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1 APPLICATIONES MATHEMATICAE 22,2 (1994), pp M. WIŚNIEWSKI (Kielce) EXTREME ORDER STATISTICS IN AN EQUALLY CORRELATED GAUSSIAN ARRAY Abstract. This paper cotais the results cocerig the wea covergece of d-dimesioal extreme order statistics i a Gaussia, equally correlated array. Three types of limit distributios are foud ad sufficiet coditios for the existece of these distributios are give. 1. Notatio ad deftios. Let {X () : {1,..., }, N} be a triagular array of d-dimesioal radom vectors whose mea values ad variaces satisfy (i) We assume that EX () V X () = (EX () i = 0 : i {1,..., d}), = (V X () i = 1 : i {1,..., d}). () the rows of the cosidered array are Gaussia equally correlated sequeces. This meas that cov(x () i, X() j ) = ϱ(0), cov(x() i, X() lj ) = ϱ() for all i, j {1,..., d},, l {1,..., }, l, N. matrices of covariace coefficiets by (i) (0) = (ϱ (0) ) 1 i,j d, We additioally assume that () = (ϱ () ) 1 i,j d. ϱ () (0, 1) for i {1,..., d}, N. We deote the 1991 Mathematics Subject Classificatio: Primary 60G70. Key words ad phrases: extreme order statistics, Gaussia array, equally correlated.

2 194 M. Wiśiewsi We also defie, for each t (0, ) d ad v (0, 1) d, t 1/ (1 v 1 ) 1/ A(t) =....., B(v) = t 1/ (1 v d d ) 1/2 We deote by M () (for {1,..., }) the d-dimesioal vector of the th extreme order statistics i the sequece Thus we have {X () l : l {1,..., }}. M () i M ( 1) i... M (1) i for i {1,..., d}, N. We wat to fid the limit distributios of the vectors of extreme order statistics ormalized by meas of sequeces of vectors a = (a,..., a ) ad b = (b,..., b ), where b = (2 l ) 1/2 ad a = b b (l l + l 4π). (Notice that all algebraic operatios are meat compoetwise.) I 1962 S. M. Berma foud the limit distributio of the first extreme order statistics built o the base of a oe-dimesioal equally correlated Gaussia sequece (see Berma [1]). Mittal s, Ylvisaer s ad Picads s papers (see [4], [5]) give a geeralizatio of this result i the statioary case. I the followig sectio the limit distributios of the th extreme order statistics built o the base of a multidimesioal equally correlated Gaussia array are foud. 2. Mai results Propositio 1. Assume that the array {X () : {1,..., }, N} satisfies coditios (i) (i). The the rows of the array ca be represeted by meas of sums of idepedet vectors i the followig way: (X () ) 1,..., X() a.s. = (Y () 0 A(r()) + Y () 1 B(r()),..., Y () 0 A(r()) + Y () B(r())), where r() = (ϱ () 11,..., ϱ() dd ), ad {Y() : {0} N} is a idepedet Gaussia sequece with covariace matrices ( cov(y () ϱ () ) (1) 0 ) =, (ϱ () ϱ () jj )1/2 1 i,j d (2) cov(y () ) = ( ad with vectors of mea values ϱ (0) ϱ () [(1 ϱ () )(1 ϱ () jj )]1/2 EY () 0 = EY () = 0 ), 1 i,j d

3 Extreme order statistics 195 (see the oe-dimesioal case i Berma [1], Galambos [2], Sectio 3.8, Picads [5]). P r o o f. Fix N. We deote by {X () : N} a d-dimesioal, Gaussia, equally correlated sequece with [ ] cov(x (), (0) X() m ) = () () (0) for m, ad with EX () = 0 for N. (Thus {X () : N} cotais the th row of the cosidered array.) For i {1,..., d} the Gaussia sequeces of radom variables {X () i : N} are equally correlated with parameters ϱ (). Hece they have the followig represetatio (see Berma [1], Galambos [2]): X () i = Y () 0i (ϱ () ) 1/2 + Y () i (1 ϱ () ) 1/2 for i {1,..., d}, N, where the sequeces {Y () i : {0} N} cosist of idepedet radom Gaussia variables with mea 0 ad variace 1. The radom variables Y () 0i ca be obtaied from the ergodic theorem i the followig way: (3) Y () 0i = (ϱ () ) 1/2 1 l.i.m. Because the radom vector Z () EZ () where O () j=1 = 1 = 0 its characteristic fuctio Ψ () = (o () X () ji for i {1,..., d}. j=1 X() j A 1 (r()) is ormal ad is Ψ () (w) = exp( 1 2 wo() w ) for w R d, [ (4) o () 1 (p, q) = ϱ(0) pq + (p, q)) 1 p,q d. It is easy to see that ( 1 1 ) ϱ pq () Notice that if Y () 0 = (Y () 01,..., Y () ) the P ( Z () 0d Y () 0 > ε) = P (max{ Z () i From (3) we obtai P ( Z () P ( Z () i ] (ϱ () pp ϱ () qq ) 1/2. Y () 0i : i {1,..., d}} > ε) Y () 0i > ε) Y () 0 > ε) E Z () i Y 0i 2 ε 2. 0 for all ε > 0. Hece for each w R d we have Ψ () (w) Ψ () 0 (w), where Ψ () 0 is the characteristic fuctio of Y () 0. From (4) it results that Ψ () 0 (w) = exp( 1 2 wo() 0 w ), where O () 0 = (ϱ ) (ϱ() ϱ () jj ) 1/2 ) 1 i,j d. We have show that Y () 0 is ormally distributed with covariace matrix (1).

