# Loop-free Markov chains as determinantal point processes

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aihp115.dvi www.imstat.org/aihp Annales de l’Institut Henri Poincaré - Probabilités et Statistiques 2008, Vol. 44, No. 1, 19–28 DOI: 10.1214/07-AIHP115 © Association des Publications de l’Institut Henri Poincaré, 2008 Loop-free Markov chains as determinantal point processes Alexei Borodin Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: [email protected] Received 1 June 2006; accepted 25 September 2006 Abstract. 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. Résumé. 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. MSC: 60J10; 60G55 Keywords: Markov chain; Determinantal point process Introduction Let X be a discrete space. A (simple) random point process P onX is a probability measure on the set 2X of all subsets of X. P is called determinantal if there exists a |X| × |X| matrix K with rows and columns marked by elements of X, such that for any finite Y = (y1, . . . , yn) ⊂X one has P{X ∈ 2X|Y ⊂ X}= det[Kyiyj ]ni,j=1. The matrix K is called a correlation kernel for P . A similar definition can be given for X being any reasonable space; then the measure lives on locally finite subsets of X. Determinantal point processes (with X = R) have been used in random matrix theory since the early 60s. As a separate class, determinantal processes were first singled out in the mid-70s in [9] where the term fermion point processes was used. The term “determinantal” was introduced at the end of the 90s in [2], and it

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