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2. 30VT ACCUSION NO. J.
REPORT NUMBER
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~
4.
I
~
5. TYPE OF REPORT
Tt T L E (_d Subtltl.)
On limiting distributions of intermediate order
statistics from stationary sequences
a
PERIOD COVERED
rrECHNICAL
6.
PERFORMING ORG. REPORT NUMBER
Mimeo Series No. 1135
7.
AUTHOR(e)
8. CONTRACT OR GRANT NUMBER(.)
Vernon Watts and M. R. Leadbetter
9.
NOOO14-75-C-0809
10. PROGRAM ELEMENT, PROJECT, TASK
PERFORMING ORGANIZATION NAME AND ADDRESS
AREA
Department of Statistics
Universi ty of North Carolina
. Chapel IIi 11, North Carolina 27514
til. CONTROLLING OFFICE NAME AND ADDRESS
Office of Naval Research
Arlington. VA 22217
a
WORK UNIT NUMBERS
12. REPORT DATE
September. 1977
13. NUMBER OF PAGES
15
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18.
SUPPLEMENTARY NOTES
'9.
KEY WORDS (Contl"u.
on tevet•• • Id. II ".c•••.", _d Id."tlfy by bloc" """,b.t)
Stationary sequences. intermediate order statistics. mixing conditions. normal
sequences
T (Contlnu. on t.v.t••• Id. II ".c•••.", _d Id."tlfy fly block "umb.t)
n
X
1 be a strictly stationary sequence of random variables and x.cKn ) the
n n2:
kn -th largest order statistic for Xl •...• X.
For the case kn+ 00 but kn In + 0
n
20.
ABSIRA
Let
1
dependence conditions are obtained which are sufficient to insure that X~n) has
the same asymptotic distribution as it would if the {~} were independentnand
identically distributed. It is shown that the conditions are satisfied by stationary normal sequences for which the covariance function tends to zero
exponentially.
DD
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On Limiting Distributions of Intermediate order
Statistics from Stationary Sequences
by
Vernon Watts and M. R. Leadbetter*
1.
Introduction
The problem of finding the asymptotic distribution of the maximum term
from a stationary, dependent sequence of random variables (r.v.'s) has been
extensively investigated in the literature.
Of particular interest is when
the dependence assumption made is that of strong mixing or related conditions,
which formulate mathematically the concept of "approximate independence."
To
say that a sequence {X } >1 on (Q,F,P) is strongly mixing of course means that
n nthere is a real sequence
g(~)~O
as
~ +
00
such that
Ip(AB)- P(A)P(B)I ~ g(~)
for any events AEa(X l , ... ,Xm) and BEa(Xm+N+
n l'X
n 2"")' for each
m+N+
a(o) denotes the a-field generated by the indicated r.v.'s.
m~l,
where
Loynes (1965)
showed that under strong mixing and an additional restriction the maximum
xi n )
=
max{Xl, ... ,Xn }, suitably normalized, has the same limiting distribution
as if the sequence were actually independent and identically distributed (i.i.d),
assuming this exists, and necessarily of one of the three classical types of
extreme value limit laws.
For stationary normal sequences Berman (1964) found
two covariance conditions, one being implied by strong mixing, for which the
distribution of the maximum converges to the double-exponential limit law,
arising in the LLd. normal case.
More recently, Leadbetter (1974) obtained
the general resul t of Loynes under a substantially weaker "distributional
mixing" assumption, and he showed that with either of Berman's covariance
conditions the normal case may be placed into the general framework.
Addition-
ally, Leadbetter considered the re'lated high-level exceedance problem for
*This research was sponsored by the Office of Naval Research under contract
NOOOl4-75-C-0809.
2
stationary sequences, as well as extensions of the results for the maximum
to other extreme order statistics.
Our objective in this paper is to obtain analogous results for intermediate order statistics.
For a given sequence of r.v.'s {X }, let x~n)
n
denote the k-th largest of Xl, ... ,Xn .
l~kn~n for each n.
Then if k n
+
00
{kn}n~l
Let
but kn/n
+
be integers such that
0, {X~n)} is called a sequence
n
of intermediate order statistics and {kn } an intermediate rank sequence.
