Nonlinear Analysis: Modelling and Control, 2001, v. 6, No. 1, 3-8
Asymptotic of the Joint Distribution
of Multivariate Extrema
A. Aksomaitis and A. Jokimaitis
Department of Applied Mathematics,
Kaunas University of Technology
Studentu 50, 3028, Kaunas, Lithuania
e-mail: [email protected]
Received: 16.01.2001
Accepted: 14.03.2001
Abstract
Let Wn and Zn be a bivariate extrema of independent identically distributed
bivariate random variables with a distribution function F. in this paper the
nonuniform estimate of convergence rate of the joint distribution of the
normalized and centralized minima and maxima is obtained.
Keywords: multivariate extreme value, limit distributions, convergence rate
1 Introduction
Let { X j = (X 1, j , X 2, j ), j ≥ 1 } be independent and identically distributed
bivariate random variables with a distribution function
F (x1 , x 2 ) = P (X 1, j < x1 , X 2, j < x 2 ) ∀j ≥ 1 ,
and let x = (x1 , x 2 ) be an arbitrary point of the Euclidean space. We define by
x + y = (x1 + y1 , x 2 + y 2 ) ,
xy = (x1 y1 , x 2 y 2 ) ,
x  x1 x 2 

= ,
y  y1 y 2 
the arithmetical operations on vectors. By the inequality x < y we mean the
system of inequalities x i < y i (i = 1,2 ) .
We form bivariate maxima and minima
3
Z n = (max (X 1,1 , , X 1, n ), max (X 2,1 , , X 2,n )) ,
W n = (min (X 1,1 , , X 1, n ), min (X 2,1 , , X 2, n )).
Let
{a n = (a1,n , a 2,n ), n ≥ 1}, {bn = (b1,n , b2,n ) > (0,0), n ≥ 1}
be sequences of centralizing and normalizing vectors. If the distribution
function F and sequences of centralizing and normalizing vectors are such that
the following limit exits:
lim n(1 − F (a n + bn y )) = u ( y1 , y 2 ) > 0 ,
n →∞
then
lim P(Z n < a n + bn y ) = H ( y ) = e −u ( y1 , y2 ) ,
(1)
n →∞
where H ( y ) is a nondegenerate distribution function of two variables.
Let
{c n = (c1,n , c 2,n ), n ≥ 1}, {d n = (d1,n , d 2,n ) > (0,0), n ≥ 1}
be sequences of centralizing and normalizing vectors. If the distribution
function F and sequences of centralizing and normalizing vectors are such that
the following limits exist
lim nF1 (c1, n + d 1,n x1 ) = z1 (x1 ) > 0 ,
n →∞
lim nF2 (c 2,n + d 2,n x 2 ) = z 2 (x 2 ) > 0 ,
n →∞
lim n(F1 (c1,n + d 1,n x1 )+ F2 (c 2, n + d 2,n x 2 )− F (c n + d n x )) = z (x1 , x 2 ) > 0 ,
n →∞
then
lim P(W n < c n + d n x ) = L(x ) = 1 − e − z1 (x ) − e − z2 (x2 ) + e − z (x1 , x2 ) ,
n →∞
(2)
where L(x ) is a nondegenerate distribution function of two variables, and
F1 (x1 ) , F2 (x 2 ) are marginal distribution functions of the distribution function
F(x).
The necessary and sufficient conditions for the convergence of normalized
maxima and normalized minima are formulated in [3].
Suppose that conditions (1) and (2) hold true. Since the bivariate maxima
and bivariate minima are asymptotically independent (see [2]), then
lim P(Z n < a n + bn y, W n < c n + d n x ) = H ( y )L(x ) .
