*Research sponsored by Office of Naval l1eseaTch under Contract N00014.67-A0321-002 TSK NF.-042-214.
'
ASPECTS OF EXTREME VALUE THEORY
FOR STATIONARY PROCESSES - A SURVEY
by
M. R. Leadbetter
DepaPtment of Statistics
University of North Car'olina at Chapel Hill
Institute of Statistics Mi~eo Series No. 951
Septenber, 1974
ASPECTS OF EXTREME VALUE THEORY
FOR STATIONARY PROCESSES - A SURVEY
by
M. R. Leadbetter
of statistics
University of North Carolina at Chapel Hill
Departn~nt
SUMMARY
The primary concern in this paper is with the distributional results of
classical extreme value theory, anc their developMent to apply to stationary
processes.
The main emphasis is on stationary sequences, where the theory is
\'lell developed.
:\esults available for continuous parameter processes are also
described, but with particular reference to stationary normal processes.
Research sponsored by Jffice of Naval Research under Contract NOOOl4-67 -A0321-002 TSK NR-042-214.
-1-
1.
INTRODUCTION
TI1is paper is concerned with certain aspects of extreme value theory of
independent and identically distributed (i.i.d.) randon variables, and their
generalization to apply to stationary sequences, and continuous parameter
processes.
If
{~n}
is any sequence of random variables we shall write
for the maxinum of the first
n
terms.
Specifically the following topics
from classical extreme value theory will be considered:
(i)
Gnedenko's Theorem - giving the possible types of asymptotic distribution of
Mn ,i.e. the .possible distributions
G such that·
._ P{an (Mn - bn ) s x} ~ G(x)
P{M s u } ~ e- T
TIn
(ii)
The relation
(iii)
The form of t!'c limiting distribution in the normal. case
(iv)
The Poisson limiting distribution for the number of exceedances of
un
(v)
by
~l
...
n
~n
n
where
u
n
1
> u ) ~
n
is chosen as in (ii)
The asymptotic distribution of the r
fixed
P(~.
when
th
largest of
~l ... ~n for
r .
In Section 2 these results will be stated (together with indications of
methods of proof) for i.i.d. random variables.
~ection
3 contains a brief
discussion of dependence conditions which have been used in the past, and
-2-
those which are used here, in order that the asymptotic results for the maximum may apply to stationary sequences, as will be seen in Section 4.
These
results will be compared with those of Berman [1] for stationary normal sequences.
In Section 5, \ve look at the "exceedances" of high levels by stationary
sequences, and indicate their weak convergence as point processes, to a
Poisson process, by means of a simple and useful general point process convergence theorem of Kallenberg [13]. TIle desired properties of the rth
' largest
value,
M(r)
• will fa 11 0\-1 as immediate consequences of this convergence.
n
Certain of the results are known to apply to the supremum
M(T)
~(t)
= sup{~(t):
as
T
~ w.
0
~
t
~
T}
of a continuous parameter stationary process
TIlese are described in Section 6.
properties presently known concern the case when
The most detailed
{~(t)}
is a nor,maZ process,
and these are also discussed in Section 6.
Finally we remark that other questions besides distributional ones, are
of interest in extreme value theory.
For example convergence questions akin
to laws of large numbers are clearly important.
We do not consider such
matters here, but concentrate on the distributional results indicated.
2. EXTREME VALUE THEORY FOR 1.1.0. RANDOM VARIABLES
In this section we shall look briefly at each of the topics Mentioned in
the Introduction, for i.i.d. random variables.
-3-
(i)
Gnedenko1s Theorem.
This theorem is central to classical extreme value theory and is stated
precisely as follows:
THEOREM 2.1 (GNEDENKO):
f.1
n
= max (sl
> ~2
Let
sl'~2 ...
be i.i.d. random variables, and let
... sn) ." Suppose that for some sequences of constants
{an> O} , {b } ,
n
P{a n (Mn
where
-
bn )
~
x}
+
G(x)
(at continuity points of G)
G is a non-degenerate d.f ..
Then
G belongs to one of the three
l1extreme value types":
Type I
Type II
Type III
= exp(-e -x )
G(x) = 0 x < 0
= exp(_x- a ) x
G(x) = exp(_(_x)a)
=1 x ~ 0
G(x)
(In saying that
x may be replaced by
_00
< x <
~ 0
x
<
00
(a > 0)
0
(a > 0)
G is one of these "types l1 it is to be understood that
ax
+
b for any fixed
a > 0,
b
in the functional
form. )
While we shall not prove Gnedenkois Theorem here (see [10] or [5] for
detailed proof) it does seem worthwhile to point out the main ideas of the
proof, since these are also used later.
TIle main part of the proof given by
Gnedenko may be displayed as the three lemmas given below.
-4-
Pirst note that if the common distribution function (d.f.) of the
is P,then Mn has the d.f.
P{~i ~ x, ~2 ~ x ... ~n ~ x}
If the left-hand side converges to some
G(x)
= pn(x)
then for any k
H:·}
1
and
= 1,2,3 ...
,
Stating this formally ,,,e have
~1'~2 ...
LEMMA 2.2 Let
be i.i.d . .
If
(2.1)
holds for
k
=1
, then it holds for all
k
= 1,2,3 ...
In particular if the assumptions of TIleorem 2.1 are satisfied then (2.1)
holds for all
k.
