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Loop-free Markov chains as determinantal point processes. (English) Zbl 1186.60066

A random subset \(X\) of a discrete space \(Y\) is considered. A distribution \(P\) of this subset satisfies the following condition: there exists a positive definite function \(K(x,y)\) \((x,y\in Y)\) such that for any \(n\geq2\), \(y_i\in Y\) it is fair \(P((y_1,\dots.y_n)\subset X)=\text{det}(K(y_i,y_j))_{n\times n}\). This random subset is said to be a determinantal point process with the correlation kernel \(K\). The author shows an example of such a point process. This is a homogeneous Markov chain with a transition matrix \(P_{xy}\) such that for any \(k\geq1\) and \(x\in Y\) \(P_{xx}^k=0\) (so called loop-free Markov chain). The family of realizations of this chain determines a random subset \(X\) of \(Y\), which is, by the author’s assertion, a determinantal point process. The correlation kernel of this process is derived in terms of the original Markov chain. The author shows that renewal processes and semi-Markov processes on a discrete space can be considered as loop-free Markov chains. Some transformations of such a random subset are shown to preserve determinantal property of this subset. A variant of the central limit theorem for determinantal point processes is proved.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60G17 Sample path properties
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
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References:

[1] J. Ben Hough, M. Krishnapur, Y. Peres and B. Virag. Determinantal processes and independence. Probab. Surv. 3 (2006) 206-229. · Zbl 1189.60101 · doi:10.1214/154957806000000078
[2] A. Borodin and G. Olshanski. Distributions on partitions, point processes and the hypergeometric kernel. Comm. Math. Phys. 211 (2000) 335-358. · Zbl 0966.60049 · doi:10.1007/s002200050815
[3] A. Borodin and G. Olshanski. Markov processes on partitions. Probab. Theory Related Fields 135 (2006) 84-152. · Zbl 1105.60052 · doi:10.1007/s00440-005-0458-z
[4] A. Borodin and E. Rains. Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121 (2005) 291-317. · Zbl 1127.82017 · doi:10.1007/s10955-005-7583-z
[5] O. Costin and J. Lebowitz. Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75 (1995) 69-72.
[6] W. Feller. An Introduction to Probability Theory and Its Applications , Volumes I and II. Wiley, 1968, 1971., · Zbl 0155.23101
[7] K. Johansson. Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 (2002) 225-280. · Zbl 1008.60019 · doi:10.1007/s004400100187
[8] R. Lyons. Determinantal probability measures. Publ. Math. Inst. Hâutes Études Sci. 98 (2003) 167-212. · Zbl 1055.60003 · doi:10.1007/s10240-003-0016-0
[9] O. Macchi. The coincidence approach to stochastic point processes. Adv. in Appl. Probab. 7 (1975) 83-122. JSTOR: · Zbl 0366.60081 · doi:10.2307/1425855
[10] R. Pyke. Markov renewal processes: Definitions and preliminary properties. Ann. Math. Statist. 32 (1961) 1231-1242. · Zbl 0267.60089 · doi:10.1214/aoms/1177704863
[11] R. Pyke. Markov renewal processes with finitely many states. Ann. Math. Statist. 32 (1961) 1243-1259. · Zbl 0201.49901 · doi:10.1214/aoms/1177704864
[12] A. Soshnikov. Determinantal random point fields. Russian Math. Surveys 55 (2000) 923-975. · Zbl 0991.60038 · doi:10.1070/rm2000v055n05ABEH000321
[13] 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. · Zbl 1041.82001 · doi:10.1023/A:1018672622921
[14] A. Soshnikov. Gaussian limit for determinantal random point fields. Ann. Probab. 30 (2002) 171-187. · Zbl 1033.60063 · doi:10.1214/aop/1020107764
[15] 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.
[16] C. A. Tracy and H. Widom. Nonintersecting Brownian excursions. Ann. Appl. Probab. 17 (2007) 953-979. · Zbl 1124.60081 · doi:10.1214/105051607000000041
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