(#2015)
INTERMEDIATE SUMS AND SfOCHASfIC O)MPACfNESS OF MAXIMA
University of North Carolina at Chapel Hill
and
University of Delaware
Let Xl ,n <
-
.. , -< Xn,n
be the order statistics of n independent random
-
variables with a common distribution function F and let k
numbers such that k
=\
rbk
n
1
Li=rak
n
1+1
n
~ 00
Xl'
n+ -l,n
and k In
n
~
0 as n
~ 00,
n
be positive
and consider the sums I (a,b)
n
of intermediate order statistics, where 0
< a < b.
We
find necessary and sufficient conditions for the existence of constants
An > 0 and Cn such that A-n 1(I n (a,b)-Cn ) converges in distribution along
subsequences of the positive integers {n} to non-degenerate limits and
completely describe the possible subsequential limiting distributions.
We
also disclose an unexpected connection between the convergence in
distribution of the intermediate sums I (a,b) and the stochastic compactness
n
of the maxima Xn,n
1
On leave from the University of Szeged, Hungary. Partially supported by the
Hungarian National Foundation for Scientific Research, Grants 1808/86 and
457188.
2partially supported by NSF Grant DMS-8803209 and a Fulbright Grant.
AMS 1980 subject cLassification.
Key words and phrases.
Primary 6OF05.
Sums of intermediate order statistics, asymptotic
distribution, stochastic compactness of maxima.
- 1 -
1.
Introduction and Statement of Results.
Let X. Xl'
~ •...
be a
sequence of independent non-degenerate random variables with a common
distribution function F(x) = P{X
Xl
~
.n
~
...
~
m.
x}. x €
and for each integer n
~
1 let
X
denote the order statistics based on the sample X ..... X .
1
n
n.n
Let {k } be a sequence of positive numbers such that
n
k
(1.1 )
When the k
n
-+
n
00
and
k In
n
-+
as n
0
-+
00.
are integers. many authors have investigated the asymptotic
distribution of the single intermediate order statistic X 1 k
.
n+ n
n'
(See
Pickands (1975). Balkema and de Haan (1978), Watts. Rootzen and Leadbetter
(1982) and Cooil (1985) for example. and the references therein.)
In this
paper we are interested in the problem of the asymptotic distribution of the
intermediate sum
rbk 1
(1. 2)
~
In(a,b) = In(a.b;kn ) =
i=
Xn+ 1- i . n .
rakn 1+1
0
< a < b.
where rx1 is the smallest integer not smaller than x. {k} satisfies (1.1)
n
and the empty sum is always understood as zero.
Consider the inverse or quantile function of F defined as
( 1.3)
Q(s)
= inf{x
: F(x)
~
s}.
0
<s
~
1.
It will be more convenient in our investigations to work with the left
continuous non-decreasing function
(1.4)
H(s) = -Q«l-s)-).
0
~
s
< 1.
the increments
o < a < b.
(1.5)
of which will play an important role.
Set
- 2 -
*
A (a.b)
(1.6)
n
A (a.b) •
=
if
An (a.b) > O.
if
An (a.b) = O.
n
{ 1
and for c = a.b and n large enough define the functions
cvk"
cvk"
n
n
- -2- ~ x ~ -2-'
A* (a.b)
'"n (c;x)
n
=
-
00
< x < -cvk"/2.
n
cvk"/2
'"n (cvk"/2)
n
n
<x <
00.
These are non-decreasing and left-continuous for each n. and satisfy
'"n (c;O)
= O.
We also need the functions
0
<fJ n (x)
=
x
HX:nJ - H[r~nlJ} /
[r nlj
Note that <fJ
n
k
distribution function on
follows. the symbol
x
n
=0 on m whenever
m for
rakn1
k
n
rakn 1
~ x
k
n
A*n (a.b),
bk
<fJn
<
>
fbk n
kn
~
1
A (a.b) = O. otherwise <fJ is a left-continuous
n
n
which <fJn(x)
=1
if x ~ rbknl/kn.
In what
=> will denote weak convergence of functions. that is.
pointwise convergence at every continuity point of
the interval to be indicated. and
~~
t~e
limiting function in
will denote convergence in distribution.
Finally. introduce the centering sequence
(1.7)
rbkn 1
k
n
~
fbk l/n
n
(a.b) -
and, as usual. let Wet). t
~
-n
J
n
H(s) ds
fak l/n
n
O. denote a standard Wiener process.
- 3 -
THEOREM 1.
Let {k } be a sequence as in (1.1) and Fix 0
n
< a < b.
