Loop-free Markov chains as determinantal point processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 19-28.

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.

DOI : 10.1214/07-AIHP115
Classification : 60J10, 60G55
Mots-clés : Markov chain, determinantal point process 
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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) no. 1, pp. 19-28. doi : 10.1214/07-AIHP115. http://www.numdam.org/articles/10.1214/07-AIHP115/

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

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

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

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

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

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

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

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

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

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

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

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 | Zbl

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

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

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

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