This work links the conditional probability structure of Lancaster probabilities to a construction of reversible continuous-time Markov processes. Such a task is achieved by using the spectral expansion of the corresponding transition probabilities in order to introduce a continuous time dependence in the orthogonal representation inherent to Lancaster probabilities. This relationship provides a novel methodology to build continuous-time Markov processes via Lancaster probabilities. Particular cases of well-known models are seen to fall within this approach. As a byproduct, it also unveils new identities associated to well known orthogonal polynomials.
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DOI : 10.1051/ps/2020004
Keywords: Continuous-time reversible Markov process, Lancaster probabilities, orthogonal polynomials, spectral expansion
@article{PS_2020__24_1_100_0,
author = {Mena, Rams\'es H. and Palma, Freddy},
title = {Continuous-time {Markov} processes, orthogonal polynomials and {Lancaster} probabilities},
journal = {ESAIM: Probability and Statistics},
pages = {100--112},
year = {2020},
publisher = {EDP Sciences},
volume = {24},
doi = {10.1051/ps/2020004},
mrnumber = {4071319},
zbl = {1434.60196},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2020004/}
}
TY - JOUR AU - Mena, Ramsés H. AU - Palma, Freddy TI - Continuous-time Markov processes, orthogonal polynomials and Lancaster probabilities JO - ESAIM: Probability and Statistics PY - 2020 SP - 100 EP - 112 VL - 24 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/ps/2020004/ DO - 10.1051/ps/2020004 LA - en ID - PS_2020__24_1_100_0 ER -
%0 Journal Article %A Mena, Ramsés H. %A Palma, Freddy %T Continuous-time Markov processes, orthogonal polynomials and Lancaster probabilities %J ESAIM: Probability and Statistics %D 2020 %P 100-112 %V 24 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/ps/2020004/ %R 10.1051/ps/2020004 %G en %F PS_2020__24_1_100_0
Mena, Ramsés H.; Palma, Freddy. Continuous-time Markov processes, orthogonal polynomials and Lancaster probabilities. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 100-112. doi: 10.1051/ps/2020004
[1] and , Vol. 1 of Higher transcendental functions. McGraw-Hill, New York (1953). | Zbl
[2] , Arbitrage theory in continuous time. Oxford University Press (2009). | Zbl
[3] , and , A theory of the term structure of interest rates. Theory Valuat. 53 (2005) 126–164. | MR | Zbl
[4] , and , Gibbs sampling, exponential families and orthogonal polynomials. Stat. Sci. 23 (2008) 151–178. | MR | Zbl
[5] , Polynomial expansions of bivariate distributions. Ann. Math. Stat. 35 (1964) 1208–1215. | MR | Zbl
[6] and , The transition function of a Fleming Viot process. Ann. Probab. 21 (1993) 1571–1590. | MR | Zbl
[7] and , Markov processes: characterization and convergence. John Wiley & Sons (2009). | Zbl
[8] , Stochastic processes with orthogonal polynomial eigenfunctions. J. Comp. Appl. Math. 233 (2008) 739–744. | MR | Zbl
[9] and , Diffusion processes and coalescent trees. London Mathematical Society Lecture Notes Series (2010). | MR | Zbl
[10] and , Exact simulation of the Wright-Fisher diffusion. Ann. Appl. Probab. 27 (2017) 1478–1509. | MR | Zbl
[11] , A first course in stochastic processes. Academic Press (2014). | Zbl
[12] and , The differential equations of birth and death processes, and the Stieltjes moment problem. Trans. Am. Math. Soc. 85 (1957) 489–546. | MR | Zbl
[13] , Reversibility and stochastic networks. Cambridge University Press (2011). | MR | Zbl
[14] , Lancaster bivariate probability distributions with Poisson, negative binomial and gamma margins. Test 7 (1998) 95–110. | MR | Zbl
[15] , The structure of bivariate distributions. Ann. Math. Statist. 29 (1958) 719–736. | MR | Zbl
[16] , Correlations and canonical forms of bivariate distributions. Ann. Math. Stati. 34 (1963) 532–538. | MR | Zbl
[17] , Joint probability distributions in the Meixner class. J. Roy. Statist. Soc. 37 (1975) 434–443. | MR | Zbl
[18] , Lancaster probabilities and Gibbs sampling. Stat. Sci. 23 (2008) 187–191. | Zbl
[19] , vol. 113 of Continuous time Markov processes: an introduction. American Mathematical Soc. (2010). | MR | Zbl
[20] , Obtaining generating functions. Springer-Verlag (1971). | MR | Zbl | DOI
[21] , Orthogonale polynom system mit einer besonderen gestalt der erzeugenden funktion. J. London Math. Soc. 9 (1934) 6–13. | MR | Zbl | DOI
[22] and , On a construction of Markov models in continuous time. METRON 67 (2009) 303–323. | Zbl
[23] , Special functions. Chelsea (1971). | MR | Zbl
[24] , Vol. 146 of Stochastic processes and orthogonal polynomials. Springer Science & Business Media (2012). | MR | Zbl
[25] and , Birth-death processes with killing. Stat. Prob. Lett. 72 (2005) 33–42. | MR | Zbl | DOI
[26] , and , A Fleming-Viot process and bayesian nonparametrics. Ann. Appl. Prob. 17 (2007) 67–80. | MR | Zbl | DOI
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