Suppose that there exist a subsequence {n'} of the positive
(I)
intergers {n} and non-decreasing, leFt-continuous Functions
~c(O} ~
such that
0,
~c(O+} ~
'iJ ,(c; e)
n
(l.S)
=>
~
c
~a
and
~b
on ffi
0, and
(e)
on
ffi
as
n' -.
00,
c
= a,b.
Then there exist a subsequence {n"} C {n'} and a non-negat ive, non~
decreasing, leFt-continuous Function
~(a)
(1. 9)
= 0,
either
~(oo)
=0
on ffi such that
~(b+)
or
= 1,
~n"( e)
and
=>
~(e)
on
ffi
and
r
rbk .. 1
_ _.::..1*-:--__ {
Xn"+l-i,n" - J-Ln.. (a,b}
Jkn .. An" (a,b)
i=rak .. 1+1
(1.l0)
}
n
as n" -.
OJ,
where
Y(~ '~'~b)
a
= -J
0
-W(a)
=J
-W(a)
~
0
a
(x)dx +
J
~a(x)dx + J b W(x)d~(x)
[a,bJ
+
a
W(x)d~(x)
-
J 0-W(b)
~b(x)dx
(~(b+) - ~(b»W(b) + J 0
-Web)
~b(x)dx.
Furthermore, we necessarily have
(1.11)
and the Limiting random variabLe Y(~a'~'~b) is degenerate iF and onLy iF ~a
~b
=0 on ffi and ~ =0
(ii)
on [a, b] .
IF For some subsequence {n'} C {n} and constants B ,
n
exist non-decreasing, LeFt-continuous Functions
A* ,(a,b)
( 1.12)
n
~B"""-- ~
n
,(c; e)
=>
~
(e)
, n c
~a
on
and
~b
>0
there
on ffi such that
ffi, c=a, b,
=
- 4 -
and
(1.13)
~
A ,( a, b) IB ,
n
n
0
n' ~
as
00
then
1
( 1.14)
~
as n'
~n ,(a,b) } ~~ V(~a ,O'~b)
Xn '+1.
-l,n , -
~B,
-J""n'
n
~
Furthermore. we necessariLy have
00.
( 1.15)
~a(x)
= 0,
x
~
O.
~b(x)
and
= 0,
x < 0,
and the Limiting random variabLe V(~a,O'~b) is degenerate if and onLy if ~a
~b ==
=
o.
The following converse result shows that Theorem 1 is optimal in
general.
THEOREM 2.
Suppose that there exist a subsequence {n'} C {n} and
norming and centering constants A ,
n
and C , such that
n
1
r
fbk,
(1. 16)
>0
AI, {
n
i=fak, 1+1
X '+1.
n
-l.n
,- C ,
n
n
where V* is a non-degenerate random variabLe.
Then there exist a subsequence
{n"} C {n'} and non-decreasing, Left-continuous functions ~a and ~b on JR such
that ~c(O) ~
(1. 17)
O.
~c(O+)
*
~n" ( c; -)
~
0,
and
Jk " A:,,(a, b)
n
= -~A----'-'---n"
and for some 0 ~ 0
(1. 18)
<
n ,,(c;-)
'iJ
=> ~c (e)
on JR.
c=a,b,
00,
~
n
"
(a b)/A .. ~
n
A 1",
L
n
1:.
U
as
n" ~
00.
For the Limiting random variabLe V* we necessariLy have the distributionaL
equati ty
- 5 -
(1.19)
where
~
€
~
is some constant.
Our last theorem establishes an unexpected connection between the
convergence in distribution of intermediate sums I (a.b) and stochastic
n
compactness of the maximum observation X
n.n
{Xn.n } is stochastically compact of there exist a sequence
We say that
of norming constants a(n)
> 0 and a sequence of centering constants c(n) such
that for every subsequence {n'} C {n} there exists a further subsequence {n"}
(X"
,,- c(n"»/a(n") converges in distribution to a
n .n
C {n'} such that
non-degenerate limit.
usual integer part of x
Let lxJ be the largest integer not greater than x, the
€ ~.
and for a number 0
s
set
o < x < 1/(2s).
H(sx) - H(s)
h (x) =
<s <1
{ H(1/2) - H(s) .
1/(2s)
~
x
< 00
•
Then it can be easily inferred from the Proposition of de Haan and Resnick
(1984), page 590, that {X
n.n
} is stochastically compact if and only if there
exist a sequence of constants B(n)
>0
such that for every sequence k
n
of
positive numbers satisfying (1.1) and for every subsequence {n'} C {n} there
exist a further subsequence {n"} C {n'} and a non-decreasing, left-continuous
finite function hex). x
( 1.20)
as n"
> O.
such that h(l)
h.
/ ,,(-)/B(ln"/k "J)
-"k " n
n
n
-+
00.
