×

zbMATH — the first resource for mathematics

On negative association of some finite point processes on general state spaces. (English) Zbl 1418.60042
Summary: 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 fulfills 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.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aleman, A.; Beliaev, D.; Hedenmalm, H., Real zero polynomials and Pólya-Schur type theorems., J. Anal. Math., 94, 49-60, (2004) · Zbl 1081.30005
[2] Anari, N.; Gharan, S. O.; Rezaei, A., Monte Carlo Markov chain algorithms for sampling strongly Rayleigh distributions and determinantal point processes., J. Mach. Learn. Res., 49, 103-115, (2016)
[3] Błaszczyszyn, B.; Yogeshwaran, D., Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications., Adv. Appl. Prob., 41, 623-646, (2009) · Zbl 1181.60022
[4] Błaszczyszyn, B.; Yogeshwaran, D., On comparison of clustering properties of point processes., Adv. Appl. Prob., 46, 1-20, (2014) · Zbl 1295.60059
[5] Błaszczyszyn, B.; Yogeshwaran, D., Stochastic Geometry, Spatial Statistics and Random Fields, 2120, Clustering comparison of point processes with applications to random geometric models, 31-71, (2015), Springer · Zbl 1328.60122
[6] Borcea, J.; Brändén, P.; Liggett, T., Negative dependence and the geometry of polynomials., J. Amer. Math. Soc., 22, 521-567, (2009) · Zbl 1206.62096
[7] Bulinski, A.; Shashkin, A., Limit Theorems for Associated Random Fields and Related Systems, (2007), World Scientific · Zbl 1154.60037
[8] Christofides, T. C.; Vaggelatou, E., A connection between supermodular ordering and positive/negative association., J. Multivariate Anal., 88, 138-151, (2004) · Zbl 1034.60016
[9] Hough, J. B.; Krishnapur, M.; Peres, Y.; Virág, B., Determinantal processes and independence., Probab. Surveys, 3, 206-229, (2006) · Zbl 1189.60101
[10] Hui, S.; Park, C., The representation of hypergeometric random variables using independent Bernoulli random variables., Commun. Statist. Theory Methods, 43, 4103-4108, (2014) · Zbl 1321.60019
[11] Joag-Dev, K.; Proschan, F., Negative association of random variables with applications., Ann. Statist., 11, 286-295, (1983) · Zbl 0508.62041
[12] Joe, H., Multivariate Models and Multivariate Dependence Concepts, (1997), CRC Press · Zbl 0990.62517
[13] Kulesza, A.; Taskar, B., Determinantal point processes for machine learning., Found. Trends Mach. Learn., 5, 123-286, (2012) · Zbl 1278.68240
[14] Kulik, R.; Szekli, R., Dependence orderings for some functionals of multivariate point processes., J. Multivariate Anal., 92, 145-173, (2005) · Zbl 1072.60011
[15] Last, G.; Penrose, M., Lectures on the Poisson Process, (2017), Cambridge University Press
[16] Last, G.; Szekli, R.; Yogeshwaran, D., Some remarks on associated random fields, random measures and point processes., ALEA, (2018)
[17] Li, C.; Jegelka, S.; Sra, S., Efficient sampling for k-determinantal point processes, Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS), 51, 1328-1337, (2015)
[18] Li, C.; Sra, S.; Jegelka, S.; Lee, D. D., Advances in Neural Information Processing Systems 29, Fast mixing Markov chains for strongly Rayleigh measures, DPPs, and constrained sampling, 4188-4196, (2016), Curran Associates
[19] Liggett, T., Ultra log-concave sequences and negative dependence., J. Combinatorial Theory A, 79, 315-325, (1997) · Zbl 0888.60013
[20] Liggett, T., Negative correlations and particle systems., Markov Process. Relat. Fields, 8, 547-564, (2002) · Zbl 1021.60084
[21] Lyons, R., Determinantal probability measures., Publ. Math. Inst. Hautes Études Sci., 98, 167-212, (2003) · Zbl 1055.60003
[22] Lyons, R., Determinantal probability: basic properties and conjectures, Proceedings of the International Congress of Mathematicians 2014, IV, 137-161, (2014) · Zbl 1373.60087
[23] Müller, A.; Stoyan, D., Comparison Methods for Stochastic Models and Risks., (2002), John Wiley: John Wiley, Chichester · Zbl 0999.60002
[24] Niculescu, C., A new look at Newton’s inequalities., J. Inequal. Pure Appl. Math., 1, 14, (2000) · Zbl 0972.26010
[25] Pemantle, R., Towards a theory of negative dependence., J. Math. Phys., 41, 1371-1390, (2000) · Zbl 1052.62518
[26] Poinas, A.; Delyon, B.; Lavancier, F., Mixing properties and central limit theorem for associated point processes. To appear in, Bernoulli, (2019) · Zbl 07066237
[27] Rüschendorf, L., Comparison of multivariate risks and positive dependence., J. Appl. Prob., 41, 391-406, (2004) · Zbl 1049.62060
[28] Soshnikov, A., Determinantal random point fields., Russian Math. Surveys, 55, 923, (2000) · Zbl 0991.60038
[29] Szekli, R., Stochastic Ordering and Dependence in Applied Probability, 97, (1995), Springer · Zbl 0815.60017
[30] Wagner, D. G., Negatively correlated random variables and Mason’s conjecture for independent sets in matroids., Ann. Combinatorics, 12, 211-239, (2008) · Zbl 1145.05003
[31] Whitt, W., Uniform conditional variability ordering of probability distributions., J. Appl. Prob., 22, 619-633, (1985) · Zbl 0571.60022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.