On the other hand if (2.1) holds for all
k , G must have
certain iistability properties l l indicated in the foUm'ling lemma.
LEMMA 2.3 If (2.1) holds for all k and G is non-degenerate then for each
k , there exist constants a k > 0 , Bk
such that
(2.2)
(Sometimes such a d.f. is called stable (e.g. [5]) though this is, of course,
not the usual definition of stability.)
This result is due to Khintchine and is given in [11].
A very general
result of this type (from llhich this lel1'lTIa fo11ml7s as a special case) is given
-5-
by Feller (7, p. 246].
This result states that if
{F}
n
is any sequence of
d.f. 's such that
Fn (An x
for some sequences
n
An > 0 ,
non-degenerate, then
for some
G(x) , Fn (A n* x
+ ~ ) +
An IAn*
a > 0 , b
This
* +
+ ~n)
G* (x) ,
* > 0 , ~n* ' where G , G." are assumed
,
A
n
n
+ 1
(lJ
- l.I * )/A + 0 and G* (x) = ~(ax + b)
n
n n
theorem may be applied to (2.1) with "An = lla n
~
*
*
11k , to gIve
.
the lemma.
n = lIb nK1 , G = G
The following 1elT'J'la now completes the proof of Gnedenko I s Theorem for
lJ
i.i.d. random variables.
LEMMA 2.4 If G is a non-degenerate d.f. satisfying
some constants
a
k
> 0 ,
(2.2) for each
k
(and
Sk ) then it is an extreme value d.f. of Type I, II,
or III.
The proof of this 1erJma is the major part of Gnedenko's derivation in
[10] and we do not give it here.
To summarize, if the
~
n
(a) . If (2.1) holds for
are i.i.d. it follows that
Ie = 1
ticular (2.1) holds for all
it holds for all
k = 1,2 ••..
In par-
k under the assumptions of Gnedenko's
Theorer:l.
(b)
If (2.1) holds for all
k
then
G (if non-degenerate) has the
stability properties (2.2),
(c)
If (2.2) holds for each
value d.f. and hence
k, G is a Type I, II, or III extreme
-6-
(d)
under the assumptions of Gnedenko's Theorem, (2.1) holds and hence
(2.2) does and hence
Looking
~lead,
G is an extreme value d.f . .
we note that the same proof will apply under any circum-
stances in which the truth of (2.1) for
k
=1
implies its truth for all
k.
This will be used in the next section to obtain the desired generalization of
Gnedenko's Theorem.
(ii) Convergence of
~
P{M
n
Suppose that a sequence
to
u }
n
{u}
n
e -T
when
may be chosen so that
(2.3)
for some fixed
T >
(Hote that this is not always possible - e.g. if
0
F increases only by jumps at 1,Z,3 ...
l
be checked.) Then
(2.4)
P{Mn ~ un}
= [1
-+
e
- (1 - F(Un))]n
with
F(j)
= [1
- Tin
=1
- T/(Zj - 1) , as nmy
+ o(l/n)]TI
-T
Conversely it is easily seen that (2.3) is also necessary for (2.4).
(2.4) holds then
n
10g[1 - (1 - Feun ))]
from which (2.3) follows.)
-+ -T
Thus the following result holds.
(If
-7-
THEOREM 2.5.
For the i.i.d. case
If
that
I -
u }
11
~
e -'[
1 - F(u )
n
~
'[In .
The limiting distribution in the normal case.
~.
1
are i.i.d. standard normal r.v. 's we
= '[In
I - ~(un)
Now
~
n
if and only if
('[ > 0)
(iii)
P{M
(for a given
~(x) ~ ~(x)/x
n4>(u11 )/un
~
un2
(2.5)
2
un
=2
co
where
where
cP
rI>
so that
is the standard norrral d.f.
is the standard
norr~l
density so
It follows by taking logs and rearranging that
log n - log 2n - 2 log u11 - 2 log
T
shows that log un = o(log n) . It then follows from
2
2 log n or un /(2 log n) ~ I , whence by taking logs we have
11
~
(2.6)
.
x,.,
o)
un
log u
and dividing by
(2.5) that
'[
as
T >
choose
~RY
log un
=}
log 2 + } log log n + 0(1) .
By substituting (2.6) in (2.5) and taking the square root we may obtain
_ !.
2
log
an
= bn
'[ + 0
log log n+log 4n + log T
2 log n
2 log n
+
oflog log n)2J
log n
t
{;J
where
I
a
n
= (2
2
log n)
I
1
(2.7)
b
(2 log n)
n =
2
log n)
- 1:.(2
2
-"2
(log log n
+
log 4n)
.
-8-
Hence, writing
= e -x
T
P{a (M
n
n
in Lemma 2.5, we have
- b ) + 0(1)] ~ x} ~ exp(-e -x )
n
P{a (M - b ) ~ x} -~ cx-p(-e-~~)
n n
n
so that also
t~e
Thus in the normal case
with the normalizing constants
(iv)
Li~iting
~1'~2".
Let
Any
(2.3) .
{~k}
.
Tl1e nWTIber
Nn
bn
an
~.
> u
J
n
,
~j
THEOREM 2.6.
If
~y
rando~
the i.i.d.
n
1 - F (u )
n
be chosen to satisfy
u
.
i.e.
