Loop-free Markov chains as determinantal point processes
Borodin, Alexei
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 19-28 / Harvested from Numdam
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Nous montrons que toute chaîne de Markov sans cycles sur un espace discret peut être vue comme un processus ponctuel determinantal. Comme application, nous démontrons des théorèmes limites centrales pour le nombre de particules dans une fenêtre pour des processus de renouvellement et des processus de renouvellement markoviens avec un bruit de Bernoulli.

We show that any loop-free Markov chain on a discrete space can be viewed as a determinantal point process. As an application, we prove central limit theorems for the number of particles in a window for renewal processes and Markov renewal processes with Bernoulli noise.

Publié le : 2008-01-01
DOI : https://doi.org/10.1214/07-AIHP115
Classification:  60J10,  60G55 
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Borodin, Alexei. Loop-free Markov chains as determinantal point processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 19-28. doi : 10.1214/07-AIHP115. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_1_19_0/

J. Ben Hough, M. Krishnapur, Y. Peres and B. Virag. Determinantal processes and independence. Probab. Surv. 3 (2006) 206-229. | MR 2216966

A. Borodin and G. Olshanski. Distributions on partitions, point processes and the hypergeometric kernel. Comm. Math. Phys. 211 (2000) 335-358. | MR 1754518 | Zbl 0966.60049

A. Borodin and G. Olshanski. Markov processes on partitions. Probab. Theory Related Fields 135 (2006) 84-152. | MR 2214152 | Zbl 1105.60052

A. Borodin and E. Rains. Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 (2005) 291-317. | MR 2185331 | Zbl 1127.82017

O. Costin and J. Lebowitz. Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75 (1995) 69-72.

W. Feller. An Introduction to Probability Theory and Its Applications, Volumes I and II. Wiley, 1968, 1971. , | Zbl 0155.23101

K. Johansson. Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 (2002) 225-280. | MR 1900323 | Zbl 1008.60019

R. Lyons. Determinantal probability measures. Publ. Math. Inst. Hâutes Études Sci. 98 (2003) 167-212. | Numdam | MR 2031202 | Zbl 1055.60003

O. Macchi. The coincidence approach to stochastic point processes. Adv. in Appl. Probab. 7 (1975) 83-122. | MR 380979 | Zbl 0366.60081

R. Pyke. Markov renewal processes: Definitions and preliminary properties. Ann. Math. Statist. 32 (1961) 1231-1242. | MR 133888 | Zbl 0267.60089

R. Pyke. Markov renewal processes with finitely many states. Ann. Math. Statist. 32 (1961) 1243-1259. | MR 154324 | Zbl 0201.49901

A. Soshnikov. Determinantal random point fields. Russian Math. Surveys 55 (2000) 923-975. | MR 1799012 | Zbl 0991.60038

A. Soshnikov. Gaussian fluctuation of the number of particles in Airy, Bessel, sine and other determinantal random point fields. J. Stat. Phys. 100 (2000) 491-522. | MR 1788476 | Zbl 1041.82001

A. Soshnikov. Gaussian limit for determinantal random point fields. Ann. Probab. 30 (2002) 171-187. | MR 1894104 | Zbl 1033.60063

A. Soshnikov. Determinantal random fields. In Encyclopedia of Mathematical Physics (J.-P. Francoise, G. Naber and T. S. Tsun, eds), vol. 2. Elsevier, Oxford, 2006, pp. 47-53. | MR 2238867

C. A. Tracy and H. Widom. Nonintersecting Brownian excursions. Ann. Appl. Probab. 17 (2007) 953-979. | MR 2326237 | Zbl 1124.60081