Hostname: page-component-7b9c58cd5d-dlb68 Total loading time: 0 Render date: 2025-03-15T14:04:22.025Z Has data issue: false hasContentIssue false
  • English
  • Français

Stein's method for negatively associated random variables with applications to second-order stationary random fields

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  28 March 2018

Nathakhun Wiroonsri
Nathakhun Wiroonsri*
Affiliation:
University of Southern California
*
* Postal address: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA. Email address: wiroonsr@usc.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ξ = (ξ1, . . ., ξm) be a negatively associated mean-zero random vector with components that obey the bound |ξi| ≤ B, i = 1, . . ., m, and whose sum W = ∑i=1mξi has variance 1. The bound d1(ℒ(W), ℒ(Z)) ≤ 5B - 5.2∑i≠ jσij is obtained, where Z has the standard normal distribution and d1(∙, ∙) is the L1 metric. The result is extended to the multidimensional case with the L1 metric replaced by a smooth functions metric. Applications to second-order stationary random fields with exponential decreasing covariance are also presented.

Keywords

Correlation inequalityblock dependencenegative dependencerandom field

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G60: Random fields
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018 , pp. 196 - 215
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Ahlberg, J. H. and Nilson, E. N. (1963). Convergence properties of the spline fit. J. Soc. Indust. Appl. Math. 11, 95–104. Google Scholar
[2]Banerjee, S. and Roy, A. (2014). Linear Algebra and Matrix Analysis for Statistics. CRC, Boca Raton, FL. Google Scholar
[3]Barbour, A. D. (1990). Stein's method for diffusion approximations. Prob. Theory Relat. Fields 84, 297–332. Google Scholar
[4]Birkel, T. (1988). On the convergence rate in the central limit theorem for associated processes. Ann. Prob. 16, 1685–1698. Google Scholar
[5]Bulinskii, A. (1996). Rate of convergence in the central limit theorem for fields of associated random variables. Theory Prob. Appl. 40, 136–144. Google Scholar
[6]Cai, G.-H. and Wang, J.-F. (2009). Uniform bounds in normal approximation under negatively associated random fields. Statist. Prob. Lett. 79, 215–222. Google Scholar
[7]Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein's Method. Springer, Heidelberg. Google Scholar
[8]Cox, J. T. and Grimmett, G. (1984). Central limit theorems for associated random variables and the percolation model. Ann. Prob. 12, 514–528. Google Scholar
[9]Daly, F. (2013). Compound Poisson approximation with association or negative association via Stein's method. Electron. Commun. Prob. 18, 30. Google Scholar
[10]Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474. Google Scholar
[11]Goldstein, L. and Wiroonsri, N. (2018). Stein's method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process. Ann. Inst. H. Poincaré. 54, 385–421. Google Scholar
[12]Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11, 286–295. Google Scholar
[13]Li, Y.-X. and Wang, J.-F. (2008). An application of Stein's method to limit theorems for pairwise negative quadrant dependent random variables. Metrika 67, 1–10. Google Scholar
[14]Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74, 119–128. Google Scholar
[15]Rachev, S. T. (1985). The Monge–Kantorovich mass transference problem and its stochastic applications. Theory Prob. Appl. 29, 647–676. Google Scholar
[16]Ross, N. (2011). Fundamentals of Stein's method. Prob. Surv. 8, 210–293. Google Scholar
[17]Shevtsova, I. (2011). On the absolute constants in the Berry–Esseen type inequalities for identically distributed summands. Preprint. Available at https://arxiv.org/abs/1111.6554. Google Scholar
[18]Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Prob., Vol. II, University of California Press, Berkeley, CA, pp. 583–602. Google Scholar