{~k}
by
E
c'
../lnce
for
n(l -
F (un))
t~e follo~ing
n
~1'~2
l;
n
1
~{
~
~
T
u
~
n
n
is
it £0 11m'1s
result holds.
is the number of exceedances of u
variables
n
will be called an exaeedanae of
n
as~;~totically Poisson,
is
~ay
{un}
of exceedances of
N
n
Mis of Type I
given by (2.7).
be i.i.c. and suppose
binomial wit:1 parameters
that
asyr2totic distribution for
Poisson distribution of Exceedances of u
for which
j
.
(satisfying.(2. 3))
, then
(2.8)
for any fixed
r
= 1,2,3 ••..
Thus the exceedances have this ilasynptotic Poisson
form of
F
(provided (2.3) can be
much more in Section 5 where
process.
t~le
satisfie~).
characterl~
whatever ttc'
TI1is notion will be Qeveloped
exceer;ances l',ill be regarded as a point
-9-
(v)
Asymptotic distribution of
Let
{u}
n
M~r) , the r th largest of ~1 ... ~n .
be a sequence such that (2.3) holds, and hence by l;leorem
{M(r) ~ u} is precisely the event
r-l n
n
r-l
T
{N < r} so that p{M(r) ~ u } = L peN = s) ~ e- L TS/S! . Su~aarizing
n
n
n
s=o
n
s=o
this formally we have the following result.
··2.6· , (2.8) holds.
THEORE~1
But the event
satisfies (2.3) and
2.7.
i.i.d. random variables
sl,s2 ...
~n
(2.9)
S
as
Is!
n
-j>.
co
,
r = 1,2,3... .
COROLLARY:
holds.
n
then
T
for all
is the r th largest of the
M(r)
{u }
n
If
is a sequence such that
then (2.9)
(For then by Theorem 2.5; (2.3) holds.)
We may develop this line a little further - to obtain the asymptotic
distribution of M~r)
P{an (Mn - bn )
~
x}
~
II, or III) then, if
un = x/an
(2.10)
+
from that of M
Specifically if
n
G(x)
where
G is non-degenerate (and hence of TYP9 I,
0 < G(x) < 1 , by applying the above corollary with
bn ' T = -log G(x)
P{a (M(r) - b )
n n
n
we see that
x} ~ G(x)
r-l
L
(-log G(x))s/s!
s=o
It is also easily seen (using continuity of the extreme value distribution
that the limit is zero if
result holds.
G(x) = 0 and
1 if
G(x) = 1.
G)
Thus the following
-10-
THEOREM 2.8.
Suppose that
a non-degenerate d.f. .
largest of
~1'~2'
... ~n
P{an (Mn - bn )
~
G(x)
is
n
~
Then the asymptotic distribution of
is given by (2.10).
right-hand side of (2.10) is to be taken as
3.
~
x}
(If
0
or
G(x)
1
where
00
the r th
M(r)
n
=0
or
G is
the
1
respectively.)
DEPENDENCE RESTRICnQr,lS.
~1'~2'"
We turn now to consider dependent stochastic sequences
.
Al though it is not always necessary to do so (cf. [8]), we shall concern
ourselves with strictly stationary sequences, i.e. such that the finite
dimensional distributions
property that
...
of
F.1 +p,···l. +P (xl ... x n )
1
n
= F.1
. (xl'"
1 , .. 1 n
have the
~.
.
1
n
xn )
for any choice
of n, i l ... i , xl ... X •
n
n
Certain of the properties of Section 2 were generalized by G. S.
~~atson
([25]) to apply to "m-dependent Ii stationary sequences, and by P.. M. Loynes
([19]) to "strongly mixing" sequences.
The strong-mixing assumption is the
usual one, i.e. requiring that
Ip(A n B) - P(A)P(E); < g(1<)
(3.1)
for any events
A
E a(~l
... ~ )
ill
denoting the a-field generated by
(the
Il
mixing
g(k)
=0
for
B
€
a(~m+k+l ' ~m+k+2 ... )
~l
function") tends to zero as
k
~
~
a(~l
...
etc.) for any m , l\There
In
k
(
~
00.
~m)
g (k)
(m-dependence requires
m .)
In particular Loynes obtained Gnedenko's
Theore~
WIder strong mixing, and
showed that (2.4) holds under natural further conditions.
In ad.dition heL .
-11-
showed that
Mn
~
under such conditions - the asynptotic distribution of
= max(~l"'~)
n
is the same as it would be if the
S.1
were i.i.d. with the
same marginal d.f. as when dependent.
l'lhen the sequence is strongly nixing but does not necessarily satisfy
further conditions, it is possible to obtain an
aSYT.~totic
distribution of
Mn
which is different from that of the i. i. d. sequence tvith the same marginal d. f .
Examples of this - and other related matters
cOI~pler:lenting
tile
~"ork
of Loynes,
are given in papers by 01Brien ([20], [21)).
For stationary normal sequences, conditions on the covariance function
di tions.
=0
E~.
(taking
1
,
2
E~.
1
=1
are more natural
)
tha~
mixing con-
It has been shown by S. M. Berman ([1]) that the same asymptotic
Mn
limit holds for
in this case as for an i.i.d.
~ormal
sequence (i.e. a
Type I limit with normalizing constants given by (2.7)) if one of the follo'lTing
conditions holds.
