI={10.1016/0304-4149(82)90011-4}, number={3}, journal={Stochastic Processes and their Applications}, author={Kingman, J.F.C.}, year={1982}, pages={235–248}} 

@article{bladt2021matrixdist,
  journal={arXiv},
  doi = {10.48550/ARXIV.2101.07987},
  url = {https://arxiv.org/abs/2101.07987},
  author = {Bladt, Martin and Yslas, Jorge},
  title = {matrixdist: An R Package for Inhomogeneous Phase-Type Distributions},
  publisher = {arXiv},
  year = {2021}
}

@Manual{expm,
title = {expm: Matrix Exponential},
author = {Goulet, Vincent and Dutang, Christophe and Maechler, Martin and Firth, David  and Shapira, Marina and Stadelmann, Michael},
year = {2021},
url = {https://cran.r-project.org/package=expm},
}