Subject to the mild restriction that kn increase monotonically, Wu (1966)
found that when the {Xn }
are i.i.d. the only possible nondegenerate limit
laws for the normalized sequence {a (Xk(n) - b )} are normal and lognormal.
n
n
n
(For convenience here all limit laws will be assumed continuous.)
We will
establish general conditions under which the intermediate order statistic
X(n) from a stationary, dependent sequence {X } has the same asymptotic
n
k
n
distribution as it would if the {Xn } were i.i.d.
These conditions parallel
those known for the corresponding result in the extreme order statistic problem,
a primary difference being that certain rapid "mixing" rates are assumed.
Using our procedure it is convenient to deal directly with an appropriate level
exceedance problem and to regard that of asymptotic distributions as a specialization .,
We also show that under a certain rapid "decay" rate of the covariance
function, our general conditions are satisfied by a stationary normal sequence
{Xn }; in this instance it is known (see Cheng (1965)) that the asymptotic
distribution of X(n) under independence is itself normal.
k
2.
n
The general stationary case.
First suppose that {Xn } is an i.i.d. sequence of r.v. 's with marginal dis-
tribution function (d.f.)
rank sequence.
F(x)
= P(Xl~x),
and that {kn } is an intermediate
Then for given real numbers an > 0, bn , the relation
3
P(an (X~n) - b ) ~ x)
n
(2.1)
-+
G(x)
n
holds for x such that 0 < G(x) < 1 if and only if
l-F(x/an
(2.2)
as n
-+00,
+
bn ) = kn In - u(x)~/n
n
+ o(~/n)
n
where G(x) = 4>(u(x)), 4> being the standard normal d.£. (see, for
example, Wu (1966)).
Consider the basic identity
where Wn = I:=lI[X.>u ] denotes the number of exceedances of the "level" un
1
n
by Xl"",X n '
For un satisfying
(2.3)
l-F(un ) = kn In -
u~/n + o(~/n)
n
n
for some real number u, it is easily seen upon suitable normalization of Wn
and by applying the classical central limit theorem for row sums from triangular arrays of r.v.'s independent within each row that
peW
(2.4)
as n
n
-+ 00.
< k )
n
-+
4>(u)
Of course for an arbitrary d.f. F, it may not be possible to choose u 's
satisfying (2.3),
n
but at least this can be done for every real u if X(n)
k
has
n
an asymptotic distribution, since then (2.1) necessarily holds for all real x.
Now suppose {Xn } is stationary with marginal d.f. F and that un satisfies
(2.3) for some u.
If under the appropriate normalization Wn has the limiting
standard normal distribution, then we obtain (2.4).
for some x and with u(x) = u, then so does (2.1).
Therefore if (2.2) holds
It
is evident that the
assumption that (2.2) hold with arbitrary u means equivalently that the intermediate order statistic ~~n) from the associated independent sequence {~n} ,
n
4
the independent sequence having the same marginal d.f. as {Xn }, has the
asymptotic distribution G(x) =
~(u(x)).
To establish (2.4) we present dependence conditions on {Xn } under which
Wn is asymptotically normal.
of Dvoretzky
~972),
The key to our approach is the following result
a central limit theorem for row sums from triangular arrays
of r.v.'s dependent in each row, and which generalizes the classical theorem.
Other versions of this are available, but the one given here was selected as
being most suitable for our purposes.
We state the result in its original
form, which is more general than what we actually require.
Lemma 2.1.
Let {X
.} >1 l<'<N
be an array of r.v.'s with EX . = 0 for
n,1 n- , -1- n~
n,1
all n,i, and let
sup
AEFn,m
sup
l~m<N
n
-k
BEG
andG
where Fn,m = a{X n, l""'Xn,m }
Ip(AB) - P(A)P(B)
n,m+k+l
= a{Xn,m+ k+l""'Xn, N}'
n,m+k+l
y
.
n,1
=
n
Suppose
n
there are integers 0 = j (0) < j (1) <..• < j (r ) = N
n
n
n n
n
j
I
such that, defining
(r)
I
x.
i=l, ... ,rn ,
i=j (r-l)+l n,1
n
we have
lim
I Ey2 = 1
n i odd n,i
lim
I Ey2n,i = 0
n i even
and
r
lim
n
n
2
. I 1 E {yn,1. I [I y
1=
I>]}
n,1. £
=0
for all
£
> 0 .