00
(3.2)
r
n
log n
-+
0
as
n -+
00
or
L.\ 1'2 <
n=1 n
00
•
(Mittal"and Ylvisaker ([27]) have recently shown that if the conditions (3.2)
do not hold a variety of limiting distributions are possible.
It is possible to weaken the strong mixing condition and still obtain
Gnedenko 1s Theorem and the other results indicated above.
It will further
turn out tl:at this weakening provides results which apply to normal sequences
under the conditions (3.2).
To see how the strong mixing condition should be
weakened, note first that not all events
A, B in (3.1) are of interest.
In
-IZfact we are clearly primD.rily interested in events of the form
~n ~
type (2.1) to hold for all
where
i
Z ...
u}
A
which suggests requiring a condition of the
= {i;.11
~
...
u
...
i;.
1
$
P
u}
D
= {~.Jl
jl - i p ?- k
of the finite dimensional distribution functions (and writing
for
i
F.
l
11··
<
<
i
p
< J <
1
jz
< jq
and
.
u
~
i; .
Jq
$
u}
Put in terms
. (u, u ..• u) ) this would require
.I
n
(D) :
whenever
as
k-+-oo .
However even (D) is not sufficiently weak to deal with normaZ sequences satisfying the weak covariance conditions (3.2).
l~e
may obtain a suitable concition
by modifying (D) so that it is essentially required to hold for a single
r~re
sequence of interest, rather than for every fixed value of
u.
if
D(u )
(u )
n
whenever
denotes a given fixed sequence we shall say that
i l < i Z ... < i < jl ... < jq , jl - i ~ k , where
p
p
increasing in
k
lin an k
p"-+-oo
for some sequence
kn -+-
00
with
'
o(Un)
n
00
an,k
is non-
,
= 0
n
kn/n -+- 0 .
It is apparent that (9) implies
k -+-
holds if
n
and where
(3.4)
sequence
precisely
(D(u ))
n
k In -r 0 may be chosen).
n
for any sequence
u
n
(any
However it will turn out that
is implied by either condition (3.2) in the normal case, for suitably
-13-
chosen
un
We shall see that Gnedenko 's Ti1eorem holds under conditions of the type
D(un ) .
However (as for strong mixing) an additional condition is required to
obtain results such as (2.4) and the limit law for normal sequences.
lowing condition
D' (u )
n
is a variant of one used by Watson [25] and Loynes
[19] and will be appropriate here.
Then D'(u)
n'
Again let
[un]
be a given real sequence.
holds if
n
lim sup n.L P{~l
(3.5) (D'(u )):
n
n~
>
J=2
unk ' ~.
>
unk}
= o(l/k) as k
J
This condition involves the bivar'iate distribution of the
are i.i.d. and
seen to hold for example if the
un
{~.}
J
~ ~ .
and is easily
satisfies (2.3).
It
form a stationary normal sequence satisfyinS
also holds (Section 6) if the
either condition (3.2), again if
4.
The fol-
satisfies (2.3).
{un}
EXTREME VALUE THEORY FOR STATIONARY SEQUENCES
(i)
Gnedenko's Theorem under
D(u n ) •
The following generalization of Gnedenko's Theorem holds.
THEOREM 4.1.
If
{~n}
Mn
is a stationary sequence,
= p.ax(f" ., ... sn )
having
non-degenerate asymptotic distribution,
Nan (Mn - bIi)
and if
D(un )
holds for
un
~
x}
= x/a.n
-+
G(x)
+ b
n
(a n
>
0)
(for each real
a Type I, II, or III extreme value distribution function.
x), then
G(x)
is
-14-
The proof may be found in detail in [15].
Its main pattern is that used
by Loynes [19] for the mixing case with. ho\'!ever. some essential differences.
It will be useful to indicate the general lines of proof here, without detail.
As for the i.i.d. case, it is only necessary to show that if (2.1) holds
for
k
=1
, then it holds for all
o (un)
uses
nax(s.: j
E:
]
k
= 1,2,3 ...
to "approximate by independence".
E)
for any set
T~e
following basic lenma
of integers,and a set
E
M(E)
In this
will denote
~Jill
(i, i + I ••• j)
be called an intervaZ.
LEMMA 4.2.
Let
E.
]
Let
N,r,k
be intervals,
be fixed and
j
= l ... r
hold for a given sequence
D(u )
n
, separated by at least
k
from each other.
Then
r
r
Ip{ n M(E.) ~ uN} j=l
]
IT P{M(E.) ~ u }I ~ (r - l)aN,k :
j=l
J
N
Lemma 4.2 is proved by a simple induction
is used in obtaining the following
a~d
result:
LEMMA 4.3.
N
= nk,
II
Let
=
n,m,k
be positive integers.
(1,2 ... (n - m)) , II*
for some constant
=
(n -
]I)
+
If
G(u)
n
holds then writing
1, ... >n) ,
K
This is proved (along the lines used by Loynes) by dividine the first
N
= nk
n
-
integers into intervals
m, m, n
-
m, m
... .
PH'N
*
II' 11 ' 12 ,
~ ~}
T
.1.
*
2'
...
II: '
*
1)(
of "lengths l1
is then approximated by
k
P{ n M(I.)
j =1
J
~
~}
11
-15k
approxi~ated
which in turn is (by Lemma 4.2)
by
IT P{M(I.) ~
J
uN} .
