Math. Nachr. 190 (1998),43-50
Almost Sure Convergence in Extreme Value Theory
By SHIHONG
CHENG
of Beijing, LIANGPENGof Rotterdam, and YONGCHENG
&I of Beijing
(Received April 18, 1995)
(Revised Version July 7, 1997)
Abstract. Let X I , . . .,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
and b, E R such that
P { ( M n - bn)/an 5 2) + G(I),
for
I
>0
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
Let XI,. . . ,X, be independent random variables with common distribution function F. Let M , = r n a x l ~ i ~Xi, and G(z) be one of the extreme-value distributions:
a a ( z ) = exp{-z-"},
* a ( z ) = exp{-(-z)a},
A ( x ) = exp {-e-'},
X > O ,
a > 0 ;
z < 0,
(Y
>
0;
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.
Math. Nachr. 190 (1998)
44
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}
for any 0 5 q
= 0
< 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
w : wE
R, lim
N-tm
1
-
N
N
n=l
l(-m,zl((M,, - b,)/a,) = G ( x ) for all x E S(G)
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).
(Note the G ( x ) is continuous and $ N 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
ZN =:
C [ l(-m,z,,l((M, - b,)/a,)
N
1
n= 1
- G(eo)]
+0
a.s. (as N
+
00).
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
lim Fn(anz
(2.3)
n3w
+ b,)
= G(z) for all z E R .
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#
1 .
N
n-1
n=2 m=l
for all large N
> 2no. Thus
+
Math. Nachr. 190 (1998)
46
which contradicts (2.2). This completes the proof of Theorem 2.1.
0
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 Q a , 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)
3.
=
Lz%”
(1 - F ( t ) )dt dy
/ /=*
(1 - F ( t ) )dt .
2
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 =
held for G = q a and G = A.
Set K ( N ) = Cf., and
s N ( ~ )=
1
Similar arguments are also
@a.
N 1
-C ; I ( - ~ , ~ ] ( ( M ,- b,)/an),
K ( N ) ,=I
for x E R.
0bviously,
-+
0 a.s.
Thus, to show (3.1), it suffices to prove that
(3.2)
lim
sup l S ~ ( z -G(z)l
)
= 0 a.s..
N-tm zES(G)
Noting that G is continuous, equation (3.2) is equivalent to
(3.3)
P { w : lim
N+a,
sN(X)
= ~ ( z )for
, dl z E s(G)} = I .
(as N
+
co).
Cheng/Peng/Qi, Almost Sure Convergence in Extreme Value Theory
47
First we shall prove (3.3) by assuming F = G = @-(I, a,, = nl/-(I and b, = 0. Now
fix x E S(G). Then, for 1 5 j 5 n we have
gjn :
= J q l ( - r n , 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)).
)
Then
1gjnl
5 1 and
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
n,
Set B, = { w E fl : limN-+mSN(T)= G(r)}. Then P(B,) = 1. Now write B =
BT,
where the intersection takes over all rational r in S(G). Then P(l3) = 1. Noting that
G(x) is continuous and S N ( ~
are) monotonous functions, we have (3.3), and so (3.2)
holds for F = @-(I.
Assume now ( M , - b,)/a, converges in distribution to G = @-(I 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} .
Math. Nachr. 190 (1998)
48
Then X j , 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
sup ~ S N ( Z-)G(x)l = 0 a.s.
lim
(3.4)
N-tm ZES(G)
Note that
where
lo,
otherwise
Under the assumption of the theorem we have (2.3), i. e., for all x
-nlogF(a,x+b,)
Now fix z
> 0.
3
2-O
as n
+ 00.
Then we have
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)- x l
Hence, we get from (3.5) that
< 6 for
all n 2 n o .
> 0 it holds
Cheng/Peng/Qi, Almost Sure Convergence in Extreme Value Theory
49
holds almost surely for all N 2 n o . Then for N 2 no,
and
SN(X) G(z) 2 SN(Z- S) - G(z - 6) + G(z - S)
- G(z + b) - KbO)
K(N)*
Using (3.4) and (3.6) we have
limsup ~ S N ( Z-)G(z)I
N-+w
5 limsup(ISN(Z+6)-G(z+6)1
N-+CCl
+ISN(Z--)
-G(Z -d )l)
+ (G(z + 6) - G(z - S))
I . & a.s.,
i. e., for every z
>0
lim S N ( 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), 561-574
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), 213-246
LACEY,M. T. and PHILIPP,W.: A Note on the Almost Sure Central Limit Theorem, Statistics
& Probability Letters 9 (1990), 201-205
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),
249 - 256
SCHATTE,P.: Two Remarks on the Almost Sure Central Limit Theorem, Math. Nachr. 164
(1991), 225-229
SCHATTE,P.: O n the Central Limit Theorem with Almost Sure Convergence, Probability and
Mathematical Statistics 11, (2) (1991), 237-246
50
Department of Probability and Statistics
Peking University
Beajing 100871
P.R. China
Math. Nachr. 190 (1998)
Econometric Institute
Erasmus University Rotterdam
P . O . Box 1738
3000 DR Rotterdam
The Netherlands
e -mail:
pengofew.eur.nl