4 196 M. Wiśiewsi Defie the radom Gaussia sequece Y () = [X () Y () 0 A(r())]B(r()) 1. From (3) it follows (Rudi [6], Theorem 4.6) that (5) EX () i Y () 0j = (ϱ () 1 m jj ) 1/2 lim m m Hece we obtai (for N) cov(y () i Y () j ) = [(1 ϱ () )(1 ϱ () jj )] 1/2 E[X () i p=1 (ϱ () = [(1 ϱ () )(1 ϱ () jj )] 1/2 [ϱ (0) (ϱ () (ϱ () = (ϱ (0) ) 1/2 ϱ () ϱ () (ϱ() )[(1 ϱ() EX () i X() pj jj )1/2 ϱ () ) 1/2 + (ϱ () ) 1/2 (ϱ () )(1 ϱ () jj )] 1/2. = ϱ () (ϱ() jj ) 1/2. ) 1/2 Y () 0i ][X () j (ϱ () jj )1/2 Y () 0j ] jj )1/2 ϱ () (ϱ() jj ) 1/2 (ϱ() ϱ () jj ) 1/2 ] I other words, Y () has the covariace matrix (2). The idepedece of the vectors of the sequece {Y () : {0} N} results from (5) i the followig way: cov(y () 0i Y () j ) ad cov(y () i Y () mj ) = (1 ϱ () jj ) 1/2 [ϱ () (ϱ() ) 1/2 (ϱ () = [(1 ϱ () )(1 ϱ () jj )] 1/2 [ϱ () (ϱ () (ϱ () ) 1/2 ϱ () (ϱ() ad so the proof is complete. jj )1/2 ϱ () jj )1/2 ϱ () ) 1/2 + (ϱ () ) 1/2 (ϱ () jj )1/2 ϱ () (ϱ() ϱ () jj ) 1/2 ] = 0 (ϱ() jj ) 1/2 (ϱ() ϱ () jj ) 1/2 ] = 0 Theorem 1. Suppose the array {X () : {1,..., }, N} satisfies coditios (i) (i), ad additioally the followig coditios hold: (iv) (v) The P ((M () ϱ () l τ (0, ) for i {1,..., d}, ϱ () (ϱ() ϱ () jj ) 1/2 ϱ for i, j {1,..., d}. a )/b x) (Λ t Φ t )(x) for N, x R d,

5 where t = (τ 11,..., τ dd ), deotes covolutio, d Λ t (x) = Λ (x + t), Λ (x) = Extreme order statistics 197 e e x i Φ t (x) = Φ(2 1/2 xa 1 (t)), 1 s=0 (e x i ) s, s! ad Φ is the distributio fuctio of a Gaussia vector Y 0, with cov(y 0 ) = (ϱ ) 1 i,j d ad EY 0 = 0. {Y () l where P r o o f. We deote the th extreme order statistics i the sequece : l {1,..., }} by M () (see Propositio 1). Observe that (M () a )/b = I + J (), I = (2 l ) 1/2 Y () 0 A(r()), J () = [M () a B 1 (r())]b(r())/b. Sice the vectors I ad J () are idepedet, to complete the proof it is eough to show that for all x R d, (6) (7) P (I x) Φ t (x), P (J () x) Λ t (x). Coditio (v) implies that the distributio fuctios of the vectors Y () 0 (see Propositio 1) coverge poitwise to the distributio fuctio of Y 0 ; moreover, from (iv) it follows that (2 l ) 1/2 A(r()) 2 1/2 A(t). Hece we obtai (6). Corollary 2 of Wiśiewsi [7] shows that the idepedece of the compoets of the limit maximum vector M (1) is equivalet to the idepedece of the compoets of the limit vectors of the order statistics M () for N. From Example of Galambos [2] it follows that M (1) has idepedet compoets M (1) i. Additioally, Theorems ad of Leadbetter, Lidgre ad Rootzé [3] imply that Hece, we get We ote that P (M () i P ((M () P (J () x i ) = e e x i 1 s=0 (e x i ) s. s! a )/b x) Λ (x). x) = P ((M () A )/B x)

6 198 M. Wiśiewsi where A = a B 1 (r()), B = b B 1 (r()). From a multidimesioal versio of Khichi s theorem it follows that to complete the proof of (7) we must show that (8) ad (9) Now, (9) follows from ϱ () A i a b B i b τ 1. 0 (see (iv)). Sice (1 ϱ () ) 1/2 = ϱ() + O((ϱ () ) 2 ) as ϱ () 0, we have A i a = [ 1 b 2 ϱ() + O((ϱ () ) 2 )](2 l + o(l )) τ, ad this completes the proof. Theorem 2. If the array {X () : {1,..., }, N} satisfies coditios (i) (i) ad (iv) the P ((M () ϱ () l 0 for i {1,..., d}, a )/b x) Λ (x) for N, x R d. P r o o f. Notice that (see the proof of Theorem 1) P (max{ I i : i {1,..., d}} > ε) P ( I i > ε) Hece the coditio EI 2 i ε 2 = 1 ε 2 P ( I > ε) 0 for all ε > 0 follows from (iv). Now, the proof is similar to that of (2). 2ϱ () E(Y () 0i ) 2 l. Theorem 3. If the array {X () : {1,..., }, N} satisfies coditios (i) (i), (v) ad (iv) ϱ () l for i {1,..., d},

7 the P ([M () Extreme order statistics 199 a B(r())]A 1 (r()) x) Φ(x) for N, x R d. P r o o f. We otice that [M () where (see the proof of Theorem 1) N () a B(r())]A 1 (r()) = Y () 0 + N (), = (M () a )B(r())A 1 (r()). To complete the proof it is eough to show that (10) P ( N () > ε) 0 for all ε > 0, N. It is easy to see that (11) P (max{ N () i : i {1,..., d}} > ε) ( M () i P b P ( N () i > ε) a ) > ε(2ϱ() l ) 1/2. Sice the limit distributios of the sequeces {(M () i a )/b : N} exist for i {1,..., d}, N (see for example Galambos [2]), the coditio (10) follows from (iv) ad (11). We emphasize that i the situatio cosidered i Theorem 3 all extreme order statistics have idetical limit distributios. Fially, we formulate a result which is easy to obtai by the method of proof of Propositio 1 ad Theorem 3. Theorem 4. If a d-dimesioal, ormalized, Gaussia sequece {X : N} is equally correlated with covariace matrix ( ) (0) cov(x m, X ) = (1) (1) (0) (for m) ad ϱ (1) P ([M () (0, 1) for i {1,..., d}, the a B(r())]A 1 (r()) x) Φ 1 (x) for N, x R d, where Φ 1 is the distributio fuctio of a Gaussia vector Y with ( ϱ (1) ) cov(y) = ad EY = 0. (ϱ (1) ϱ(1) jj )1/2 1 i,j d

8 200 M. Wiśiewsi Refereces [1] S. M. Berma, Equally correlated radom variables, Sahyā A 24 (1962), [2] J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley, New Yor, [3] M. R. Leadbetter, G. Lidgre ad H. Rootzé, Extremes ad Related Properties of Radom Sequeces ad Processes, Spriger, New Yor, [4] Y. Mittal ad D. Ylvisaer, Limit distributios for the maxima of statioary Gaussia processes, Stochastic Process. Appl. 3 (1975), [5] J. P i c a d s III, Maxima of statioary Gaussia processes, Z. Wahrsch. Verw. Gebiete 7 (1967), [6] W. Rudi, Real ad Complex Aalysis, McGraw-Hill, New Yor, [7] M. Wiśiewsi, Multidimesioal poit processes of extreme order statistics, Demostratio Math., to appear. MATEUSZ WIŚNIEWSKI TECHNICAL UNIVERSITY OF KIELCE AL LECIA PAŃSTWA POLSKIEGO KIELCE, POLAND Received o

Lecture 19: Convergence

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