(3)
n →∞
In this paper we are going to obtain a nonuniform estimate of the
convergence rate in (3). It will generalize the result obtained in [1].
4
2 Main result
Let us denote
∆ [cn + d n x, an + bn y ) F = F (a n + bn y ) + F (c n + d n x ) −
− F (c1, n + d 1,n x1 , a 2,n + b2, n y 2 )− F (a1, n + b1, n y1 , c 2,n + d 2,n x 2 ) ,
u1, n ( y ) = n(1 − F (a n + bn y )) ,
v1, n ( y ) = u1,n ( y ) − u ( y ) ,
u 2,n ( y, x1 ) = n(1 − F (a n + bn y ) + F (c1, n + d 1, n x1 , a 2, n + b2, n y 2 )) ,
v 2, n ( y, x1 ) = u 2, n ( y, x1 ) − u ( y ) − z1 (x1 ) ,
u 3, n ( y, x 2 ) = n(1 − F (a n + bn y ) + F (a1,n + b1, n y1 , c 2, n + d 2, n x 2 )) ,
v 3, n ( y, x 2 ) = u 3, n ( y, x 2 ) − u ( y ) − z 2 (x 2 ) ,
(
u 4,n ( y, x ) = n 1 − ∆ [cn + d n x, an + bn y ) F
)
v 4, n ( y, x ) = u 4, n ( y, x ) − u ( y ) − z (x ) .
Theorem. Suppose that (3) holds, and
a n + bn y > c n + d n x .
For all x, y , for which
v1, n ( y ) ≤ log 2 ,
v 2, n ( y, x1 ) ≤ log 2 ,
v 3, n ( y, x 2 ) ≤ log 2 ,
v 4, n ( y, x ) ≤ log 2 ,
the following estimate holds true:
P(Z n < a n + bn y, W n < c n + d n x ) − H ( y )L(x ) ≤
≤ ∆ 1,n ( y ) + ∆ 2, n ( y, x1 ) + ∆ 3, n ( y, x 2 ) + ∆ 4, n ( y, x ) .
Here
∆ 1, n ( y ) =
u12, n ( y )e
2(n − 1)
∆ 2, n ( y, x1 ) =
∆ 3, n ( y, x 2 ) =
∆ 4, n ( y , x ) =
− u1, n ( y )
u 22, n ( y, x1 )e
(
− u 2 , n ( y , x1 )
2(n − 1)
u 32, n ( y, x 2 )e
− u 3, n ( y , x 2 )
2(n − 1)
u 42, n ( y, x )e
)
+ e −u ( y ) v1,n ( y ) + v12, n ( y ) ,
−u 4, n ( y , x )
2(n − 1)
(
)
+ e −u ( y )− z1 (x1 ) v 2, n ( y, x1 ) + v 22, n ( y, x1 ) ,
(
)
+ e −u ( y )− z2 (x2 ) v 3, n ( y, x 2 ) + v 32, n ( y, x 2 ) ,
(
)
+ e −u ( y )− z (x ) v 4, n ( y, x ) + v 42, n ( y, x ) .
5
Proof. We have
P(Z n < y, W n < x ) = P(Z n < y ) − P (Z n < y, W1, n ≥ x1 )−
− P (Z n < y, W 2, n ≥ x 2 )+ P(Z n < y, W n ≥ x ) =
(
)
= F n ( y ) − (F ( y ) − F (x1 , y 2 ))n − (F ( y ) − F ( y1 , x 2 ))n + ∆ [x, y ) F n ,
if y > x , and
P(Z n < y, W n < x ) = P(Z n < y ) = F n ( y ) ,
if y ≤ x . Here W1,n , W 2,n are components of the bivariate minima Wn.
Let a n + bn y > c n + d n x . We have
P(Z n < a n + bn y, W n < c n + d n x ) − H ( y )L(x ) ≤
≤ F n (a n + bn y ) − e −u ( y ) +
+ (F (a n + bn y ) − F (c1,n + d 1,n x1 , a 2,n + b2, n y 2 ))n − e −u ( y )− z1 (x1 ) +
(4)
+ (F (a n + bn y ) − F (a1, n + b1, n y1 , c 2, n + d 2, n x 2 ))n − e −u ( y )− z2 (x2 ) +
(
+ ∆ [cn + d n x, an + bn y ) F
)n − e −u (y )− z (x ) .
Let us estimate the first summand of the right – hand side of the inequality
(4). We have
(5)
F n (a n + bn y ) − e −u ( y ) ≤
≤ F n (a n + bn y ) − e
− u1, n ( y )
+e
− u1, n ( y )
− e −u ( y ) .
Let us estimate the second summand of the right – hand side of the
inequality (5). Applying the inequality
n
x 2e −x
 x
0 ≤ e − x − 1 −  ≤
2(n − 1)
 n
we get
F n (a n + bn y ) − e
− u1, n ( y )
(0 ≤ x ≤ n ) ,
=
 u1,n ( y ) 
 − e −u1,n ( y ) ≤
= 1 −