=>
~
O. h(l+)
h(-)
on
~
O. h
~
O. and
(0. 00 )
in which case {X
} is stochastically compact with norming
n.n
constants B(n) and centering constants c(n)
= -H(1/n).
The "asymptotic
balance" condi tion (1. 20) and its equivalent forms are investigated in depth
by de Haan and Resnick (1984) who also provide many interesting examples.
THEOREM 3.
(I)
The Following two statements are equivalent:
There exists a sequence of norming constants B(n)
>0
such that for
- 6 -
every sequence {k } of positive numbers satisFying (l.l) and subsequence
n
{n'l C {n} there exist a Further subsequence {n"} C {n'} and numbers 0
b
O
< 00
such that For each 0
< a < a o < bo < b < 00
{n"'} C {n"} there is a Further subsequence {n
(1.21)
JkniV B
as n iv ~
00,
[1
In
iv
/knivJ
J{
rbk iv1
~
i=rakn
iv
iV
and subsequence
} C {n"'} For which
Xiv
l+ 1
iv +1-i,n
n
< ao <
~ iv{a,b) } ~~ Y
n
where Y is a non-degenerate random variabLe and ~ (a,b) is as in
n
(1. 7).
{Xn,n } is stochasticaLLy compact with norming constants B(n) and
(II)
centering constants -H{I/n).
We close by an example.
We say that F is in the domain of attraction of
~~
an extreme value distribution if (X
-c )/a
n,n n
n
and c
n
€
mare
Y as n
~
some constants and Y is non-degenerate.
~
where a
00,
n
>0
As pointed out in
Csorgo, Haeusler, and Mason (1990), with earlier references, this happens if
and only if for some constant
~
€
m,
-,.
-~
(1. 22)
lim H(sx)-H(sy) =
slo H( sv ) -H (sw)
for all distinct 0
(x
{
A (y) = P{Y
~
~
n
-w
~
# 0,
)
= 0,
~
if
In this case we write F € A , where, with
00.
~
and c '
n
,
[ exp(-y 1/, )
exp(-exp{-y»,
y} =
exp(-(-y)
1/~
It is easily checked that if F € A for some
~
~n(c;x) ~
if
-~-'Y
)/(v
log{x/y)/log(v/w) ,
< x,y,v,w <
appropriate choices of a
-y
0 for any choice of c
>0
and
y
if
~
m;
if
~=O,
y €
), y
~
> 0;
€
< 0;
m,
if ~
> 0,
< O.
then for all x €
~n(x) ~ ~~(x),
a
~
x
~
m,
b, for any
- 7 -
choice of 0
<a <b
as n
~
where
00.
--r -'1
--r--r
(x -a )/(b -a ).
~'1 (x)
= ~'1
b(x) = {
•a •
log(x/a)/log(b/a)
•
'# O.
if
'1
= O.
'1
then by Theorem 1.
€ ffi.
1
~
as n
'1
So if F € A for some
The last convergence follows. of course. from (1.22).
'T
if
X
' n n+ l-1.
< a < b < 00
for any choice of 0
00
2.
~n (a.b) } ~~ J_rba W{x) d~
;u
'1
(x)
and sequence {k } as in {I. 1).
n
Proofs.
Let U ..... U be independent random variables uniformly
1
n
distributed on (0.1) with corresponding order statistics U1
~ ... ~ U
.
.n
n.n
Consider the uniform empirical and quantile processes a (t) = vh(G (t)-t) and
n
n
'V
~n{t)
'V
= vh{t-Un{t».
' V _ I
0 ~ t ~ 1. where Gn{t) = n
o
~
t
~
#{1 ~ k ~ n:
Uk ~ t}.
'V
'V
~
1. and Un(t) = inf{O
so that Un{t) = Uk . n if (k-1)/n
process (cf. Mason (1988»
~
s
'V
~
1 : Gn(s)
<t
~
t}. 0
<t
~
= U1 . n .
1. Un(O)
The tail empirical
kin. k=l ..... n.
and the tail quantile process (cf. Cooil (1985»
pertaining to the given sequence {k } satisfying (1.1) are defined as
n
'V
wn(s) = (n/kn )
~
'V
an(skn/n) and vn(s) = (n/kn )
Mason (1988) points out. for any T
> O.
~
~n{skn/n).
0
~
s
~ n/k
n.
As
w (.) converges weakly in the
n
Skorohod space D[O.T] to a standard Wiener process.
Then by a Skorohod
construction there exist versions w (.) of w (.) on a suitable probability
n
space such that for any T
>0
n
we have the distributional equalities
'V
(2.1)
{wn(s) : 0
~
s
~
T}
for each n large enough to have n/k
n
sup
(2.2)
O~s~T
where W(s). s
~
=~
~
{wn(s) : 0
~
s
~
T}
T and
Iwn (s) - W{s)1 ~ 0
a.s.