N
Then lAm rngn(jn) = 0 where jn
=
min
l~r~r
is asymptotically normal (0,1).
n
{jn(r) - jn(r-l)} implies that Li~lXn i
'
5
Let {a } and {8 } be sequences of positive integers such that. as n
n
n
-+
OJ
•
(2.5)
Then for each large n an integer t n is determined by the relation
o
s;
Rn < a
n
+
28n
it is readily checked that
t
(2.6)
n
-1
n
na
(n)
(n)
(n)
Now partition the first n positive integers into "intervals" II • 1 2 •...• I 2t •
n
of respective alternating lengths an' Sn • an • Sn •...• an • Rn .
By interval we
mean a finite set of consecutive integers. and the length of any interval is
merely the number of integers it contains.
For an interval I let WI be the
number of exceedances of the level un by {Xi: iEI} • where for simplicity we
delete explicit dependence upon n.
Thus
2t
Wn =
n
i~l \~n)
.
1
With this notation we now state our main dependence assumption. which we
denote by A(un ).
{un}
This is given in a form depending explicitly on the levels
satisfying (2.3) for some real u. and for the asymptotic distribution
problem the integer sequences assumed to exist may vary with different values
of u = u(x).
A(un ):
There are sequences of integers subject to (2.5) such that.
as n
-+
00.
(2.7)
for which
a -1
(2.8)
a
-1 n
I
n j=l
(an-j){p(Xl>u .x. l>U )
n J+
n
o(kn In)
6
and
8 -1
-1 n
8
n
(2.9)
L (8 -j){P(X 1>un .x.J+ l>un )
j=l n
2
- (l-F(un )) }
= o(k
n
In).
and such that
n
g
ex
n
n
(2.10)
-
+ 0 •
where
sup
1:5;m<2.Q. -1
n
sup
Ip(AB) - P(A)P(B)
AEF
,n.m
BEG
n.m+2
I •
Fn •m = cr(WI(n)·····WI(n))· and Gn.m+2 = cr(W (n).···.W (n))
I m+2
m
1
IZ.Q.
n
We note that the intervals len) and len) are "separated" by at least 8 .
n
m
m+2
It
is then evident that a condition sufficient for (2.10). which is stronger
in that more events must be considered. is
(2.11)
as n
+ 00.
where
h (.Q.) =
n
sup
sup
Ip(AB) - P(A)P(B)
l:5;m<n-.Q. AEF'
n.m
BEG'n.m+.Q.+1
I •
F'
n,m
Now. for fixed n,m let l ~ {l, ...• m} and i ~ {m+8n +1, ...•n}.
A. = {X.>u • iEl; X.
1
1
n
1
,:5;U
n
•
i'd1 •... ,m}\l},
(2.12)
B. = {X.>u , jEi; X.,:5;U , j 'dm+8 +1, ... ,n}\iL
n
J n
J
n
J
Define events
7
Then we can see that each AEF'
is some disjoint union of at most 2m events
n,m
each of the form A., and similarly each BEG'
B 1 is a disjoint union of
n-m-S
~
n,m+ n+
at most 2
n events of the form B.. Hence a condition sufficient for
1.
(2.11) is
n
(2.13)
2
--!!. h' .... 0 ,
a n
n
where
h'n
=
sup
/P(A.B.) - P(A.)P(B.)
~ 1.
~
2
l~m<n-B
n
I.
One may show that the "distributional mixing" condition (2.13) is satisfied
by strongly mixing sequences for which
g(t)~O
at some particular exponential
rate, but (2.13) and especially (2.10) are potentially much weaker, in that
fewer pairs of events need to be "tested."
Unfortunately, (2.10) may be hard
to verify directly, as is the case for stationary normal sequences considered
in Section 3, where with the presently known technical results pertaining to
normal sequences we will be content to deal with (2.13).
In this section, however, we work with the assumption A(un ) as stated, and
we now suppose for the given stationary sequence {Xn}and levels {un} satisfying
(2.3), that integer sequences {an } and {Bn
} for which
A(u ) holds have been
n
chosen.
Then we define a triangular array of r.v. 's {X .} for n large and
n,l
xn,l. =
W
- EW
I~n)
I~n)
1
1
;r-n
(J
n
where
an -1
1.2. 14)
+ 2
L (a
. 1 n
J=
,\2t
-j){P(X 1>u ,X. l>u )
n J+
n
The following lemma shows that L. n X . is the appropriate normalization of W .