Eac~
term
j=l
P{M ~ ~} .
n
it can be shown that
is finally approximated by
From this
each
k.
If
le~ma
uN} • P' (M ~ ~) ~ 0 for
n
holds with u =.!.- + band P{a (M - b ) ~ x} ~ G(x)
na
n
n n
n
n
D(u)
n
(i. e. if (2.1) holds for
k{Mn
k
= 1 ) then
P{~ ~
uN}
~
G(x)
and hence
(
b)
}
11k
f or all 1,( = 1,2 ...
nk Mn - nk ~ x ~ G (x)
k = 1,2 ... ) as required to finish the proof of
}
G( )
nk ~ , x or
(i.e. 2.1 holds for all
p.
1(
P{~~ ~
P{ a
~ U
Theorem 4.1.
(ii)
Convergence of
P{Nn
~
u }
n
under (2.3),
It is shown in [19] that under strong mixing if
P(Mn ~ Un(T)) ~ ~(T)
= e -cn for some a with
when U (T)
satisfies (2.3) for each
o~
It may similarly be shown that this 2.)plies if, instead of strong
n
a
~
1.
mixing, the conditions
using the fact that
D(u (T))
2n
then
~(T)
hold for each
n
[1 - F(u
T
T
(T))] ~ (T/2) /n
(This is seen simply by
so that
P{Mn ~ u 2n (T)} ~ HT/2)
~(T) = ~2(T/2) from "lhich the exponential
and thus, by Lemma 2.5 of [15],
limit may be deduced as in [19]).
0' Brien ((20]) gives examples in which
interested in the "usual" case
DV (u)
n
of the last section.
a
=1
a
<
l.
We are here particularly
which may be guaranteed by the condition
Specifically we obtain the =ollO\'!ing result,
again generalizing one given in [19] under strong mixing.
It should be noted
that the limit is shown to exist under the stronger conditions (rather than
assumed to exist and then shown to have a given form as above, when just
-16-
THEOREM 4.4.
where
{u}
n
Let
be a stationary sequence and let D(u)
, DI(u) hold,
n
n
{!;n}
satisfies (2.3) for sone fixed
P{M
n
~ u } + e- T
as
n
n
T >
O.
+
•
00
(i.e.
The proof of this result is given in [15] and will not be repeated here.
It
is based on the standard inequalities
I
Ip(!;· > u} -
.
1
1
(iii)
"
1<)
Pi!;. > u , ~. > u} ~ P{U(i;.
to denote the maximum of n
n
distribution function
{!;.}
)
(0 < 0 < 1)
.
,
and that
Then P{M
n
> u)}
= P(Mn
> u)
:<:;;
{u}
n
i.i.d. r.v.
IS
with the same
u } +
n
{~i}
.
"-
P{M
n
~iJ}+e
n
D(u ) , Di eu )
n
n
e
for some sequence
un and sone
hold for the stationary sequence
.
PROOF: By Theorem (2.5) 1 - F(un )
with this
1
F as the terms of the stationary sequence
THEORH1 4.5. Suppose that
,
.
1
Relations with the i.i.d. case.
We write M
e
J
1
~
TIn with
T
=-
log
e , i.e. (2.3) holds
Hence by Theorem 4.4
P{M ~
n
u }
n
+
e- T
=e .
Thus the sa.'1le limit holds for the naximum in the dependent sequence as
would apply if the random variables were i.i.d. with the same marginal distribution function.
As a corollary we may consider the asymptotic distribution
-17-
of M
n
under the standard normalization.
THEOREM 4.6.
If
P{an(Mn - bn )
~
x}
+
G(x)
(non-degenerate) then
D(un ) ,Dl(Un ) hold for un = x/an + bn
This follows at once from Theorer:: 4.5 for 0 < G(x) < 1 and by continuity
P{an(Mn - bn )
of
~
x}
+
G(x)
G, at points where
(iv)
If
if
G(x)
= 0 or 1 .
Stationary norMal sequences.
{~.}
J
is a (zero mean, unit variance) stationary normal sequence
whose covariances
{r}
n
satisfy either condition (3.2), then it is known
(Berman, [1]) that
P{an (Mn - bn )
where an
and
bn
Sec~ion
x}
are given by (2.7).
quence of Theorem 4.6.
shown in
~
+
exp(-e -x )
This result also follm·!s as a conse-
For this liBit applies to i.i.c. random variables (as
2) and it follows by Leoma 4.3 of [IS] that either condition
\'/hen u = x/a + b , in this particular
n
n
n
case.
The ultimate amount of work in t!1is route to the as)'!llptotic distribution
in the normal case is not less than that of the original proof in [1].
it does indicate the satisfactory nature of the D, D\
However
conditions, since they
are implied by the very weak conditions (3.2) when the sequence is normal.
-13-
5. EXCEEDANCES OF HIGH LEVELS BY A STATIONARY SEQUENCE AND THE DISTRIBUTION
OF r TH LARGEST VALUES.
9
(i)
Exceedances of high levels.
As previously noted, any point
an "exceedance of the level
U"
j
where
E; •
J
by the sequence
exceeds
{E;.} •
J
u will be cal lee
The exceedances form
a point process i. e. a series of events occurring in ""time" according to sone
probabilistic law.
fying (2.3).
As
Suppose now that
n
{un}
is a sequence of constants satis-
increases, the probability of an exceedance becomes
smaller and the exceedances consequently tend to become rarer.