0 is a standard Wiener process.
as n
~ 00.
Then by the obvious
- 8 -
left-continuous version of Lemma 1 of Vervaat (1972) we obtain the versions
v (e) of v (e) defined on the same suitable space such that
n
n
(2.3)
~
for all n such that nIk
n
sup
(2.4)
O~s~T
T and
Iv (s) - W(s)1 ~ 0
a.s.
as n ~
00.
n
From the definition of H in (1.4) and (1.3) we have
(Xl
.n
) =611 (-H(U
) • . . .• -H(U 1
n.n;.u
n.n
.n
•...• X·
».
n
~
1.
If we integrate with respect to a right-continuous or a left-continuous
function. we accordingly use the symbol fY to mean f(
x
x.y ] or f[ x.y ).
Using
this convention (in fact already used in (1.7) and in the formulation of the
theorems). the notations in (1.2) and (1.7). integration by parts and some
elementary rearrangements. from the last distributional equality we obtain
I n (a.b) -
~ (a.b)
n
=611
n
;.u
- n
+ n
I
fbk lin ""'
n
(G (u)-u)dH(u)
fak lin
n
I
I
n
""'
U (ak In)
n
n
fak n lin
""'
U (bk In)
n
n
fbkn lin
""'
n
(G (u)-u)dH(u)
""'
(Gn (u)-u)dH(u).
Substituting u=sk In and going over to the probability space of relations
n
(2.1)-(2.4) . we arrive at
(2.5)
I n (a.b) -
~
M -Rn (a) + Rn (b).
n (a.b) =611
;.un
where
M
n
=
rj{
"~n
I
fbk 11k
[Sk ]
n n w (s)dH -E.
fak 11k
n
n
n
n
- 9 -
and
_ 1- K
Rn(c)
rckn 1
Ck
(r
1]
n
n
1
v
kn
+
k
n
c
= a.b.
rckn 1
k
n
Aiming at a proof of Theorem 1. we handle these terms in separate lemmas.
LEMMA 1.
For any subsequence {n'} C {n} there exist a further
subsequence {n"} C {n'} and a non-negative. non-decreasing. left-continuous
function
~
Mn "
f'k'":"
,J~n"
A*,,(a.
b)
n
almost surely as n"
Proof.
(1.9) holds and
such that
~
J
[a.b]
W(s)d~(s) =
1: W(s)d~(s)
(~(b+)-~(b»W(b)
~ 00.
We distinguish two cases.
In the first one there exists a
=0
subsequence {n"} C {n'} such that An,,(a. b)
=0 on ffi.
+
a
and hence we have
(1.9) with
~
for all n".
=0 on ffi.
In this case
~n"
implying the second
statement with zero limit.
The second case is when A ,(a.b)
n
>0
for all n' large enough.
In this
case. since~n ,(a) =~n ,(rakn ,11kn ,) =0. ~n ,nbkn ,11kn ,) = 1, rbkn ,11k,
n ->
b. rbk
n
.11kn ,
~ b as n' ~
00.
by a Helly selection we can choose a subsequence
{n"} C {n'} such that (1.9) holds with
~(b+)
= 1.
To prove the second statement in the present second case. notice that
Mn "
,Jfk":"
~n"
A*,,(a.b)
n
=
[w
o
n
,,(s)d~n lIes)
and
[ o wn
,,(s)d~n
lIes) - [ W(s)d~ lIes)
On
Iwn lIes) - W(s)1
- 10 -
for all large enough n". and this bound goes to zero almost surely as n"
by (2.2).
~
n
,,(a)
< d < b is a continuity point of
Furthermore. if a
=0
and
~(a)
=0
~.
-+
<Xl
then. since
and since for
J
rbk
n
= .b
T (b)
n
11kn
W(b»d~
(W(s) -
n
(s)
we have
almost surely as n
[o
-+
<Xl.
we obtain that
W(s)d~n ,,(s) = Jrd
a
+
(W(s) - W(a»
Jfd
d~n ,,(s)
(W(s) - W(b»
+ Weal
~n .. (d)
d~n .. (s) + W(b)(l-~n ,,(d»
+ T .. (b)
n
converges almost surely to
r (W(s)-W(a»d~(s)
a
=
+
1: W(s)d~(s)
a
LEMMA 2.
W(a)~(d) + Jf
d
+
d~(s) + W(b)(l-~(d»
(W(s) - W(b»
W(b)(l-~(b»
=
J[a.b] W(s)d~(s).
If (1.12) holds for some sequence B ,
n
A~, (a.b)) along {n'} with limiting functions ~a
and
listed above (1.12). then for c=a.b.