1=1 n,l
n
8
This result will constitute the major portion of establishing (2.4).
Lemma 2.2.
Proof:
L2~n X . is asymptotically normal (0,1).
Under A(un ),
Clearly
EX
i=l
n,1
. = 0 for all n,i.
n,1
Upon showing that
(2.15)
2
lim
EX
= 1
L
n,i
n i odd
(2.16)
2 .
EXn,1
lim
= 0 ,
L
n i even
and
2~
lim
n
(2. 17)
n
2
i~lE(Xn,iI[IXn, i l >£])
= 0
for every
E:
> 0,
then along with (2.10) we will have fulfilled the requirements of Lemma 2.1 and
the conclusion will follow.
o
:0;
r
:0;
2~
n
Here we are taking Nn = r n =
2~n
and jn(r) = r,
.
First we have that
Var W
I ~n)
2
1
= 1
L EXn,i = L
2
i odd
1 odd
~ a
n n
for each n by stationarity, so that (2.15) obviously holds.
For convenience we write pn = l-F(u).
n
relation pn
~
Then from (2.8) and the obvious
kn In we have
a2
n
(2.18)
and moreover, if
~n
~
a k In
n n
> 1,
Sn -1
Var WI(n) = SnPn(1- Pn)
+
2
L (Sn -j){P(X 1>un ,
. 1
J=
2
by (2.5) and (2.9).
- Pn }
Thus
(~
(2.19)
2
X. 1>u )
J+
n
2
Ex
n,
i =
i even
L
i<2~
n
n
-l)Var W ( )
Var WI(n)
I n
2
2
<
2
2
~ (J
0
n n
n
-+
0
.
= 0
2
(an) ,
9
This combined
Also,
with (2.19) gives (2.16).
Ixn,l. I
Finally,
is bounded by
a. +28
n n
I r (J
n
so for given
Remark:
£
> 0, p(IX
.1
n,l
>
£)
= 0(0.n Ilk)
,
n
n
=0
for all large n, which leads to (2.'17).
o
If {B n } satisfies 8 k = O(n), then (2.9) automatically follows from
n n
(2.8) (see Watts (1977)).
However, for the derivation of a covariance condition
sufficient for A(un ) in the normal case, the requirement that Bn kn = O(n) is
unnecessarily restrictive, whereas (2.8) and (2.9) are verified with the same
calculation.
We now give the main result.
Theorem 2.3.
Let {Xn } be a stationary sequence of r.v.'s with marginal d.f.
F and {k n } an intermediate rank sequence.
Suppose the level un satisfies
(2.3) for some real u and let Wn be the number of exceedances of un by
Xl' ... ,Xn .
Proof:
If A(un ) holds, then P(Wn<kn )
+ ~(u)
as n
+
00.
For large n we have
2R.
k -np
n)
< n
peW <k ) = P( In X
n, i
n n
I r (J
i=l
n n
where p
n = l-F(un )·
From (2.3), (2.6), and (2.18) we see that
kn -np n
+ u ,
~(J
n n
and the conclusion follows by Lemma 2.2.
o
From this result we may state the following theorem giving sufficient
conditions for
~n)
n
to have an asymptotic distribution, which is the same as
10
if the {Xn } were i.i.d.
Theorem 2.4.
Let {Xn } be stationary and suppose for some constants an > 0, b n ,
Pea (~(n)
n
as n
k
n
A
+
00
for all x, where {Xn }
If A(un ) is satisfied for un
bn ) S x)
+
G(x)
= ~(u(x))
is the independent sequence associated with {Xn }.
= x/an+bn
with u
= u(x)
for all x for which u(x)
is finite, or equivalently, for which 0 < G(x) < 1, then also
Pea (x(n) - b ) S x)
n k
n
+
G(x)
n
for all real x.
Proof:
The conclusion for x such that 0 < G(x) < 1 follows from the previous
theorem and is easily established as well for all x, using the continuity of G.
o
3.
The normal case.
In this section we develop a covariance condition under which A(un ) holds
for stationary normal sequences.
As previously indicated, in doing this we
work with (2.10) replaced by (2.13), from which will arise the requirement of
convergence of covariances at an exponential rate.
For simplicity we deal
exclusively with standard (that is, zero mean, unit variance) normal sequences,
since results obtained for this case can be easily transformed to other normal
sequences.
