A Central Limit Theorem for the Volumes of High Excursions of Stationary Associated Random Fields

  • Vadim Demichev Lomonosov Moscow State University, 119991 Moscow, Russia; The University of Sheffield, S10 2TN Sheffield, UK
  • Judith Sarah Olszewski Institute of Stochastics, Ulm University
Keywords: Association, excursion set, central limit theorem, Gaussian random field, shot-noise.


We prove that under certain conditions the excursion sets volumes of stationary positively associated random fields converge after rescaling to the normal distribution as the excursion level and the size of the observation window grow. In addition, we provide a number of examples.


J. M. Azaïs and M. Wschebor. Level Sets and Extrema of Random Processes and Fields. Wiley, New York, 2009.

R.J. Adler and J.E. Taylor. Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York, 2007.

D. Meschenmoser and A. Shashkin. Functional central limit theorem for the volume of excursion sets generated by associated random fields. Statistics and Probability Letters, 81(6):642–646, 2011.

A. Bulinski, E. Spodarev, and F. Timmermann. Central limit theorems for the excursion sets volumes of weakly dependent random fields. Bernoulli, 18(1):100–118, 2012.

V. Demichev. Functional central limit theorem for excursion set volumes of quasi-associated random fields. Journal of Mathematical Sciences, 204(1):69–77, 2015.

E. Spodarev, Limit theorems for excursion sets of stationary random fields, In Modern Stochastics and Applications, Springer Optimization and its Applications, vol. 90, pp. 221–244, 2014.

A. V. Ivanov and N. N. Leonenko. Statistical Analysis of Random Fields. Kluwer, Dordrecht, 1989.

Y. Bakhtin. Poisson limit for associated random fields. Theory of Probability and its Applications, 54(4):678–681, 2010.

C. M. Newman. Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Y. L. Tong and S. Gupta, editors, Inequalities in Statistics and Probability, volume 5 of Lecture Notes–Monograph Series, pages 127–140. Institute of Mathematical Statistics, 1984.

A. Bulinski and A. Shashkin. Limit Theorems for Associated Random Fields and Related Systems. World Scientific, Singapore, 2007.

J. Karamata. Sur un mode de croissance régulière. Théorèmes fondamentaux. Bulletin de la Société Mathématique de France, 61:55–62, 1933.

A. Bulinski and E. Spodarev. Introduction to random fields. In E. Spodarev, editor, Stochastic Geometry, Spatial Statistics and Random Fields. Asymptotic Methods, volume 2068 of Lecture Notes in Mathematics, pages 277–335. Springer, Berlin, 2013.

K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2007.

A. Bose, S. Gangopadhyay, A. Sarkar, and A. Sengupta. Asymptotic properties of sums of upper records. Extremes, 6:147–164, 2003.

B. C. Arnold and J. A. Villaseñor. The asymptotic distributions of sums of records. Extremes, 1(3):351–363, 1999.

W. Vervaat. Success Epochs in Bernoulli Trials, with Applications in Number Theory. Mathematisch Centrum, Amsterdam, 1972.

G. Last and M. D. Penrose. Poisson process fock space representation, chaos expansion and covariance inequalities. In Probability Theory and Related Fields, volume 150, pages 663–690. Springer, Berlin, 2011.

How to Cite
Demichev, V., & Olszewski, J. S. (2015). A Central Limit Theorem for the Volumes of High Excursions of Stationary Associated Random Fields. Statistics, Optimization & Information Computing, 3(2), 138-146. https://doi.org/10.19139/soic.v3i2.137
Research Articles