An obvious
question is whether one nay thus obtain a convenient "limiting point process".
We have seen in the i.i.d. case that the number of exceedances in
has an asymptotic Poisson distribution and one may expect
t~1at
true for stationary sequences under appropriate assumptions.
(1,2 •.. n)
this will be
We shall indicate
how this Hay be shown - and indeed that the sequence of point processes fOrtlee
from the exceeclances of u
n
converges weakly - as random elements of a natural
metric space - to a Poisson process.
First we make a time scale change to avoid degeneracy.
discrete-parameter process defined on the points
nn (j In)
= E;j
•
by nn
and
let
set
B.
Let
be a
~,j = 0,1,2 ... , by
Consider the point process consisting of e;weedances of un
N (B)
n
denote the number of such exceedances in the Borel
(i. e. the number of exceec:ances of un
while we "lose exceedancesl!
by increasing un
way by this change of time scale.
by
for
j
E
nE .)
Thus
we gain then in a compensating
-19-
Nn
is a point process for each
n
- i.e. a randon clement in the space
N of non-negative integer-valued Borel measures on the real line R.
M
becomes a metric space with the i;vague topologyti (cf. [13]) and we rlay therefore consider convergence in distributions of these random elements.
(That is
a sequence
process
{I;n} of point processes converges in distribution to a point
+
I; , I; n V
if Ef(l;)
+ Ef(l;) for every bounded, vaguely continuous
11
real
on fJ • The following is a useful criterion (due to Kallenberg (13]
f
and modified according to a remark of Kurtz [14]) for such convergence.
THEOREM 5.1.
Let
let
(Kallenberg)
I;n' n
= 1,2 ...
be point processes on the positive real line and
be a point process without multiple points and such that
1;
a.s. for every fixed real
a
O.
~
r;({a})
=0
If
(i)
(ii)
~ E~(a,b]
lim sup EI; (a,b]
n
In our case
n
1;
n
= Nn
, and
I;
for all finite
a
<
b
then
is a Poisson process with parameter
T ,
so that e.g.
where
B
is any Borel set and
We shall suppose that
But
m denotes Lebesgue measure.
D (u ) , D 9 (u )
n
11
{M
n
~
u 1n'
both hold and thus by Theorem 4.4
is the same as
{Nn(CO,IJ)
= O}
so that
-20e -T
-+
i. e. Condition (i) holds when
= P{~((O,l]) = O}
B = (0,1] .
It may be shown from this that Condition (i) holds for intervals
and then for intervals
U(a i , b i ] .
(a s b]
and fina.lly for finite disjoint unions
(ine details of this development may be found in [17].)
Thus Condition (i) holds.
[xl
(O,a]
for the integer part of
~
Condition (ii) is easy to verify since
(~Titing
x)
neb - a)T/n
= E1;((a,bJ)
= T(b
- a)
.
Thus Theorem 5.1 applies and we obtain the following result.
THEOREM 5.2. Let
{t.}
J
be a stationary sequence such that
satisfying (3.2).
points
j/n
the Borel set
meter
for which
B.)
~j >
Then
Nn
un
V
-+
Let the point process
(NnCB)
s
where
D(u) , D'(u )
n
n
Nn consist of those
being the number of such points in
~
is a Poisson process with para-
T .
COROLLARY. Under the conditions of the theorem, if B is any Borel set whose
boundary has Lebesgue measure zero
P { N (B)
n
=r}
-+
(m(3B)
= 0)
then for any
e -Till (D) (Tm(B) ]r/ r!
r
= 0,1,2 ..•
,
-21(with similar convergence for the joint distribution of any
This follows at once since the random variables
distribution to
when
1; (B)
Nn V
-+
Nn(B)
Nn(B1) .•• Nn(B ) ).
k
converge in
(cf. [13]), and similarly for joint
l;;
distributions.
(ii)
Asymptotic distribution of
THEOREM 5.3.
hold for
{~.}
Let
{u}
be a stationary sequence such that
J
satisfying (2.3).
n
~l '"
n
n
~n ).
PROOF: This result follows simply from Theorem
n
n
Then
th e r th 1argest among
·
b elng
p{M(r) ::;; u } = P{N ((0,1]) < r}
D(u) , DI(u )
5.2 by noting that
(cf. Theorem 2.7).
n
Finally we may generalize Theorem 4.6 to obtain (2.10) for our stationary
sequences.
will again denote the maximum of
M
n
with the same distributions as
THEOREM 5.4.
Let
P{a (M - b ) : ;
n n
n
u
n
= x/a n
+
b
n
{~.}
G(x)
{u }
n
1
Suppose that
(non-degenerate) and that
D(U ) , D' (u )
n
n
hold for
Then (2.10) holds, i.e.
r-l
Ph (M(r) - b )
n n
n
PROOF:
i. Ld. random variables
~ ..
be a stationary sequence.
1
x} -+
n
= {x/an
+
b }
n
follows by Theorem 5.3 with
x}
-+
G(x)
I (-
log G(x))s/S! .
5=0
satisfies (2.3) by TI1eorem 2.5.
T
=-
log Sex)
(where '0
<
The result no\'!