R ,(c)
n
~B,
n
..J~n'
almost surely as n'
Proof.
-+
-+
J
0
-W(c)
<Xl.
By the change of variables
~c(x}
dx
>0
0
(that can be
~b having the properties
- 11 -
s
=
fekn
k
n
1
X
+-
K
where e is either a or b. we obtain
R • (e)
n
Ik":" Bn ,
~"n'
Therefore. using (2.2) and (2.4) for T=2e. say. and the almost sure uniform
continuity of W(o) on [O.2e]. it follows that
R . (e)
_n_ _
Ik":"B.
n
=
~n.n'
=
=
J-W(e)+o(l) {Wee) +
0
-Wee)
Jo
-Wee)
Jo
*
0(1) + x} d'" ,(c:x)
n
*
{Wee) + x} d'" .(e:x) + 0(1)
n
*
*
x d'" .(e:x) + Wee) '" ,(e;-W(e»
n
n
+ 0(1)
-Wee)
=
=
almost surely as n'
~~.
Jo
J
x d'" (x) + Wee) '" (-W(e»
e
e
+ 0(1)
0
"'e(x) dx + 0(1)
-Wee)
where we also used the faet that -Wee) is almost
surely a eontinuity point of '"e (e).
The next lemma is needed for the proof of the degeneraey statements in
Theorem 1.
LEMMA 3.
Let g and h be two Borel. measurabl.e functions on IR and Zl and.
o
- 12 -
Z2 be two independent non-degenerate normat random variabtes.
If g(Zl) =
h(Zl+Z2) atmost surety, then g and h are constant atmost everywhere on
Proof.
ffi.
The condition implies that P{g(Zl) = h(Zl+ Z2)lz =z} = 1 for
1
almost all z €
ffi,
almost all z €
ffi.
which in turn implies that P{g(z) = h(z + Z2)} = 1 for
Thus g(z) = hex) for almost every z and x, which occurs
o
only if both g and h are the same constant almost everywhere.
Proof of Theorem 1.
All the statements in (1.9), (1.10) and in (1.14)
follow directly from the distributional equality (2.5) and Lemmas 1 and 2.
To prove (1.11) in case (i), let x
€ ffi
be arbitrary and consider a
continuity point s of 'P in (a,b).
Then, since H is non-decreasing, 'II ,,(a;x)
~ 'Pn,,(s)
Hence, letting n" -+
for all n" large enough.
'P(s), which letting now s
all x
n
€ ffi
00,
lim sup 'IIn,,(a;x)
! a, implies the statement for'll.
a
~
Simi lar ly, for
and n" large enough.
'IIn ,,(b;x) ~ {H(skn"/n") - H( rbkn "l/n")}/A*..
n (a. b)
= 'P ,,(s) - I,
n
and the statement for 'lib follows by letting first n" -+
00
and then s
r b.
The corresponding statement (1.15) in case (ii) follows in exactly the
same way, using also (1.13).
Finally we prove the claims about the degeneracy of the limiting random
variables.
If all 'IIa ' 'P, and 'lib are identically zero, then V('IIa,'P,'II ) = 0
b
almost surely.
To prove the converse statements, set
g (y)
c
=
-y
J0
'II (x)dx, y
c
€ ffi,
c =a,b.
Assume that in case (ii), V('IIa,O,'II ) = C almost surely for some C
b
Writing Zl = Weal. z2 = W(b)-W(a), g = ga' and h=gb+C , the degeneracy
statement follows directly from Lemma 3.
€ ffi.
- 13 -
Assume now that
V(~a'~'~b}
= C almost
surely in case (i).
Using the
notations just introduced, this means that
(2.6)
almost surely, where
~1
=
~(b+)
-
~(a),
~2 = I
d~(u)/(b-a),
(u-a)
[a,b]
and
Z3
= IW(U} d~(u)
-
~1
Weal -
-
~l
Zl -
[a,b]
=
I
[a,b]
d~(u)
W(u)
so that Zl' Z2' and Z3 are independent.
both Z3 and ga(ZI) +
since
~l
~(a)
= 0,
= ~2 = 0,
in case (ii).
~l
~2(W(b}
~2
- Weal}
Z2'
Therefore, (2.6) can happen only if
Zl + gb(ZI+Z2) - C +
Z3 can degenerate only if
~
~2
Z2 are degenerate.
=0 on [a,b].
and hence Lemma 3 implies again that
~a
= ~b
The theorem is completely proved.
However.
