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1 Math-Net.Ru All Russian mathematical portal N. R. Mohan, S. Ravi, Max Domains of Attraction of Univariate Multivariate p-max Stable Laws, Teor. Veroyatnost. i Primenen., 1992, Volume 37, Issue 4, Use of the all-russian mathematical portal Math-Net.Ru implies that you have read agreed to these terms of use Download details: IP: November 14, 2018, 05:09:31

2 ТЕОРИЯ ВЕРОЯТНОСТЕЙ Том 37 И ЕЕ ПРИМЕНЕНИЯ Выпуск г. MOHAN N. R., RAVI S. 1} MAX DOMAINS OF ATTRACTION OF UNIVARIATE AND MULTIVARIATE p-max STABLE LAWS 1. Introduction. Let F be a distribution function (d.f.) on R^, d ^ 1. Suppose that there exist norming constant a n (i) > 0, b n (i) real, 1 < i ^ d, n ^ 1, a d.f. К on H d that with nondegenerate univariate marginals such lim F n (a n (i)xi + b n (t)-. 1 <» < d) = ЛГ(яг), (1.1) n *oo ж =..., х^) G C{K), where C(K) is the set of all continuity points of К. We shall call К as max stable d.f. under linear normalization or simply l-max stable d.f. In the univariate case it is well known that an /-max stable d.f. can be only one of the three types of extreme value d.f.'s of Gnedenko, namely,,, f 0, if x ^ 0, Ф «[ Х ) -\ехр{-х- а }, if x > 0, ф ( x ) = l e x P{-(-x) a }, ifa?<0, a V ; \l, ifx^o, Л(ж) = exp { exp( ж)}, -оо < ж < оо; where a is a positive parameter. If (1.1) holds, we write F E T>i(K) to indicate that F belongs to the max domain of attraction of К under linear normalization. A d.f. F on R is said to belong to the max domain of attraction of a d.f. H on H d with nondegenerate univariate marginals under power normalization if there exist norming constants a n (i) > 0 /3 n (i) > 0, 1 < i < d, n > 1, such that hm F n (a n (i)\x i \ Mi) sgn(x i ), 1 < i ^ d) = x C(#), (1.2) n >oo where sgn(a;j = 1 if ж г - < 0, = 0 if ж г - = 0, = 1 if X{ > 0. We shall denote this as F V p (H). We shall call H as max stable d.f. under power normalization or simply p-max stable d.f. if (1.2) holds. We say that two d.f.'s F G are of the same p-type if there exist A > 0 В > 0 such that jf(a?) = С(А ж В sgn (x)) for ж E R. It has been shown in [4] that a x ) Research supported by Junior Research Fellowship of the University Grants Commission, New Delhi, at the University of Mysore, Mysore, India. 709

3 univariate p-max stable d.f. can be ар-type of one of the following six d.f.'s, namely, -oo < x < oo; - oo < x < сю; if x < 1, if x > -1, where a is a positive parameter. It is easily seen that p-max stable laws on R d are continuous. Criteria for a d.f. F to belong to V\(K) are well known can be found in [1] [6]. In the next section we obtain necessary sufficient conditions for a univariate d.f. to belong to V p (-) for each of the six p-max stable laws. In section 3 we compare the max domains of attraction of univariate /-max stable laws with those of p-max stable laws. We show that every d.f. attracted to an /-max stable law is necessarily attracted to some p-max stable law that p-max stable laws, in fact, attract more. The results of section 2 are generalized in section 4 to obtain Marshall Olkin [3] type necessary sufficient conditions for a d.f. on R d, d > 1, to belong to P p (#), when H has the same p-type of univariate marginals. In the last section we show that if a d-variate d.f. F Vi(K), d > 1, then there necessarily exists a p-max stable d.f. on R d to whose domain F belongs under power normalization. In view of this result the result of section 3 it appears that it is desirable to go for power normalization rather than linear normalization. In [4] a criterion is given for a univariate d.f. to belong to the max domain of attraction of a max stable law under nonlinear normalization. It has been shown that for some invertible continuous function h(-) a regularly varying function ( ), if a univariate d.f. F satisfies h(x) lim * M~F{x)) = 1, then lim F n (G n ( I )) = exp(-e-' l ( x ) ), with G n (x) = h (h(x) + log(nl(logn))) conversely. As pointed out by a referee, in the case of F, a stard Normal d.f., Pancheva's condition 710

4 2 л X 1 Is satisfied with h(x) = sgn x L(x) =. However, It may 2 2VTT Z be noted that F G Р/(Л), as will be seen later, F G Р р (Ф) also. But Pancheva's criterion does not reveal these facts. For a d.f. F on ft, let r(f) = sup{x: F(x) < 1} denote its right extremity F = 1 F. For any nondecreasing function К define К (у) = = mf{x: K(x) > y}. We denote max(a, b) by a V 6. If F G 2> (Ж) on R then we denote the norming constants by a n b n so that (1.1) holds with d = 1, a n (l) = a n b n (l) = 6 n. Similarly, if F G V p (H) on R, then we denote the norming constants by a n (3 n so that (1.2) holds with d = 1, «n(l) = «n /? n (l) = p n. 2. Criteria for d.f.'s to belong to V p (-): univariate ease* Throughout this section Theorem from [1] will be referred to as G for brevity. The norming constants given at the end of each theorem of this section are defined In some cases for large n only. Theorem 2.1. A d.f. F G V p (H lta ) iff (1) r(f) = oo, ( И ) 1-^(ехр(0) " У ' In this case we may set a n = I /З п = log F~~(l 1/n). Proof. Let F G V p (H 1, a ). Observe that F n (0) -> 0 so r(f) > 0. Define for some a, 0 < a < r(f), V > 0 ' Then we have G n {j3 n y + loga n ) -» Ф а (у) hence by G 2.4.3, r(g) = = logr(f) = oo, proving (i). 1 Setting U = - -, W = U V = logw whenever W > 0, 1 r proceeding as In the proof of Proposition 1.11 (see [6, p. 55]), we show that V Is regularly varying with exponent 1/a. Therefore (1 F(e t )) Is regularly varying with exponent ( a), which Is (ii). Conversely, suppose that (I) (Ii) hold. If (3 n log F~~(l - 1/n), then n{l - F(exp(/3 n x))} -> x~ a, x > 0. Thus F G V p (H ha ) with the stated norming constants. Theorem 2.2. Л d.f. F G V p (H 2, a ) iff (1) 0 < r(f) < oo, г л r l-f(r(f)exp(-y/q) e ^l-f(r(f)ex P (-l/0) =^ / r(f) \ Here we may choose a n = r(f) /З п log ( = - j. ^F (1-1/n)' Proof. Suppose F G V p (H 2^). Then F n (0) -» 0 so r(f) > 0. 1 Let U -, W U y > * V = log W whenever W > 0. As in the proof of Proposition 1.12 (see [6, p. 60]), we show that r(f) = W(oo) < oo 711

5 (1) is proved. Proceeding as in the proof of the same Proposition, we show that (V(oo) - F(-))" 1 is regularly varying with exponent 1/a. This implies (1 F(r(F) exp(!/ ))) is regularly varying with exponent ( a), (ii) is proved. Now let (i) (ii) hold. Define ^ W - \F(r(F)ex V (-l/y)), iiy>0. Then from (ii) it follows that (1 - G(-)) is regularly varying with exponent (-a). Setting l/a n = log { F -^\/ n ))^ we have a n = G~(l - 1/n). Then from Proposition 1.11 [6] it follows that G n (a n y) Ф а (у), so that F n (r(f)exp(-l/(a n y))) Ф а (у). Thus F V p (H 2, a ) with the stated norming constants. Theorem 2.3. A d./. F V p (H 3^a) iff (i) r(f) = 0, l-f(-exp(-%)) _ e (n) hm ( у = y, i/ > 0. t_+oo 1 - F[ - exp(- )J The norming constants in this case may be chosen as a n = 1 /З п = = -log(-f"(l-l/n)). Proof. Define G(y) = F(-exp(-y)). Suppose that F V p (H 3, a ). This implies that r(f) < 0 G n (/3 n y - loga n ) > Ф а (у). Hence G Т>1(Ф а ) by G 2.4.3, we have r(g) = oo (1 G(-)) is regularly varying with exponent ( a). Thus r(f) = exp( r{g)) = 0 (1 -F(- exp(- ))) is regularly varying with exponent (-a), proving (i) (и)/ Now let (i) (ii) hold. Then (1 - <?( )) is regularly varying with exponent (-a) hence G G Х>,(Ф а ) by G Thus G n {a n y) where o n = G~(l - 1/n) = -log(-f~(l - 1/n)). Therefore - (-*)"") -* Ф а ( - log(-x)) - Я 3, а (х), -1< x < 0. -> Ф а (у), Since F(0) = 1 from (i), a n > oo, we have F n ( a: an sgn(x)) ^ F n ( 1)» -> 0 if ж < -1; F n (x a ") ^ F n (0) = 1 if x > 0. So, F V p (H 3, a ) as claimed. Theorem 2.4. A d.f. F V p (H 4, a ) iff (i) r(f) < 0, I 11 ) bm, у =», y>0. In Ш$ case a choice for the norming constants is a n /Г(1-1/п)\ = ( r(f)) /j " = ' g ( r(f) J' Proof. Defining G(y) = F(-exp(-y)) arguing as in the proof of Theorem 2.3 the theorem is established. Theorem 2.5. A d.f. F > р (Ф) iff (i) r(f) > 0, 712

6 exp (»/(*))) (II) lim =-г-г = exp( у), for some positive valued functlr(f) 1 r \t) tion f. r(f) / a /I - F(x)\ у dx < 00 for 0 < a < r(f) r(f) 1 / /1 F(x)\ (ii) /xo/ds ш'й choice f(t) = -^^y у ^- -^-J dx. Т/ге norming constants here may be chosen as a n = F~(l 1/n) j3 n = /(a n ). Proof. Suppose F G Х> р (Ф). This implies that r(f) > 0 a n > -+ r(f). Define G as in (2.1). Then G n {f3 n y + loga n ) = F n (a n (e y )^) -> Л(у), у real, so G 2?j(A). From Theorem 11 [2] we have for some positive valued function g. Also, if for some g, (2.2) holds, then r(g) / (1 G(x)) dx < oo, (7 may be chosen as r(g) 5 ( 0 = г г щ / ( x - G ^ ) ) d x = t Since r{g) = log(r(f)), (ii) follows from (2.1) (2.2). Now let (i) (ii) be true. With G defined as in (2.1) we have by Theorem 11 Corollary 13 [2] G n (a n x + b n ) > A(x), where b n = G~{\ - -1/n) = log F~(l - 1/n) = loga, say, a n = /(exp(6 )) = /(a n ) = /?, say. Thus F n (exp(b n )y a ") F 2? Р (Ф) with the stated norming constants. Theorem 2.6. A d.f. F G Р Р (Ф) ijff (i) r(f) < 0, -+ exp(-l/j/), у > 0. Since F n (0) -+ 0, we have 1-F(<exp (y/(t))) (ii) ^lim^ - = e, /or some positive valued function /. r(f) / a /1 F(x)\ ^ i-ij dx < 00 for a < V(F) r(f) / 1 \ (ii) Ao/ds ш*й Йе choice f(t) = ^- p(ty / J /1 - \ F(x)\ ) forming t constants in this case may be chosen as a n = F~(I 1/n) /З п = f{-<xn)- 713

7 Proof. Suppose that F G Р р (Ф). Then F n (0) -> 1 so r(f) ^ 0, proving (I). Define <7(y) = F(-exp(-y)). Then G n (@ n y - loga n ) -» A(y). The necessity part is proved now using Theorem 11 [2] arguing as in the proof of Theorem 2.5. Sufficiency follows as in the proof of Theorem Comparison of V p (-): univariate case. In the following theorem we compare the three max domains of attraction under linear normalization with the six max domains of attraction under power normalization. Theorem 3.1. Let F be a d.f. Then (a) (i) F G 2>,(Ф в ) 1 F e V m. (ii) F e V,(A), r(f) = 00 J * 6 U ^ h (b) F G V,{A), 0 < r(f) < 00 4=^ F G 2> р (Ф), r(f) < 00; (c) F G Z>,(A), r(f) < 0 FGP p ($), r(f) < 0; (d) (i) FG2MA), r(f) = 01 (ii) r(f) = 0 J ' (e) FGP ( (* A ), r(f)>0 FGX> p (tf 2, Q ); (f) F 6 P, ( U r(f) < 0 <=> FeV p (H 4<a ). Proof. We make use of the fact that the convergence is uniform in (1.1) (1.2). (a) (i) If F G Т>[(Ф а ) then b n = 0. Set a n = a n, (3 n = 1/a; define A «W = A<;> w = {0, «*< - [ ж, if 0 < x. Then lim F n (a n a; /3 " sgn(a;)) = lim F N (а Л^1) П (ж) + 6 N ) = = Ф а (Л (1) (ж)) = Ф(х). (3.1) (ii) Suppose F Х>/(Л) with r(i 7 ) = сю. Then 6 n > 0 for n large a n /b n 0. Set o n = 6 N, /? n = n n /6 n, define Л2Ъ ч f -l//? n > if ж < 0, { {x Pn - l)//? n, If0<z; л(2)/ ч = j -oo, if Ж ^ 0, ^ ' \ log ж, if 0 < x. Since A()i } (x)) = Ф(ж)> proceeding as In (3.1) we have F G Р р (Ф). (b) -If F G P (A) 0 < r(f) < oo then the proof that F G Р р (Ф) Is the same as that for the case r(f) = сю above. So let F G 2> Р (Ф) with 7.14

8 r(f) < oo. Then a n -> r(f) (3 n 0. Set a n = a n /3 n, b n = a n ; define so that (ib v _ /0, if -l//? n, U n { Х ) -\(1 + 0 п х)^-, if-l//? n <x; (1)/ \ x гг \x) e lim F n (a n x + b n ) = lim F n (a n \u^\x)f n sgn(a)) = = ф(и (1) (х)) = Л(х). (3.2) (с) If F Р ( (Л) r(f) < 0 then b n < 0 a n -> 0. Set a n = -6 n, /3 n = -a n /b n ; define л(зь ч f (l-(-x/")//? n, ifx<0, П ( ] ~.ll//3n, if 0 < x; л(з)( х ) = / -log(-x), if x < 0. 1 oo, if 0 ^ x. Note that Л(Л (3) (х)) = f (x). The claim now follows as in (3.1). Now let F G 2>р(Ф) with r(f) < 0. Then a n -* (-r(f)) /? я -» 0. Set a n = a n (3 nj b n = ( a n ); define л f-(l-/3 n x) 1 / / J -, ifx<l//? n, " V ' 10, ifl/pn^x; ts 2 \x) = exp( ж). Then proceeding as in (3.2) we get the result since (d) (i) Suppose F G Р/(Л) r(f) = 0. Then b n < 0 a n /b n Proceeding as in the proof of (c) above we show that F G Р Р (Ф). (ii) Now let F G Х>/(Ф<*) with r(f) = 0. Note that 6 n = 0. Set a n = a n, /3 n = 1/a; define A (4) (x) = AL 4) (x) = -(- a ; ) 1 / Q ' j *< ж. Observing that Ф а (Л (4) (х)) = Ф(х) we prove the claim as in (3.1). (e) Let F G 2?/(Ф a ) ' r(f) > 0. Then b n = r(f) a n -» 0. Set «n &п> /3n = Q>n/b n- Define -1//J n, if ж ^ 0, K 5} (%)= { (ал - 1)//J n, if0<s<l, 0 5 if 1 < ж; 715

9 { -oo, if x ^ 0, logs, if 0 < ж < 1, 0, if К x. Note that Ф a (X^5\x)) = #2,0(2). The rest of the arguments is as in (3.1). Suppose now that F G V p (H 2>a ). Ther a n = r(f) 0 n > 0. Set a n - а п (З п, b n = a n. Define f0, ifa:<-l//? n, <\x)= l(l + p n x) 1/p \ if-l//? n <*<0, 1 1, if0<z;,(3)^ _ j e*, if x < 0, if 0 ^ x. Then #2,а(и' 3^(ж)) = Ф а (ж) the result follows as in (3.2). (f) Let F G Х>,(Ф а ) r(f) < 0. Then b n = r(f) a n -> 0. Set «n = -b n, Pn = -a n /b n ; define А ( 6 Ь ) = f (1-(-х) Л )//З п, ifz<-l, w l0, if-1 < ж; д(б)/ т ч _ / - log(-ar), if x < -1, Since Ф а (А^(ж)) = #4, а (ж) proceeding as in (3.1) we conclude that F 6 If F e 2> р (Я 4, а ), then r(f) < 0, c* n = -r(f) /3 n -> 0. a n = a n p n, b n = -a n. Define Set n W l-l 5 if 0^ж; 14 1 j v f-exp(-s), ifx<0? l-l, if0<s; so that H 4^(u^4\x)) = Ф а (ж). The claim follows as in (3.2). The proof of the theorem is complete. 4. Some examples. Let if ж < 1, exp ( - (log ж) 2 ), if 1 ^ ж. It can be verified that F t Р р (Ф) with a n = exp( v /log га) /3 n ~ l/(2i/logn). However F x does not belong to 2?/(Ф а ) or to 2?/(Л). 716

10 Let ГО,. if x < -1, F 2 (x) = I 1 - exp ( - yj- log(-z)), if ~1 ^ ж < 0, I 1 if 0 < x. It can be seen that F 2 E Р р (Ф) with a n = exp(-(log n) 2 ) /? n = 2(logn). Note that F 2 neither belongs to Х>/(Ф а ) nor to 2?/(Л). The d.f. F / т ч /0, if x < e% 3K } \l-l/loglogx, ife e^z; neither belongs to ad (-) nor to a V p (-). Note that d.f.'s belonging to X> p (# 1?a ).or V p (H 3^a) max domain of attraction of any /-max stable law. do not belong to the 5. Criteria for d.f.'s to belong to P p (-): multivariate case. Throughout this the next chapter F G denote d.f.'s on R d ; К H denote /-max stable p-max stable laws on R d x = (ж 15, ж^) у = (vii - -> yd)- For a d.f. F on R d its г-th univariate marginal. For any two functions / g let fog respectively; /(*(*)). Theorem 5.1. Zei Я; = tf 1)Qj, 1 <» ^ d. Л d./. F G Р р (Я) iff (i) r(fi) = oo for all i, let F^ denote denote V ' t^oo 1 - F x (f) 6 W for all у such that H{y) > 0, where 9i(t) = log t, = log((f)~ о ofi(t)), 2< * < rf. Proof. Let F e Р Р (Я) so that F< G 2> Р (Я;) for all г. So (i) holds by Theorem 2.1. Define for some a > 0,.. _ f 0, if г/i < log a for at least one i,. \ F( ехр( й ), 1 ^ i< d), if log a ^ ю for all i; ( 5 Л ) G ( 2 / ) = = Я(ехр(ю), 1 < t < d) for all y. (5.2) Then G n (/? n (i)s/i+loga n (0, 1 ^ i ^ d) -> A'^) G G X>,(A'). Note that Ki{Vi) = #i,a,( ex P(2/»)) = Ф а > (^). Hence by Proposition 3.1. [3] we have ' = " l o S *M ( 5 " 3 ) for all у such that > 0, where fi\{t) t, jii(t) = {Gi)~ о Gi(t) = = 0<(е*), 2^i^d. Now (ii) follows from (5.3) using (5.1) (5.2). Now let (1) (ii) hold. Defining G К as in (5.1) (5.2), we note that (ii) implies (5.3). Hence by Proposition 3.1 [3] there exist a n (i) > 0, b n (i) real, 1 ^ i ^ d, such that G n (a n (i)yi + 6 п (г), 1 ^ г ^ d) -> AT(y). 717

11 Hence F G V P (H) with a n (i) = exp(6 n (i)), p n (i) = a n (i) Щ = H 1^i for all i. Theorem 5.2. Let H { = # 2? Q., 1 < i ^ d. A d.f. F V p (H) (i) 0 < r(itj) < oo /ог а// г, hi) lim -,.. v - = - log Я (j/) tn l-f x (r(f x )t) ^ k ; 6 /or a// у suc/i that H(y) > 0, where в х (t) = log t, for 2 ^ i ^ d e i{ t) = io g ( _ r L Fi) ). Proof. Similar to that of Theorem 5.1 makes use of Proposition 3.2 [3]. Theorem 5.3. Let H { = # 3, a., 1 ^ i ^ d. A d.f. F V p (H) (i) r(fi) = 0 for all i, iff iff = a" r ^ F o ~ l o g /or all у such that H{y) > 0, where 9i(t) log t, for 2 ^ i ^ d ^(0 = log(-(f,)~of 1 (-0). Proof. Define /0, if 2/ г - ^ 0 for at least one г, ^ j j?( _ 1 ^ г ^ d), if 0< y i for all г; JO, if yi ^ 0 for at least one г, K{y) = j я(- 1 < t < d), if 0 < y t - for all i. Note,that К is p-max stable that K\ = Я 1? а. for all г. Also, if F G G P p (#), then G G V p (K) conversely. The proof is similar to that of Theorem 5.1. The details are omitted. Theorem 5.4. Let Н { = Я 4^., 1 < i ^ d. A d.f. F G V p (H) (i) r(fi) < 0 /or all i, l-f( r(f0!^ ' l ( t ) sgn(y 2 ), K i ^ ) (ii) lim ^ / f ч \ = ~ loff iff /or all у such that H(y) > 0, where 0i(t) log t, for 2 < г < d The proof is similar to that of Theorem 5.3 hence is omitted. The following lemma, whose proof we omit, is a reformulation of Proposition 3.3 [3]. It is this form which is more suitable for use in the next two theorems. Note that the univariate analogue of this lemma is Theorem 11 [2]. Lemma. Let K { = А, 1 < г < d. A d.f. G V t (K) iff 1 - G{ Ci (t) yi + di(t), 1 ^ i < d) 1 718

12 for all у such that K(y) > 0, where d t (t) = t, di(t) (Gi)~ о Gy(t)^ 2 ^ i ^ d\ C((t) = fi(di(t)), for some positive function fi, 1 ^ i ^ d. r(a 5) If the condition holds for some 1 ^ i ^ d, then J (1 Gi(x)) dx < oo, a < r(gi)j for all г, й /го/ds ш Л Ле choice a r(ai) m = i _ / (i - ад) i Theorem 5.5. Let Я; = Ф, 1 < i ^ d. A d.f. F V p (H) (i) r(fi) > 0 for all i, iff for all у such that H(y) > 0, where rji(t) = t, rji(t) (Fi) 2 ^ i ^ d; fi is a positive function for each i. ofi(t)), a < r(fj), (ii) Ло/ds ш Л Йе choice / a /1 Fi(x)\ у J dx < oo /or r(fi) f 1-1 < i ^ d. Proof. Define G as in (5.1) with 0 < a < min r(fj), if as in (5.2). Note that ifj is Л for all i. The rest of the arguments is similar to that of Theorem 5.1 makes use of the above lemma. Theorem 5.6. Let Hi = f, 1 ><C % ^ d. A d.f. F 2> p (#) i/f (1) r(fi) ^ 0 /or all г, Г Л.. L-F(7 /,(Q Y T / - O R " ( < ) SGN( Y,), ttr i) ГПЩ = - ь ё я ы /or a// у such that H(y) > 0, w/геге 771 ( ) = Tji(t) = (Fi) о Fi(t), 2 ^ гd; /^ is a positive function for each i. r(fi) If (ii) Isolds /or some /;, 1 ^ г ^ d, Йеп J (~ d#^ < то /or a < r(fi)^ (ii) Ло/ds with the choice a r(fi) t Proof. Defining Cr(y) = F(- ехр(-^г), 1 ^ г ^ d) K(y) = = H( exp(?/ г ), 1 ^ г ^ d); proceeding on lines similar to that of Theorem 5.3, the theorem is proved. 719

13 6. Comparison of V p (-): multivariate case. The theorem below is a generalization of Theorem 3.1. If F E V { {K) for some /-max stable d.f. К then we denote the norming constants by a n {i) > 0 b n (i) real so that (1.1) holds. Similarly, if F V P (H) then a n (i) (3 n (i) denote the norming constants so as to satisfy (1.2). For a d.f. F on вд let 1^(1). denote the... z(fc))th fc-variate marginal, 1 < <... < i(k) ^ d, 2 < ^ d. Theorem 6.1. Let F Vi(K) for some l-max stable law K. Then there exists a p-max stable law H such that F 6 V p (H). Proof. Let F e Vi{K) so that (1.1) holds for some a n (i) > 0 b n (i) real. Then F { <E V t {Ki) for au i. Hence by Theorem 3.1, F { V p {Hi) for some p-max stable law Hi, which must necessarily be one of the four p- types of # 2? a, H Aoi, Ф, Ф. The norming constants a n (i) > 0 /? п (г) > 0 are determined by a n (i) Ь п (г) as in the proof of Theorem 3.1. Further, it follows from the proof of Theorem 3.1 that there exists 0^\xi) such that lim F?(a n (i)\xif n{i) sgn(xi)) = n >oo = JKm F?(a n {i)e { :\ Xi ) + bi(i)) = Ki( ( Xi )), where 0\ } is one of the A^'s, 1 < j < 6, defined as in the proof of Theorem 3.1 depending upon which one of the conditions is satisfied by F 2 e (i \xi) = lim e^\xi). So, НЛхЛ = КАв {1) (хл) for all i. Now fix n»>oo ж = (ж ь... If for some j, 1 < j < d, Hj(xj) = 0, then by Theorem 3.1, we have ^ п (а п (г') ж/ л ( 0 sgn(^), 1 <» < d) < < F^aniJ^x/^hgnixj)) - 0. (6.1) Suppose now that for some integers k, г(1),..., i(k), we have 0 < < Нц^(хц^) < 1, 1 < j < Ar, #;(ж;) = 1 for г ^ г(1),..., i(k). Since the convergence in (1.1) is uniform we have Mm F n (a n (i)\xif n(i) sgji(xi), n»oo 1 < i < d) ^ ^ Um F n (a n (i)e { :\ Xi ) + 6 n (i), 1 ^ < d) = - K{e (i \ Xl ), = = KiW...i{k){^{i) (x iu) ), l^j^k), (6.2) since Hi(xi) = Ki(e^\xi)) = 1 if i ф г'(1),..., %[k). Again, ПпТ F n (a n (t) x/" (0 sgn(z,), 1 < i < d) < < Jirr^ ^i)... i ( f c ) (a n (i(j))e { : U) \x i(j) ) + b n (i(j)), 1 < j < *) = 720 = K i(1) A{k) {9 i{j) (x iu) ), Kja). (6.3)

14 The claim now follows from (6.1), (6.2) (6.3) with Remark. In view of Theorem 3.1 Theorem 6.1 it is clear that p- max stable laws collectively attract more distributions than do /-max stable laws collectively. So we would conclude that it is preferable to use power normalization rather than linear normalization. REFERENCES 1. Galambos J. The Asymptotic Theory of Extreme Order Statistics. New York: Wiley, de Haan L. Slow variation characterization of domains of attraction. In: Statistical Extremes Applications. Proc. of the NATO Advanced Study Inst., Vimeiro, Ed. by Tiago de Oliveira, 1984, p Marshall A., Olkin I. Domains of attraction of multivariate extreme value distributions. Ann. Probab., 1983, v. 11, 1, p Pancheva E. Limit theorems for extreme order statistics under nonlinear normalization. Lect. Notes in Math., 1984, v. 1155, p Pancheva E. Max-stability. Теория вероятн. и ее примен., 1988, т. XXXIII, в. 1, с Resnick S. I. Extreme Values, Regular Variation, Point Processes. New York: Springer-Verlag, Поступила в редакцию 13.XII.1989 исправленный и сокращенный вариант 29.VII