G(x) < 1 ) and by
-22continuity arguments where
G
=0
1.
or
6. CONTINUOUS PARAMETER STATIONARY PROCESSES
In this section we consider a continuous parameter stationary process
{~(t): t 2:: O}
M(T) =
(assumed to have continuous saJ:Jple functions), and "'Tite
0
sup{~(t):
(i)
~
T} .
~
t
Strongly mixing processes.
The definition of strong mixing given is easily adapted to the continuous
parameter case requiring, for any
P(A)P(B)I
Ip(A n B) -
for
A € a{~
S
~
: s
t} ,
E € a{~
s
t,
: s
~
t
+
< geT)
T}
where
geT)
T
as
+ 0
+
00
•
Gnedenko's Theorem nmy be extended as follows.
THEOREM 6.1.
{~(t)}
Suppose the stationary process
Ph·T(M(T) - bT) ~ x}
+
of constants.
Then
G(x)
PROOF: tTite
Z.
1
G(x) , non-degenerate, for sor~e families
M(n)
= sup{~(t):
= max{Z.:
1
a..,.. > () , b
l.
T
has one of the three extreme value forms.
i-I
~
t
~
i} .
7hen
seen to be a stationary and strongly mixing sequence.
an integer,
is strongly mixing and
1
~
i
~
{Z.:
i
1
Further, putting
~ay
be
T
=n
nl , and
P{a (M(n) - b ) ~ xl
n
n
+
G(x)
so that by Gnedenko's Theorem ror strongly mixing sequences
the three extreme value forms.
= 1,2 .•. l
G(x)
has one of
-23-
Other partial results may be shown under strong mixing or
conditions.
DI
type
For example we may see under appropriate conditions, that
P{M(n)
(Zl
~,
= sup{~(t):
~
u } -+ e -T
n
0 ~ t ~ l})
if
P{Zl
n
Tin
and thence
P{M(T) ~ U'T'} -+ e
(6.1)
> U } '"
-T
t
if
(6.2)
Tnis may be made rigorous under sufficiently strong conditions.
Eowever
natural, simple conditions need to be found to make the result satisfying and
useful in the general context.
For the normal case there are such conditions,
as will be seen below.
(ii)
Stationary normal processes.
Extreme values of stationary normal processes have been considered in
detail by a number of authors, including S. M. 3errsn, J. Pickands, C. Qualls
and E. l':!atanabe.
Let
{l;(t)}
then be such a process with (for convenience)
zero mean, unit variance and covariance function
is usually assumed that
reT)
reT)
= E~(t)~(t
has the following expansion as
+
T)
It
T -+ 0 •
(6.3)
for some
ex with
0
<
ex
the case \qhere a function
~
2 , though the theory can be extended to apply to
G(T)
of slow "growth il is inCluded as a l!lul tiplica'::iv
-24factor together with
a
=2
1"'•1 a
(cf. r
~24
] ).
gives the "regular il case - where
~(t)
has a quadratic mean
derivative and where the I'lean number of "upcrossings"of any level per unit
time, is finite.
If
0 < a < 2 , the sample functions are continuous, but the
mean number of upcrossings of a level
there is no q.m. derivative.
u
TIle case
in any interval is infinite, and
a
=1
corresponds to the Ornstein-
Uhlenbeck Process.
The developMent indicated under (i) above May be applied to stationary
normal processes satisfying (6.3) and in so doing \Ale may replace strong
mixing by either of the conditions
(6.4)
ret) logt
(i. e. the continuous analogues of
LEr~
6.1.
process
if Uy
f:r~.(t)dt <
-+ 0"
(3.2)).
00
Indeed the follo\'Ting result holds.
Suppose (6.3) and (6.4) are satisfied, for the stationary normal
~(t)
considered above.
Then
is chosen so that (6.2) holds, i.e.
P{sup(~(t): 0 ~ t ~
1)
> Uy} ~ 'tIT.
This lemma may be obtained e.g. from the discussion in (22] or [3).
these references we may also obtain the asymptotic form of
P{sup(~(t):
a~
t ~
1)
>
u}
as follows.
From
-25-
LEMMA 6.2.
Under the conditions of Lenma 6.1, as u
P{sup(~(t): 0 ~ t ~ 1)
where II
is a certain constant and
a
density function.
if
~
> u} -
Thus the conclusion
~
-+
ClO
,
C1/~au2/a~(u)/u
is the standard normal prohability
P(M(T) ~ liT)
e-
-+
T
of Lemma 6.1 holds
is chosen to satisfy
(6.5) .
By taking logarithiEls in (6.5) we E1ay obtain, after some manipulation
(6.6)
~ =
(2 log T)1/2 - (2 log T)-l12[!Og
T
[i - ~)
+
log log
T
and Lemma 6.2 at once gives the following theorem.
THEOREM 6.3.
(6.3).
Let the stationary normal process
[~(t)]
satisfy (6.2) and
Then
where
~ =
b
T
(2
log T)1/2
= (2 log T) 1/2
- (2 log T) -1/2 [(} +
~)J
log log T
log [(2TI)1/2 c1/a 2 (a-2)/2a H-l]
II
-26-
PROOF:
Put
= e -x
't
in (6.6) to give
H
The value of the constant
a
u
T
= x/aT
+
tr
and apply
is known only for
Le~~a
= 1,2
a
Its form is given in [22], and rlepencs on
~(t)
otherwise on the process
6.2.
a, but not
.