Thus suppose {Xn } is stationary with marginal d.f.
rn
=
EXIXn +l ,
n~l.
and let
Let {kn } be an intermediate rank sequence and for a given
real x, suppose the level un satisfies (2.3) with u
relabeling,
~(x),
= u(x) = x,
or upon
11
(3.1)
4>(b ) = 1- k /n and a = n4>' (b )/v'i<
n
n
n
n
n
One such un is un = x/an +b n , where
Again let Wn be the number of exceedances of u by
n
(see Cheng (1965)) .
Xl' ... , Xn ·
Then we seek a condition on {rn } under which A(un ) holds.
We assume that r
n
0 as n
+
+ 00,
which is essential for the required
"approximate independence" between X. and X. when Ij-il is large, and it is
J
1
convenient to define the quantities
and
0
= sup Irml
m~n
n
As has been indicated by Berman (1964), the assumption r n
Also, it is clear that for each n
+ 0
implies 0 < 1.
1 there is an integer mn ~ n such that
on =Irm I, so if t n~O and r n = O(t n ), then since 0n t-n l :;; Irm It-ml , it
n
n
n
follows that also 0n = OCt n ).
~
We begin the development with some useful technical results pertaining to
normal sequences.
First, we have for un defined by (3.1),
1
I2TI
Taking logarithms gives
(3.2)
u
n
e
2
-u
n
~
u
n
12 log nlk , so that
n
2
41T(kn In) log nlkn
Also required is the following result, various forms of which have been
used for consideration of the maximum term from normal sequences by a number
of authors, including Berman (1964) and Leadbetter (1974).
use of a now standard technique and will be omitted.
The proof makes
12
Lemma 3.1.
r
n
O.
+
Let {Xn } be a standard stationary normal sequence with covariances
Let I, I', J, and J' be four mutually disjoint sets of distinct
positive integers.
Then
\P(X.>u, iEl;
1
X.,~u,
i'd'; X.>u, jEJ;
1
)
X.,~u,
j'EJ')
)
- P(X.>u, iEl; X.,~u, i'd')P(X.>u, jEJ; X.,~u, j'EJ')1
1
1
)
2
(3.3)
~ K
Lip 1,)
.. Ie
. .
_u /(I+lp . .
)
1)
1,)
1,)
for some finite constant K depending only on 0, where p 1,)
. . = r, )-1
. . I for
iEIuI', jEJUJ', and the summation is taken over all such possible pairs i,j.
In particular for i
(3.4)
<
j and any real u,
(l-~(u))
rp(X.>u, X.>u) 1
)
2
2
-u /(I+lr . . J)
I
~
)-1
Klr . . I e
)-1
Finally,we require a measure of how rapidly k
n
precisely as follows.
Define 8
= inf{81:
8
Then it is evident that 0
(3.5)
= 8({kn })
kn
~
8
~
k
1,
n
= O(
+00
, which we formulate
by
8
n I)}
and moreover
= o(n 8+£ )
for every
E > 0 .
As part of our covariance condition we will restrict what intermediate rank
sequences can be considered, in terms of values of 8.
Having the preceding results and notation we now suppose for the stationary
normal sequence {Xn }, that the covariances {r } satisfy
n
(3.6)
r
n
=
O(e-
n
p
)
for some p > 2.
4It
13
Let
{k }
n
be subject to the restriction 2/p <
e<
sequences of integers {an} and {Sn} for which an ~ n
o < !p<"
(3. 7)
11
<
1.
Then we may choose
and Sn ~ nll , where
A
A < ~2 .
(We note this choice of sequences does not depend on x.)
It is clear the
relation (2.7) holds, and as the next lemma shows, so do (2.8) and (2.9).
Lemma 3.2.
Suppose {rn } satisfies (3.6) and {kn } is such that
e
< 1.
Then
for {an } and {Sn } as chosen above, (2.8) and (2.9) hold for un given by (3.1),
for any real x.
In fact, for each such un we have
[/k]
n
k
n
2
n
L Ip(x 1>u,
j=l
n
X.
J+
l>u) - (1-~(u)) I
n
n
0
+
as n
+
00
•
([e] denotes the greatest integer function.)
Proof:
Let y be such that 0 < y < (1-8)(1-8)/(1+8).