n


n
≤
u12,n ( y )e
(6)
− u1, n ( y )
2(n − 1)
.
Let us estimate the first summand of the right – hand side of the inequality
(5). Applying the inequality
e − y − e − x ≤ e − x x − y + (x − y )2 ( x − y ≤ log 2 ) ,
(
)
we get
6
e
− u1, n ( y )
(
)
− e −u ( y ) ≤ e −u ( y ) v1,n ( y ) + v12,n ( y ) ,
(7)
if v1,n ( y ) ≤ log 2 .
Taking into account inequalities (6) and (7), from the inequality (5) we get
(8)
F n (a n + bn y ) − e −u ( y ) ≤ ∆ 1, n ( y ) ,
if v1,n ( y ) ≤ log 2 .
Analogously, we get
(F (a n + bn y ) − F (c1,n + d1,n x1 , a 2,n + b2,n y 2 ))n − e −u ( y )− z (x ) ≤
1 1
≤ ∆ 2,n ( y, x1 ) ,
(9)
if v 2,n ( y, x1 ) ≤ log 2 ;
(F (a n + bn y ) − F (a1,n + b1,n y1 , c2,n + d 2,n x2 ))n − e −u ( y )− z (x ) ≤
2
≤ ∆ 3,n ( y, x 2 ) ,
2
(10)
if v3,n ( y, x 2 ) ≤ log 2 ;
(∆ [c +d x,a +b y )F )n − e −u (y )− z (x ) ≤
n
n
n
n
≤ ∆ 4, n ( y , x 2 ) ,
(11)
if v 4,n ( y, x 2 ) ≤ log 2 .
Taking into account inequalities (8) – (11), from the inequality (4) we get
the estimate in (3).
Theorem is proved.
3 The Example
Let
{X j , j ≥ 1}
be independent identically distributed bivariate random
variables with a distribution function
F (x ) = 1 − e − x1 − e − x2 + e − x1 − x2 ,
x1 > 0, x 2 > 0 .
For the chosen centralizing and normalizing vectors
a n = (log n, log n ) ,
bn = (1,1) ,
c n = (0, 0 ) ,
d n = (1 n , 1 n ) ,
we have
lim P(Z n < a n + bn y, Wn < c n + d n x ) = H ( y )L(x ) ;
n →∞
here
(
)
H ( y ) = exp − e − y1 − e − y2 ,
y1 , y 2 ∈ R ,
7
(12)
L(x ) = 1 − e − x1 − e − x2 + e − x1 − x2 ,
x1 > 0, x 2 > 0 .
We shall estimate the convergate rate in (12). We have
u1,n ( y ) = e − y1 + e − y2 −
v1,n ( y ) = −
e − y1 − y2
,
n
e − y1 − y2
,
n
u 2,n ( y, x1 ) = e − y1 + e
− y2 − x1 n
−
e − y1 − y2
− x1 

+ n 1 − e n  ,
n


− y − y2
 − x1
 e 1
v 2,n ( y, x1 ) = e − y2  e n − 1 −
n


u 3,n ( y, x 2 ) = e − y2 + e
− y1 − x2 n
−
− x1

+ n 1 − e n


 − x1 ,

e − y1 − y2
− x2 

+ n1 − e n  ,
n


− y − y2
 − x2
 e 1
v3,n ( y, x 2 ) = e − y1  e n − 1 −
n


− x2 

+ n 1 − e n  − x 2 ,


(
)
e − y1 − y2
+ n 1 − e −(x1 − x2 ) n ,
n
−y −y
x2
x1
−
−
 e 1 2



+ n 1 − e −(x1 + x2 ) n − x1 − x 2 .
v 4,n ( y, x ) = e − y1  e n − 1 + e − y2  e n − 1 −
n




Evidently, the order of the convergence rate with respect to n equals 1 n .
u 4, n ( y , x ) = e
− y1 − x2 n
+e
− y2 − x1 n
−
(
)
4 References
1. Aksomaitis A., Jokimaitis A., “An asymptotic of the distribution of extrema
of independent random variables”, Liet. Mat. Rinkinys, 35, p.1-13, 1995
2. Galambos J., The Asymptotis Theory of Extreme Order Statistics, Wiley,
New York, 1978
3. Marshall A., Olkin I., “Domains of attraction of multivariate extreme
value distributions”, Ann. Probab., 11, p. 168 - 177, 1983
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