Author manuscript, published in "Statistics & Probability Letters 78, 11 (2010) 1301" DOI : 10.1016/j.spl.2007.11.025 Accepted Manuscript How to compute the extremal index of stationary random fields peer-00622144, version 1 - 12 Sep 2011 H. Ferreira, L. Pereira PII: DOI: Reference: S0167-7152(07)00409-9 10.1016/j.spl.2007.11.025 STAPRO 4841 To appear in: Statistics and Probability Letters Received date: 16 November 2005 Revised date: 10 July 2007 Accepted date: 6 November 2007 Please cite this article as: Ferreira, H., Pereira, L., How to compute the extremal index of stationary random fields. Statistics and Probability Letters (2007), doi:10.1016/j.spl.2007.11.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. of stationary random fields H. Ferreira∗ and L. Pereira SC Department of Mathematics RI P How to compute the extremal index T ACCEPTED MANUSCRIPT University of Beira Interior AN U Abstract: We present local dependence conditions for stationary random fields under which the extremal index and the asymptotic distribution of the maximum M(n1 ,...,nd ) can be calculated from the joint distribution of a finite number s1 s2 of variables. 1 Introduction TE DM keywords: Extremal index, local and long range dependence, random field. Let X = {Xn }n≥1 be a stationary random field on Zd+ , where Z+ is the set of all positive integers and d ≥ 2. We shall consider the conditions and results for d = 2 since it is notationally simplest and the proofs for higher dimensions follow analogous arguments. For i = (i1 , i2 ) and j = (j1 , j2 ), i ≤ j means ik ≤ jk , k = 1, 2. For a family of real levels {un }n≥1 and a subset I of the rectangle of points Rn = {1, . . . , n1 } × {1, . . . , n2 }, we will denote the event {Xi ≤ un : i ∈ I} by Mn (I) or simply by Mn when I = Rn . If I = ∅ then Mn (I) = −∞. EP For each i = 1, 2, we say the pair I and J is in S i (l) if the distance between Πi (I) and Πi (J) is great or equal to l, where Πi , i = 1, 2 denote the cartesian projections. ∗ Helena Ferreira, Department of Mathematics, University of Beira Interior, 6200 Covilh˜ a, Portugal. E-mail:[email protected] AC C peer-00622144, version 1 - 12 Sep 2011 Portugal 1 ACCEPTED MANUSCRIPT The distance d(I, J) between sets I and J of Zd+ , d ≥ 1, is the minimum of distances T d(i, j) = max{|is − js |, s = 1, . . . , d}, i ∈ I and j ∈ J. RI P Suppose that X satisfies the coordinatewise-mixing ∆(un )-condition introduced in Leadbetter and Rootz´en (1998) which exploit the past and future separation one coordinate at a time. Therefore, we suppose that for X there exist sequences of integer valued constants {kni }, {lni }, i = 1, 2, such that, as n = (n1 , n2 ) → ∞, we have (kn1 , kn2 ) → ∞, ( kn1 ln1 kn2 ln2 n1 , n2 ) → 0 and (kn1 ∆1 , kn1 kn2 ∆2 ) → 0, where ∆i are the components of the SC mixing coefficient defined as follows: ∆1 = sup|P (Mn (I1 ) ≤ un , Mn (I2 ) ≤ un ) − P (Mn (I1 ) ≤ un )P (Mn (I2 ) ≤ un )|, AN U n1 kn1 , ∆2 = sup|P (Mn (I1 ) ≤ un , Mn (I2 ) ≤ un ) − P (Mn (I1 ) ≤ un )P (Mn (I2 ) ≤ un )|, where the supremum is taken over pairs of I1 and I2 in S 2 (ln2 ) such that Π1 (I1 ) = Π1 (I2 ) and |Π2 (I2 )| ≤ n2 kn2 . TE DM Under the coordinatewise-mixing ∆(un )-condition we have the following asymptotic independence for maxima over {(i − 1)[ knn1 ] + 1, . . . , i[ knn1 ]} × {(j − 1)[ knn2 ] + 1, . . . , j[ knn2 ]}, 1 1 2 2 i = 1, . . . , kn1 , j = 1, . . . , kn2 : P (Mn ≤ un ) − P kn1 kn2 (M([ n1 kn1 n2 n2 ],[ k ]) ≤ un ) −→n→∞ 0. (1.1) Accordingly Choi (2002), we shall say that X has extremal index θ if for each τ > 0 (τ ) (τ ) (τ ) there exists {un }n≥1 such that, as n → ∞ , n1 n2 P (X1 > un ) → τ and P (Mn ≤ un ) → EP e−θτ . This paper is concerned with describing how to compute the extremal index of the stationary random field under local mixing conditions analogous to those considered in Chernick et al. (1991). Those local restrictions on the oscillations of the values of the random field enable to AC C peer-00622144, version 1 - 12 Sep 2011 where the supremum is taken over pairs of I1 and I2 in S 1 (ln1 ) such that |Π1 (I2 )| ≤ compute the extremal index from the joint distribution of a finite number s1 s2 of variables. 2 ACCEPTED MANUSCRIPT Condition D(s) (un ) T 2 In the following consider R∗i,j = {i1 , i1 + 1, . . . , j1 } × {i2 , i2 + 1, . . . , j2 } − {i}. In par- RI P ticular, for i = 1 we write simply R∗j . For sake of simplicity we write [n/k] for ([ knn1 ], [ knn2 ]). 1 2 Definition The random field X satisfies the condition D(s) (un ), for some s ∈ Z2+ , if there exist sequences of integer valued constants {kni }, {lni }, i = 1, 2, such that, as X n1 n2 max i≤[n/k] kn1 ln1 kn2 ln2 n1 , n2 ) → 0 and SC n → ∞, we have (kn1 , kn2 ) → ∞, ( P (Xi > un , Mn (R∗i,i+s−1 ) ≤ un , Xj > un ) → 0. j∈R[n/k] \Ri AN U conditions considered in Pereira and Ferreira (2005) and Pereira and Ferreira (2006). We shall pursue the direction of this dependence conditions and extend to spatial processes some known formulas to obtain the extremal index of time series. By applying stationarity and the D(s) (un )-condition, the asymptotic distribution of the maximum Mn can be calculated from the joint distribution of the s1 s2 variables X1 TE DM and Xi , i ∈ R∗s . Different local dependence conditions can be considered by changing the definition of the region R∗s . Therefore the parameter ν in the following proposition could be computed from a different set of variables. Proposition 2.1 Let X be a stationary random field satisfying the ∆(un )-condition and D(s) (un )-condition for some s. As n → ∞, we have n1 n2 P (X1 > un , Mn (R∗s ) ≤ un ) → ν > 0 if only if EP P (Mn ≤ un ) → e−ν , ν > 0. Proof: From (1.1) and the stationarity we get P (Mn ≤ un ) − P (M[n/k] AC C peer-00622144, version 1 - 12 Sep 2011 For the cases s = 1 = (1, 1) and s = 2 = (2, 2) we find in the above definition the local 3 kn kn 2 1 ≤ un ) = RI P P (Mn ≤ un ) − exp −(1 + o(1))kn1 kn2 P (M[n/k] > un ) = T ACCEPTED MANUSCRIPT X ∗ P (Mn ≤ un ) − exp −(1 + o(1))kn1 kn2 P (Xi > un , Mn (Ri,[n/k] ) ≤ un ) + o(1) = i≤[n/k] SC X ∗ P (Mn ≤ un )−exp −(1+o(1))kn1 kn2 P (X1 > un , Mn (R[n/k]−i+1 ) ≤ un )+o(1) = o(1). i≤[n/k] The result follows now by applying the D(s) (un )-condition, since for [ knn1 ] − i1 + 1 ≥ s1 or [ knn2 ] − i2 + 1 ≥ s2 it holds AN U P (X1 > un , Mn (R∗s ) ≤ un ) − P (X1 > un , Mn (R∗[n/k]−i+1 ) ≤ un ) ≤ X P (X1 > un , Mn (R∗s ) ≤ un , Xj > un ) = o(n1 n2 ). j≤[n/k] d(1,j)≥min{s1 ,s2 } TE DM Therefore ∗ P (Mn ≤ un ) − exp −(1 + o(1))n1 n2 P (X1 > un , Mn (Rs ) ≤ un ) + o(1) = o(1). (τ ) As a corollary we provide a relation between the limit of n1 n2 P (X1 > un , Mn (R∗s ) ≤ (τ ) un ) and the extremal index. (τ ) Proposition 2.2 Let X be a stationary random field satisfying the ∆(un )-condition and (τ ) EP D(s) (un )-condition for each τ > 0. The extremal index of X exists and is θ if and only if, as n → ∞, it holds n1 n2 P (Mn (R∗s ) ≤ un |X1 > un ) → θ. AC C peer-00622144, version 1 - 12 Sep 2011 2 1 4 ACCEPTED MANUSCRIPT From the Normal comparison lemma we can prove that the stationary normal random (τ ) T fields such that ρn = E(Xj Xj+n ) satisfies ρn log(n1 n2 ) → 0, as n → ∞, satisfy ∆(un )(τ ) condition and D(1) (un )-condition (Pereira e Ferreira (2005)). (τ ) (τ ) some s. RI P The m-dependent random fields satisfy ∆(un )-condition and D(s) (un )-condition for It is an open and interesting question to know if some autoregressive random fields can (τ ) satisfy D(s) (un )-condition for some s with different coordinates. SC Here we present a simple example to apply the results. 2 , let b (i), s = 1, 2, . . . , 8, be the neighbors of i defined Example: For each i = (i1 , i2 ) ∈ Z+ s as b1 (i) = (i1 + 1, i2 ) , b2 (i) = i + 1, b3 (i) = (i1 , i2 + 1) , b4 (i) = (i1 − 1, i2 + 1) , b5 (i) = AN U 2. bs (i) ∈ / Z+ Let Y = {Yn }n≥0 be an i.i.d. random field with common distribution function FY and define Xn = max Yn , Yb5 (n) , Yb6 (n) , Yb7 (n) , n ≥ 1. (τ ) Let {un }n≥1 be such that n1 n2 (1 − FY (un )) −−−→ τ > 0. Then, we have un ≡ un 1 for X with τ1 = 4τ . (τ ) TE DM n→∞ (τ ) X satisfies the ∆(un )-condition and D(2) (un )-condition for each τ > 0 and n1 n2 P (Mn (R∗2 ) ≤ un |X1 > un ) → 41 . Therefore X has extremal index θ = 14 . We present a simulation of this random field to illustrate its predisposition to form clusters of high values. In the figure 1 we present the values of Xi and in figure 2 the EP contours of values higher than the 97-percentile. AC C peer-00622144, version 1 - 12 Sep 2011 (i1 − 1, i2 ) , b6 (i) = i − 1, b7 (i) = (i1 , i2 − 1), b8 (i) = (i1 + 1, i2 − 1) and Xbs (i) = −∞ if 5 ACCEPTED MANUSCRIPT T figure 1 RI P 1 0.9 0.8 X(i,j) 0.7 0.6 SC 0.5 0.4 0.3 0.2 30 15 AN U 20 30 25 20 15 10 10 5 5 0 j 0 i figure 2 30 20 15 10 5 TE DM 25 EP 5 Acknowledgements: AC C peer-00622144, version 1 - 12 Sep 2011 25 10 15 20 25 30 We are grateful to the referee for his corrections. Work partially supported by FCT/POCI2010/FEDER (EXMIXMOD project). 6 T ACCEPTED MANUSCRIPT RI P References Chernick, M.R., Hsing, T. and McCormick, W.P. (1991). Calculating the extremal index for a class of stationary sequences. Adv. Appl. Prob. 23, 835-850. Choi, H. (2002) Central limit theory and extremes of random fields. PhD Dissertation, SC Univ. of North Carolina at Chapel Hill. Leadbetter and Rootz´en, H. (1998) On extreme values in stationary random fields Sto- Preprint. Univ. of Lisbon. AN U Pereira, L. and Ferreira, H. (2005) On extreme values in non stationary random fields. Pereira, L. and Ferreira, H. (2006) Limiting crossing probabilities of random fields. Journal EP TE DM of Applied Probability 43, 3. AC C peer-00622144, version 1 - 12 Sep 2011 chastic processes and related topics, 275-285, Trends Math. Birkhauser Boston, Boston. 7

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