Then by (3.4)
and by (3.2) the right side does not exceed
Kn
l+y
~e
2
-u / (1+8)
n
<
-
K1 (1 og n ) ny- (1-8) (1-8) / (1+8) [k ] (1-8) / (1+8)
~
n
for some constant K1 .
sion tends to zero.
Then from (3.5) and the choice of y, this last expresThe proof is complete if it is possible to choose y such
that [~] ~ n Y for all large n.
Otherwise, if [~] > n Y we have
[~]
L
Ip(x1>un , X.J+ l>un ) - (l-~(un ))2 1
k
n j=[nY]+l
n
8
14
which tends to zero since
°
[n Y]
= O(e -
Y
n PI
) for every PI < p.
The conclusion
o
then follows upon combining the two summations.
We now show that (2.13) holds with the above choices of {a } and {S }.
n
Fix n and m with 1
~
m< n < S
n
and let A. and B. be events of the form
n
2
1
Then by (3.3) we have
(2.12).
2
Ip(A.B.) - P(A.)P(B.)
-1 -J
-1
1.
I
~ K
I Ir..
J-1
. .
1,J
-u /(l+lr .. 1)
Ie n
J-1,
where the summation is taken over all pairs i,j with 1
j
~
n.
~
i
~
m and m
+
Sn + 1
The right side above does not exceed
2
-un/(l+OS )
e
n
2
Kn
Os
n
IIp
But
Os
=
O(e- n
1
) for every PI < P, and by (3.7), we may suppose II
n
Then taking the supremum over all choices of indices
1
~
m < n-Sn' we see that (2.13) holds.
l, i,
PI
and all m,
> 1.
Summarizing, we have proved the
following theorem.
Theorem 3.3.
covariances
Suppose {X } is a standard stationary normal sequence with
n
rn
= EX 1Xn+ l'
. f Y1ng
.
n >- 1 , sat1s
rn
= 0 (e- n
P
) f or some P > 2 .
Let {kn } be an intermediate rank sequence for which 2/p <
the level un satisfies (3.1) for some real x.
e<
1, and suppose
Then A(un ) holds, so by Theorem
2.3,
peW
n <k)
n
as n -+
00,
-+~(x)
where Wn is the number of exceedances of un by Xl, ... ,X .
n
Moreover,
it follows that
Pea (x(n)_ b ) ~ x) -+ ~(x)
n k
n
n
as n
-+00
for all real x, where an and bn are given by
~(bn)
=I
- kn/n and
~
15
a
n
= n~' (b )
n
/Ikn .
Finally, we mention that the conclusion of Theorem 2.3 has been obtained
under an alternative set of conditions by Watts (1977).
There a construc-
tive procedure is given for actually calculating an approximation to the
value of peWn <k)
n
which has the same limit.
Moreover it is shown that for
stationary normal sequences the exponential convergence rate for the covariances again arises.
However, the precise rate required is less severe than
that indicated in Theorem 3.3, for consideration of intermediate rank sequences
{k n } for which 8
rn
= O(e -n
P
= 8({kn })
is "small," or close to zero.
Specifically, if
) for some p > 0 and if 8 satisfies the two restrictions
8 < 2/5(1+ l/p) and 8 < (1-0)/2, where 0
= sup
n~l
Ir
n
I,
then the conclusion of
Theorem 3.3 holds.
REFERENCES
Berman, S.M. (1964).
Limit theorems for the maximum term in stationary sequences.
Ann. Marh. Srarisr., 35, 502-516.
Cheng, B. (1965). The limiting distributions of order statistics.
Marh., 6, 84-104.
Chinese
Dvoretzky, A. (1972). Asymptotic normality for sums of dependent random
variables. Proc. Sixrh Berkeley Symp. Marh. Srar. Prob., 2, 513-535.
Leadbetter, M.R. (1974). On extreme values in stationary sequences.
verw. Geb. 28, 289-303.
Z. Wahr.
Loynes, R.M. (1965). Extreme value in uniformly mixing stationary stochastic
processes. Ann. Marh. Srarisr., 36, 993-999.
Watts, J.H.V. (1977). Limit theorems and representations for order statistics
from dependent sequences. Ph.D. dissertation, University of North Carolina
at Chapel Hill.
Wu, C.Y. (1966). The types of limit distributions for some terms of variational
series. Scienria Sinica, 15, 749-762.