But in this case
=0 on ffi exactly as
o
In the proof of Theorem 2 we will use the representation in (2.5) in the
form
(2.7)
I n (a,b) -
~
n (a,b)
=~
M + Rn (b) - Rn (a),
;un
where for c = a,b, as seen in the proof of Lemma 2,
and, transforming back the tail empirical process,
M =
n
where
neG (u)-u)dH(u),
n
- 14 -
0 ~ u ~ 1,
Gn (u) = u + ..J""n
rj{ Wn (nu/kn )/n.
'"
for which we have {Gn(u) : 0 ~ u ~ I} =~ {Gn(u) : 0 ~ u ~ I}.
We begin by establishing several lemmas needed in the proof.
LEMMA 4.
For Dn (a.b)
= IRn (b)
- Rn (a)I/(rj{
..J""n A*(a.b»
n
< M} > O.
lim lim inf P{Dn (a.b)
M~
ProoF.
we have
n~
In the course of the proof of (1.11) we have already shown that
(2.8)
lim sup
~
n~
n (a;x) ~ 1
>0
for x
and
(2.9)
lim sup(- ~n(b;x»
~
1 for x
< O.
n~
Also. as in the proof of Lemma 2.
(2.10)
-W(b)+o(l)
D (a.b) =
{W(b)+x+o(l)} d~ (b;x)
n O n
J
-W(a)+o(l)
-
Jo
almost surely as n ~
{W(a)+x+o(I)}
d~
n
(a;x)
Hence. introducing the event A
00.
= {-I < -Web) < -1/2.
~ t {4 +
A
A
o(I)} almost surely for all n large enough. where AA is the indicator of A.
1/2
< -Weal < I} and using (2.8) and (2.9). we see that Dn(a.b) t
Noting that P{A}
LEMMA 5.
An .(a.b)
(2.11)
>0
> O.
the lemma follows.
o
Whenever there exist a subsequence {n'} C {n} such that
For aU. n
-~n I
(a;x)
Large enough and
I
~
00
For some
x
<0
n'
as
~ 00
or
(2.12)
~n'
(b;x)
~
00
For some
x
>0
and a sequence of positive constants A , such that
n
as
n
~oo
- 15 -
Dn ,(a,b) ~nn
I'k, An ,(a,b) / (I'k,
An ,(a,b) V An ,) = 0p(l)
~nn
(2.13)
as n'
where D (a,b) is as in Lemma 4 and x V y
~ 00,
n
I'k, A ,(a.b)/A , ~ 0 as
n
n
(2.14)
~"'n
and for all x €
= max(x.y).
n' ~
(X)
m.
lim sup I'k, A*.(a.bHI.J1 ,(a;x)1 + 1.J1 .(b;x)I}/A .
n'~
~"'n
n
n
n
n
(2.15)
Proof.
< O.
First assume that (2.11) holds for some x
only have to consider negative arguments.)
the event B
then
= {-Weal < 2y.
-1
~
-Web)
Choose any y
< -1/2}.
<
00.
(By (2.8) we
<x <0
and consider
Using (2.10), (2.9) and
(2.11), we see that with probability 1 for all n' large enough.
~
>0
Since P{B}
and y
nB{(y+o(l»
~
n
,(a;y)
~n.(a;y)
~ w
as
n'
- 2(l+o(l»}.
~ w.
in order for (2.13) to hold we
obviously must have
<
lim• sup -.J1n ,(a;x)l'k.
An ,(a.b) V An ,)
-.Inn An .(a.b)/(I'k,
~nn
(X)
n~
for all x
€
mby
the monotonicity of
~n,(a;·).
which of course can only
happen if (2.14) holds and
lim sup 1.J1 . (a; x) 11'k. A , (a. b) / A .
n'~
n
-.Inn n
n
<w
for all
x
€
m.
A similar argument shows that if (2.12) holds then again (2.14) must be true
and
lim• sup 1.J1n ,(b;x) 1I'k,
An ,(a. b)/An ,
~nn
<w
for all
x
€
m.
n~
Putting everything together and noting also that by (1.6) we presently have
- 16 -
An ,(a,b)
= A*n ,(a,b)
for all large n', it is now routine to argue that
whenever (2.11) or (2.12) hold along with (2.13), we must have (2.14) and
o
(2.15) .
A slight variation of the proof of Lemma 5 also gives the following.
LEMMA 6.
=0
An ,(a,b)
Whenever there exist a subsequence {n'} C {n} such that
for every n' and positive constants A , such that
n
{Rn , {b)-Rn,{a)}/A ,
n
= 0p{l)
as n'
~oo,
then we have {2.15} for all x € ffi.
Next we look at the term M in (2.6).
n
{1.5} and the notation x/\y
~
n
for any n
=
~
f
rbk l/n
n
rak l/n
J
n
= min{x,y}.
Since. using the definition in
we clearly have
rbk l/n
n
n{u/\v-uv) dH{u)dH{v) ~ rbk 1 A2 {a.b)
rak l/n
n
n
n
1. we obtain the following.
LEMMA 7.
On any subsequence {n'} C {n} on which A ,{a.b}
n
>0
for all n
large enough,
Finally we quote a variant of Lemma 2.10 of Csorgo, Haeusler. and Mason
{1988}. which is obtained by the original proof.
LEMMA 8.
Let Y
and Y
be two sequences of random variabLes such
I .n
2 .n
that YI . n + Y2 . n
= 0p{l)
as n ~
00.
the sequences IY I . n ' and IY2 . n l are
asymptoticaLly independent. and for at least one of i = 1 or i
(2.16)
lim lim inf p{IY i n l
n~
.
< M} > O.
M~
Then both sequences Yl
.n
and Y
are stochastically bounded .
2 .n
= 2,
- 17 -
Proof of Theorem 2.
Assume (1.16).
To get rid of the centering, for
each n' let M', + R' ,(b) - R' ,(a) be an independent copy of M , + R ,(b) n
n
n
n
n
R ,(a) in the representation (2.7).
Then, since (1.16) can be written as
n
A-1,{I ,(a,b) n
n
(2.17)
as n'
~
00,
IJ,
n
,(a,b)} + A-1,(IJ, ,(a,b) - C ,) ~6lI V*
n
n
n;.u
it implies that the sequence
{M,
-R',{a»}
/ {A,
n +Rn ,(b) -Rn ,(a) - (M',
n +R',(b)
n
n
n V
-JI'k",
non An ,(a,b»
Since by Lemmas 4 and 7 the sequence Yl
is stochastically bounded.
,n
,= {Mn ,
I'k", An ,(a,b}) obviously satisfies (2.16), Lemma
+ Rn ,(b) - Rn ,(a}} / (An , V
-J&~n
8 therefore forces
which by Lemma 7 implies that
as n'
~ 00.
We shall now show that (2.17) and (2.18) imply the existence of a subsequence
{n"} C {n'} such that (1.17) and (1.18) hold along {n"}.
We must consider
three separate cases.
Case 1:
(2.19)
A ,(a,b)
n
lim sup
n'~
(I",
n
>0
,(a;x)
for all n' large enough and
I
+
I'"
n
,(b;x)
I) <
00
for aU x € IR.
In this case, by a Helly selection one can choose a subsequence {n"'} C {n'}
such that
(2.20)
'"n ",(c;o)
=> '"*c (0)
on
IR
as
n'"
~ 00,
c = a,b,
* and "'b* satisfying the usual conditions.
for some functions "'a
its proof we know that for a function
{n"} C {n"'} we have (1. 9),
~,
By Lemma 1 and
along perhaps a further subsequence
- 18 -
Mn "
y " :=
n
rF:"
~&~n"
*
An .. (a,b)
~
Z(cp) :=
f
W( s )dcp( s)
[a,b]
and
y " =
n
rbkn"l/kn " W(s)dcp
J
n .. (s) + 0(1) =: Zn " + 0(1)
a
almost surely as not
~
00,
where Z .. is easily seen to be a normal random
n
variable with mean zero and
Jr
bk
2
E Zn" =
n
"l/kn " Jrbkn "l/kn "
a
(u/'w)dCPn,,(u)dCPn"(v) ~ a.
a
This implies that the normal random variable Z(cp) is non-degenerate, which in
turn implies that cp t O.
the non-degenerate limit
Then by part (i) of Theorem 1 we have (1.10) with
*
*
V(~a,CP'~b)
and this, (2.17). and the convergence of
types theorem yield (1.18) and
(2.21)
{11 ,,(a,b) -
n
for some constant
*
o ~c(-)'
Cn .. }/An "
'Y €
m.
~ 'Y
as
not
~ 00
Now (1.18) and (2.20) imply (1.17) with
~
c
(-)
=
c=a,b, and we see that for V* in (1.16) we have (1.19) with the
'Y
from (2.21).
Case 2:
(2.12).
A ,(a,b)
n
> 0 For aLL n' Large
enough and we have (2.11) or
In this case (2.18) means exactly condition (2.13) of Lemma. 5 and
hence we have (2.14) and (2.15).
Therefore, by (2.15) and a Helly selection
we see that there exist an {not} C {n'} and two functions
usual properties such that (1.17) holds.
~a
and
~b
with the
Now (2.14) means that we have
(1.18) with 0 = 0 along the original {n'} in fact.
Thus by part (ii) of
Theorem I,
{In ,,(a,b) - 11n ,,(a,b)}/An "
~6lI V(~ ,O'~b)
~
a
as
which in conjunction with (2.17) gives (2.21) for some
of types.
