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›› 2010, Vol. 27 ›› Issue (5): 577-583.DOI: 10.7523/j.issn.2095-6134.2010.5.001

• Research Articles •     Next Articles

Convergence rate in a martingale CLT for percolation clusters

JIANG Jian-Ping, ZHANG San-Guo, GUO Tian-De   

  1. School of Mathematical Sciences, Graduate University, Chinese Academy of Sciences, Beijing 100049, China
  • Received:2010-03-09 Revised:2010-04-16 Online:2010-09-15
  • Supported by:

    Supported by Knowledge Innovation Program of the Chinese Academy of Sciences(kjcx-yw-s7), the National Natural Science Foundation of China (10831006) and Presidential Foundation of GUCAS(O85101BM03) 

Abstract:

Consider bond percolation on Zd with parameter p. Let Kn be the number of open clusters in [-n,n]d. We investigate the convergence rate in the martingaleCLT for Kn. Generally, the best convergence rate for classicalmartingale CLT is O(n-d/2), and our result is Pp((Kn-Ep(Kn))/(Varp(Kn))) ≤x =x-∞(1/(2π)) e(-y2)/2dy+o(n-d/2 +ε0) for all x, where ε0 is any constant real number in 0, d/2 . As far as we know, this is the first convergence rate in CLTs for percolation.

 

Key words: percolation, martingale, central limit theorem, rate of convergence

CLC Number: