Almost Sure Convergence in Extreme Value Theory

Transcription

1 Math. Nachr. 190 (1998), Almost Sure Convergence in Extreme Value Theory By SHIHONG CHENG of Beijing, LIANG PENG of Rotterdam, and YONGCHENG &I of Beijing (Received April 18, 1995) (Revised Version July 7, 1997) Abstract. Let XI,..., Xn be independent random variables with common distribution function F. Define M, := max Xi lsiln and let G(z) be one of the extreme-value distributions. Assume F E D(G), i.e., there exist an > 0 and b, E R such that P{(Mn - bn)/an 5 2) + G(I), for I E R. Let.) denote the indicator function of the set (-w,z] and S(G) =: {z : 0 < G(z) < 1). Obviously, l(-m,il((mn - bn)/an) does not converge almost surely for any z E S(G). But we shall Drove 1. Introduction..., X, be independent random variables with common distribution func- Let XI, tion F. Let M, = rnaxl~i~, Xi and G(z) be one of the extreme-value distributions: aa(z) = exp{-z-"}, X > O, a>0; *a(z) = exp{-(-z)a}, z < 0, (Y > 0; A(x) = exp {-e-'}, XER. If F is in the domain of attraction of extreme value distribution G, then there exist a, > 0 and b, R such that (M, - b,)/a, converges in distribution to G(x) (see [5]). But for mathematical statistics it may be of some interest whether assertions are possible for almost every realization of the random variables Xi. In this 1991 Mathematics Subject Classification. Primary: 60F05. Keywords and phmses. Extreme value distribution, almost sure convergence, arithmetic means, logarithmetic means.

2 44 Math. Nachr. 190 (1998) paper we shall consider the sequence l(-m,zl((m, - bn)/un), where l(-w,zl(.) denotes the indicator function of the set (-m,x]. Obviously, l(-m,zl((mn - b,)/a,) does not converge almost surely for any x with 0 < G(x) < 1. Thus we consider the limitation of l(-m,zl((mn - b,)/a,) by arithmetic means and the limitation of l(-m,zl((m, - b,)/an) by logarithmetic means. Similar problems for the sum S, = ZE, Xi were discussed by many authors, for example, G. A. BROSAMLER, A. FISHER, P. SCHATTE, etc. (see 111, [3], [4], [6],[7] and [$I). 2. Arithmetic Means Set S(G) = {x : 0 < G(x) < 1). Theorem 2.1. If (M, - bn)/a, converges in distribution to G(x), where G is one of the extreme -value distributions defined in the introduction, and there exists a rational xo E S(G) such that {anx,-, + bn} is a non-decreasing sequence for n large enough. Then =o} = 0 for any 0 5 q < 1. Proof. We only need to prove (2.1) for q = 0 because the proof is similar for 0 < q < 1. Let Since G(z) is continuous and therefore convergence for all 5 and uniform convergence are equivalent, the set B can be written in the form 1 N w : w E R, lim - l(-m,zl((m,, - b,)/a,) = G(x) for all x E S(G) N-tm N n=l In the last definition of B we can restrict ourselves to all rational x E S(G) because the convergence for all rational x E S(G) implies the convergence for all 2 E S(G). N (Note the G(x) is continuous and $ l(-m,zl((mn - b,)/a,) are monotonous.) Hence, B is the intersection of countable many events and is itself an event. Suppose that B has positive probability. Since B is a permutable event relative to the i. i. d. random variables Xj, I3 must have probability one according to the Hewitt -Savage zero-one law (see [2]). Thus we have 1 N ZN =: - C [ l(-m,z,,l((m, - b,)/a,) - G(eo)] + 0 a.s. (as N + 00). n= 1

3 Cheng/Peng/Qi, Almost Sure Convergence in Extreme Value Theory 45 Now P{IZnl > c} + 0 since convergence with probability one implies convergence in distribution. Further from lz,l 5 2 it follows that Noting that the assumption (M, - bn)/an converges in distribution to G is equivalent to (2.3) lim Fn(anz + b,) = G(z) for all z E R. n3w Fix 61 E (G(zo), 1) and 62 E (0, G(z0)). From (2.3) we can choose an integer no such that 62 < Fn(anzO + bn) 5 61 and {anso + bn} is non- decreasing for all n 2 no. Then we find that EIZ, - EZ# - N n-1 1. n=2 m=l for all large N > 2no. Thus

4 46 Math. Nachr. 190 (1998) which contradicts (2.2). This completes the proof of Theorem Remark 2.2. Define F-(s) = inf{z : F(y) 2 z}. If F is in the domain of attraction of am, we can choose a, = (1/(1- F))-(n), b, = 0 and 20 = 1. If F is in the domain of attraction of Qa, we can choose a, = z* - (1/(1- F))-(n), b, = z* and zo = -1 where z* = sup {z : F(z) < 1). If F is in the domain of attraction of A, we can choose a, = f(bn), b, = (1/(1- F))-(n) and zo = 0 where f(5) = Lz% (1 - F(t)) dt dy / /=* (1 - F(t)) dt Logarithmetic Means Here we shall consider the logarithmetic means because of Theorem 2.1. Theorem 3.1. Assume (M, - b,)/a, converges in distribution to G, one of the extreme -value distributions defined in the introduction. Then we have Proof. We prove the theorem by assuming G Similar arguments are also held for G = qa and G = A. Set K(N) = Cf., and 0 bviously, 1 N1 sn(~) = - C ;I(-~,~]((M, - b,)/an), for x E R. K(N),=I Thus, to show (3.1), it suffices to prove that (3.2) lim N-tm zes(g) sup ls~(z) -G(z)l = 0 a.s.. Noting that G is continuous, equation (3.2) is equivalent to -+ 0 a.s. (as N + co). (3.3) P {w : lim sn(x) = ~(z), for dl z E s(g)} = I. N+a,

5 Cheng/Peng/Qi, Almost Sure Convergence in Extreme Value Theory 47 First we shall prove (3.3) by assuming F = G a,, = nl/-(i and b, = 0. Now fix x E S(G). Then, for 1 5 j 5 n we have Then 1gjnl 5 1 and gjn : = Jql(-rn,4 (4/P) - G(x)) (1(-rn,5](Mn/na)- G(4) = G(x)Gn-j(nl/"x) - G2(x) = G(~)(G~-j(n'/~x) - Gn(nl/-(Ix)). Therefore, it follows where Cl(x) is a positive constant which depends on x. Now we put Nk = 2". Then, by using standard arguments, we conclude from Chebyshev's inequality and the Bore1 - Cantelli lemma that with probability one. On the other hand, for Nk < N < Nk+l, where C2 is a positive constant. Thus we have lim SN(Z) = G(z) a.s.. N-bm Set B, = {w E fl : limn-+m SN(T) = G(r)}. Then P(B,) = 1. Now write B = n, BT, where the intersection takes over all rational r in S(G). Then P(l3) = 1. Noting that G(x) is continuous and SN(~) are monotonous functions, we have (3.3), and so (3.2) holds for F Assume now (M, - b,)/a, converges in distribution to G with general F and let Y1,..., Y,, be independent random variables with common distribution function 9". Set F-(z) = inf {g : F(y) 2 x}.

6 48 Math. Nachr. 190 (1998) Then Xj, j 2 1, are distributed the same as F-(G(Y,.)), j 2 1. For simplicity assume Xj = F-(G(Y,)), j >_ 1. Then M, = F-(G(maxl<j<,Y,)) _ - for all n 2 1. Keep in mind that we have proved (3.4) lim sup ~SN(Z) - G(x)l = 0 a.s. N-tm ZES(G) Note that where lo, otherwise Under the assumption of the theorem we have (2.3), i. e., for all x > 0 it holds Now fix z > 0. Then we have -nlogf(a,x+b,) 3 2 -O as n lim sn(z) = z. *+ 00 For any given e > 0, pick a 6 E (0,x) such that (3.6) G(z+6) - G(z-6) < E At this stage, there exists no such that Isn(x) -xl < 6 for all n 2 no. Hence, we get from (3.5) that

7 Cheng/Peng/Qi, Almost Sure Convergence in Extreme Value Theory 49 holds almost surely for all N 2 no. Then for N 2 no, and SN(X) - G(z) 2 SN(Z Using (3.4) and (3.6) we have KbO) K(N) * - S) - G(z - 6) + G(z - S) - G(z + b) - - limsup ~SN(Z) - G(z)I N-+w 5 limsup(isn(z+6)-g(z+6)1 +ISN(Z--) -G(Z-d)l) N-+CCl I.& a.s., i. e., for every z > 0 + (G(z + 6) - G(z - S)) lim SN(X) = G(z) a.s.. N+w Thus we can prove (3.3) a9 before. This completes the proof. 0 Corollary 3.2. Let h(y) be a bounded and almost everywhere continuous function on S(G). Then under the conditions of Theorem 3.1 we have Proof. It is easy to prove the Corollary according to the proof of Corollary 1 of [S]. 0 References BROSAMLER, G. A.: An Almost Everywhere Central Limit Theorem, Math. Proc. Cam. Phil. SOC. 104 (1988), CHOW, Y. S., and TEICHER, H.: Probability Theory, Springer-Verlag, 1988 FISHER, A.: Convex-Invariant Means and a Pathwise Central Limit Theorem, Adv. in Math. 63 (1987), LACEY, M. T. and PHILIPP, W.: A Note on the Almost Sure Central Limit Theorem, Statistics & Probability Letters 9 (1990), RESNICK, S. I.: Extreme Values, Regular Variation, and Point Processes, Springer -Verlag, 1987 SCHATTE, P.: On Strong Versions of the Central Limit Theorem, Math. Nachr. 137 (1988), SCHATTE, P.: Two Remarks on the Almost Sure Central Limit Theorem, Math. Nachr. 164 (1991), SCHATTE, P.: On the Central Limit Theorem with Almost Sure Convergence, Probability and Mathematical Statistics 11, (2) (1991),

8 50 Math. Nachr. 190 (1998) Department of Probability and Statistics Econometric Institute Peking University Erasmus University Rotterdam Beajing P.O. Box 1738 P.R. China 3000 DR Rotterdam The Netherlands e -mail: pengofew. eur. nl