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On negative association of some finite point processes on general state spaces

Part of: Special processes Markov processes

Published online by Cambridge University Press:  12 July 2019

Günter Last  and
Ryszard Szekli
Günter Last*
Affiliation:
Karlsruhe Institute of Technology
Ryszard Szekli*
Affiliation:
University of Wroclaw
*
*Postal address: Department of Mathematics, Karlsruhe Institute of Technology, Englerstr. 2, D-76131, Karlsruhe, Germany.
**Postal address: University of Wrocław, Mathematical Institute, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland. Email address: ryszard.szekli@uwr.edu.pl
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Abstract

We study negative association for mixed sampled point processes and show that negative association holds for such processes if a random number of their points fulfils the ultra log-concave (ULC) property. We connect the negative association property of point processes with directionally convex dependence ordering, and show some consequences of this property for mixed sampled and determinantal point processes. Some applications illustrate the general theory.

Keywords

Finite point processmixed sampled point processnegative associationstrong Rayleigh measureULC

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 139 - 152
Copyright
© Applied Probability Trust 2019 

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References

Aleman, A., Beliaev, D. and Hedenmalm, H. (2004). Real zero polynomials and Pólya–Schur type theorems. J. Anal. Math. 94, 49–60.CrossRefGoogle Scholar
Anari, N., Gharan, S. O. and Rezaei, A. (2016). Monte Carlo Markov chain algorithms for sampling strongly Rayleigh distributions and determinantal point processes. J. Mach. Learn. Res. 49, 103–115.Google Scholar
Błaszczyszyn, B. and Yogeshwaran, D. (2009). Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications. Adv. Appl. Prob. 41, 623–646.CrossRefGoogle Scholar
Błaszczyszyn, B. and Yogeshwaran, D. (2014). On comparison of clustering properties of point processes. Adv. Appl. Prob. 46, 1–20.CrossRefGoogle Scholar
Błaszczyszyn, B. and Yogeshwaran, D. (2015). Clustering comparison of point processes with applications to random geometric models. In Stochastic Geometry, Spatial Statistics and Random Fields (Lecture Notes Math. 2120), Springer, pp. 31–71.CrossRefGoogle Scholar
Borcea, J. Brändén, P. and Liggett, T. (2009). Negative dependence and the geometry of polynomials. J. Amer. Math. Soc. 22, 521–567.CrossRefGoogle Scholar
Bulinski, A. and Shashkin, A. (2007). Limit Theorems for Associated Random Fields and Related Systems. World Scientific.CrossRefGoogle Scholar
Christofides, T. C. and Vaggelatou, E. (2004). A connection between supermodular ordering and positive/negative association. J. Multivariate Anal. 88, 138–151.CrossRefGoogle Scholar
Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2006). Determinantal processes and independence. Probab. Surveys 3, 206–229.CrossRefGoogle Scholar
Hui, S. and Park, C. (2014). The representation of hypergeometric random variables using independent Bernoulli random variables. Commun. Statist. Theory Methods 43, 4103–4108.CrossRefGoogle Scholar
Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables with applications. Ann. Statist. 11, 286–295.CrossRefGoogle Scholar
Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts. CRC Press.CrossRefGoogle Scholar
Kulesza, A. and Taskar, B. (2012). Determinantal point processes for machine learning. Found. Trends Mach. Learn. 5, 123–286.CrossRefGoogle Scholar
Kulik, R. and Szekli, R. (2005). Dependence orderings for some functionals of multivariate point processes. J. Multivariate Anal. 92, 145–173.CrossRefGoogle Scholar
Last, G. and Penrose, M. (2017). Lectures on the Poisson Process. Cambridge University Press.CrossRefGoogle Scholar
Last, G., Szekli, R. and Yogeshwaran, D. (2018). Some remarks on associated random fields, random measures and point processes. ALEA, to appear. arXiv: 1903.06004.Google Scholar
Li, C., Jegelka, S. and Sra, S. (2015). Efficient sampling for k-determinantal point processes. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS) (Proc. Mach. Learn. Res. 51), pp. 1328–1337.Google Scholar
Li, C., Sra, S. and Jegelka, S. (2016). Fast mixing Markov chains for strongly Rayleigh measures, DPPs, and constrained sampling. In Advances in Neural Information Processing Systems 29, eds Lee, D. D. et al., Curran Associates, pp. 4188–4196.Google Scholar
Liggett, T. (1997). Ultra log-concave sequences and negative dependence. J. Combinatorial Theory A 79, 315–325.CrossRefGoogle Scholar
Liggett, T. (2002). Negative correlations and particle systems. Markov Process. Relat. Fields 8, 547–564.Google Scholar
Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98, 167–212.CrossRefGoogle Scholar
Lyons, R. (2014). Determinantal probability: basic properties and conjectures. In Proceedings of the International Congress of Mathematicians 2014, Vol. IV, pp. 137–161.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Niculescu, C. (2000). A new look at Newton’s inequalities. J. Inequal. Pure Appl. Math. 1, 17, 14pp.Google Scholar
Pemantle, R. (2000). Towards a theory of negative dependence. J. Math. Phys. 41, 1371–1390.CrossRefGoogle Scholar
Poinas, A., Delyon, B. and Lavancier, F. (2019). Mixing properties and central limit theorem for associated point processes. To appear in Bernoulli.CrossRefGoogle Scholar
Rüschendorf, L. (2004). Comparison of multivariate risks and positive dependence. J. Appl. Prob. 41, 391–406.CrossRefGoogle Scholar
Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55, 923.CrossRefGoogle Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer.CrossRefGoogle Scholar
Wagner, D. G. (2008). Negatively correlated random variables and Mason’s conjecture for independent sets in matroids. Ann. Combinatorics 12, 211–239.CrossRefGoogle Scholar
Whitt, W. (1985). Uniform conditional variability ordering of probability distributions. J. Appl. Prob. 22, 619–633.CrossRefGoogle Scholar