Whence we also have (1.19) with 0=0.
n"
'Y
€
~
00,
m by
convergence
- 19 -
Case 3:
There exi.sts a. subsequence {n'''}
{n'} such that Ii '" (a. b)
n
=0
Then by (2.18) and Lemma 6 there exist an {n"} C {n"'} and two
for a.1I. n"'.
functions
C
~a
and
~b
with the usual properties such that (1.17) holds. and
since (1.18) now trivially holds with 0=0. the proof can be finished exactly
as in Case 2.
o
This completes the proof of the theorem.
First we assume (I).
Proof of Theorem 3.
Let k
n
be an arbitrary
positive sequence satisfying (1.1) and choose any subsequence {n'} C {n}.
Then there exist a subsequence {n"} C {n'} and 0
< a < a o < b0 < b < ClO
any choice of 0
subsequence {n
iv
< a o < b 0 < ClO
such that for
and {n"'} C {n"} there is a further
} C {n"'} such that (1.21) holds.
This implies by Theorem 2
v
iV
that for some {n } C {n }.
Ii*v(a.b)
~nV{c; 0)
n
= il-ln;;;;'V-/k--v-Jr ~nV{c;
0)
=>
~c{o)
on IR.
c=a.b.
n
and
~
(2.22)
where 0
~
o{a.b)
< ClO and
~a
and
~b
o{a,b)
a n
v
~ ClO.
are some functions satisfying the
properties listed above (1.17).
Clearly. for any a
< a * < a o < b o < b* < b
. a further
we can fInd
vi
v
subsequence {n } C {n } such that
Ii Vi{a*.b*)IB[lnVi/k ViJ]
n
where 0
~
n
o(a* .b* )
I~
n
for all n
vi
< ClO.
~
o(a*.b*)
as
n
vi
~ ClO.
Noticing that for any x € IR.
VI.(d;x)1 -< Ii VI.(a.b)lB[lnVi/k VI.J].
n
n
d
= a*.
b*.
sufficiently large. we see by Theorem 1 that if o(a,b)
= O.
then
- 20 -
n
which contradicts (I).
>0
Thus o(a,b)
vi
-+ 00,
in (2.22).
A diagonal selection procedure now shows that a subsequence
vii
vii
{n
} C {n'} can be chosen so that (1.20) holds true along {n
} with some
finite function h(-) which is not identically zero on
(0,00),
that is the
de Haan-Resnick condition for (II) is satisfied.
Now assume (II) and choose any sequence {k } satisfying (1.1).
Let
n
{n'} C {n} be an arbitrary subsequence.
Then there exist a further
subsequence {n"} C {n'} and a finite function h(-) on (0. 00 ) such that h
and we have (1.20) for some {B(n)}.
Thus we can find 0
< a o < b0 <
(X)
~
0
such
that
" (b ) - ~ / " (a )} / B(ln"/k
{~n" /n
On" n o n
as n"
-+ 00.
>0
This implies that A ,,(a, b)
n
and all n" large enough.
< a < ao
Fix 0
"J)
-+
for all 0
and b
0
>0
h(b ) - h(a )
0
0
< a < a 0 < b0 < b <
< b < 00,
(X)
and let {n"'} be an
Then we can find a subsequence {nv } C {n"'}
arbitrary subsequence of {n"}.
such that
(2.23)
A v(a,b)/B[lnV/k vJ]
n
n
-+
0,
and we also see that for any x €
I~n ,,(c;x) I ~ {hk " / n,,(b)
n
for all n" large enough.
0
m,
0
< 0 < 00,
< a < a,
- h_ /
Uk " n
n
as n v
-+
and b
< b < 00,
,,(a)} /
(X)
A ,,(a, b),
n
c=a, b,
Hence we can choose a further subsequence
vi
v
vi
{n } C {n } such that (1.8) holds along {n } with some functions
But then, since A .(a.b)
n
subsequence {n
iv
VI
>0
for all n
vi
~a and ~b.
large enough, there exists a final
vi
} C {n } such that by part (i) of Theorem 1 we have (1.9)
- 21 -
along {n iv} with some
~ ~
Since
~
satisfying
~(b+)=l
and also (1.10) along {n iv }.
O. by (2.23) we finally obtain (1.21) with the non-degenerate
o
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CSORGO. S .. HAEUSLER. E. and MASON. D.M. (1988). The asymptotic distribution
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" S .. HAEUSLER. E. and MASON. D.M. (1990). The asymptotic distribution
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DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
CHAPEL HILL. NC 27599-3260
DEPARTMENT OF MATHEMATICAL SCIENCES
UNIVERSITY OF DELAWARE
NEWARK. DE 19716