Short communication On the extremal behavior of a nonstationary normal random field Luı ´sa Pereira University of Beira Interior, Department of Mathematics, Portugal article info Article history: Received 2 December 2007 Received in revised form 16 November 2009 Accepted 27 April 2010 Available online 6 May 2010 Keywords: Random field Dependence conditions Nonstationarity Normal random field abstract Let X ¼fX n g n1 be a nonstationary random field satisfying a long range weak dependence for each coordinate at a time and a local dependence condition that avoids clustering of exceedances of high values. For these random fields, the probability of no exceedances of high values can be approximated by expðtÞ, where t is the limiting mean number of exceedances. We present a class of nonstationary normal random fields for which this result can be applied. & 2010 Published by Elsevier B.V. 1. Introduction Let X ¼fX n g n Z1 be a random field on Z d þ , where Z þ is the set of all positive integers and d Z2. We shall consider the conditions and results for d =2 since it is notationally simplest and the results for higher dimensions follow analogous arguments. Random fields in two and three dimensions are encountered in a wide range of sciences such as hydrology, agriculture and geology. For i =(i 1 ,i 2 ) and j =(j 1 ,j 2 ), i rj means i k rj k , k =1,2, and n ¼ðn 1 , n 2 Þ-‘ means n k -1, k ¼ 1, 2. For a family of real levels fu n, i : i rng n Z1 and a subset I of the rectangle of points R n ¼f1, ... , n 1 gf1, ... , n 2 g, we will denote the event fX i ru n, i : i 2 Ig by fM n ðIÞ rug or simply by fM n rug when I = R n . We say the pair I and J is in S i ðlÞ, for each i =1,2, if the distance between P i ðIÞ and P i ðJÞ is greater or equal to l, where P i , i ¼ 1, 2, denote the cartesian projections. The distance d(I,J) between sets I and J of Z 2 þ , is the minimum of distances dði, jÞ¼ maxfji s j s j, s ¼ 1, 2g, i 2 I and j 2 J. The dependence structure used here for nonstationary random fields is a coordinatewise-mixing type condition as the Dðu n Þcondition introduced in Leadbetter and Rootze ´n (1998), which restrict dependence by limiting jPðM n ðI 1 Þ ru, M n ðI 2 Þ ruÞPðM n ðI 1 Þ ruÞPðM n ðI 1 Þ ruÞj with the two indexes sets I 1 and I 2 being ‘‘separated’’ from each other by a certain distance along each coordinate direction. Definition 1.1. Let F be a family of indexes sets in R n . The nonstationary random field X on Z 2 þ satisfies the condition Dðu n, i Þ over F if there exist sequences of integer valued constants fk n i g n i Z1 , fl n i g n i Z1 , i =1,2, such that, as n ¼ðn 1 , n 2 Þ!‘, we have ðk n 1 , k n 2 Þ!‘, k n 1 l n 1 n 1 , k n 2 l n 2 n 2   !0, Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/jspi Journal of Statistical Planning and Inference ARTICLE IN PRESS 0378-3758/$ - see front matter & 2010 Published by Elsevier B.V. doi:10.1016/j.jspi.2010.04.049 E-mail address: lpereira@mat.ubi.pt Journal of Statistical Planning and Inference 140 (2010) 3567–3576