The d.erivation of Lemmas 6.1 and 6.2 naturally involves considerable
approac~es
calculation, and different
May be usee (cf. [22], [3]).
use is made of the 1Ie:-upcrossings" of a level.
uparossing of
for
t
0
$ t
<
u at
t
0
+ <5
to
if
, for SOlile
are no other upcrossings in
t
(to - e:
to) .
0
$
t
$
However the number of
a < 2.
-
0
As
1
is said to have an
~(t)
0 < t
, and this is
0 > 0
(6.3) the number of upcrossings in
is not so for
for
~(t) $ U
In [22]
$
an
note~
t
f;(t)
and.
0
u
:?:
e:-upcY'ossing if t!1ere
before if
a
=2
in
has a finite nean, whereas
e:-t~1crossings
t~is
(for any fixed
e: > 0 ) in any finite interval is a bounded random variable, which turns out
to be useful in the case
a < 2 .
The (e:-)upcrossings in the continuous case naturally replace the
in the discrete case.
It is also possible to show that tneir
exceedance~
li~!iting
bution (for suitably chosen hifh levels) is Poisson (cf. [4], [2],
distri-
[23]).
In
fact it has recently been shown by Lindgren et al. ([18]) that these (e:-)
upcrossings converge weakly to a Poisson process.
The pl'oof is similar to
that described here in the sequence case (Sec. 5) for exceedances.
Further
...l
'.r.
questions concerning the asymptotic l..dstributlol1
o.c r th largest maX1IDa
an d
closely related
~~tters
are also discussed in [IS].
Finally we note that it has also been
recently shown by Mittal and
Ylvisaker ([27]), that if the conditions (6.4) do not hold, a variety of other
limit laws are possible for the m'.zimum.
-27-
REFERCNCES
(Including some related
~OT~S
not referred to in text)
[1]
Berman, S.M. Limit theorems for the maximum term in stationary sequences.
Ann. Math. Statist' 3 35 (1964) pp. 502-516.
[2]
Berman, S.M. Asymptotic independence of the numbers of high and low
level crossings of stationary Gaussian processes. Ann. Math.
Statist' 3 42 (1971) pp. 927-945.
[3]
Berman, S.M. Maxioa and hig)l level excursions of stationary Gaussian
processes. Trans. Amer. Math. Soc'$ 160 (1971) pp. 67-85.
[4]
Cramer, H. & Leadbetter, M.R. Stationary and Related Stochastic
Processes, John ~i1ey & Sons, New York, (1967).
[5]
De Haan, L. On regular variation and its application to the weak convergence of sample extremes. Math. Centre Tpact 3 ~ (1970) Amsterdam.
[6]
Deo, C.M. A note on strong mixing Gaussian sequences.
(1973) pp. 186-187.
[7]
Feller, W. An Introduction to Probability Theory and its Applications 2.
John Wiley &Sons, New York, (1966).
[8]
Galamhos, J. On the distribution of the maximum of random variables.
Ann, Mlth. Statist' 3 43 (1972) pp. 516-521.
[9]
Galambos, J. A general Poisson limit theorem of probability theory.
Duke Math. J' 3 40 (1973) pp. 581-586.
Ann. Pr>ob' 3 1
[10]
Gnedenko, B.V. Sur 1a distribution limite du terme maximum d'une serio
aleatoire, Ann. Math' 3 44 (1943) pp. 423-453.
[11]
Gnedenko, B.V. & Kolmogorov, A.N. Limit Distributions for Sums of Independent Random Variables. Addison t~sley, New York, (1954).
[12]
Gumbel, E.J.
(1958).
[13]
Kallenberg, O. Characterization and convergence of random measures a.nd
point processes. Z. Wahr. verw. Geb' 3 ~ (1973) pp. 9-21.
[14]
Kurtz, T. Point processes and completely monotone set functions.
Rept. Dept. Hath, Univ. of Wisconsin (1973).
Statistics of Extremes.
Columbia Univ. Press, New York,
Tech.
-23-
REFERENCES (cant.)
[15]
Leadbetter, M.R.
verw. Geb.,
an
~
extreme values in stationary sequences.
(1974) pp. 289-303.
Z. Wahr.
[16]
Leadbetter, M.R. Lectures on extreme value theory.
Math. Stat., Lund Univ. Sweden, (1974).
[17]
Leadbetter, M.R. Weak convergence of high level exceedances by a
stationary sequence. Inst. of stat. ftnmeo Series No. 933 s Univ. of
N.C., (1974).
[18]
Lindgren, G., de Mare, J., &Rootzen, H. Weak convergence of high level
crossings and maxi.ma for one or more Gaussian processes. Tech. Rept.
4 Dept. Ma.th. Stat., Lund Univ., Sweden, (1974).
[19]
Loynes, R.M. Extr~~e values in uniformly mixinr. stationary stochastic
processes. Ann. Math. Statist., 36 (1965) pp. 993-999.
[20]
O'Brien, G.L.
process.
[21]
O'Brien, G.L. The maximum term of uniformly mixing.stationary processes.
To appear in Z. Wahr. verw. Geb.
[22J
Pickands, J. Asymptotic properties of the maximum in a stationary
Gaussian process. Trans. Amer. ,~th. Soo.~ 145 (1969) pp. 75-86.
[23]
Pickands, J. Upcrossing probabilities for stationary Gaussian processes.
Trans. Amer. Math. Soo's 145 (1969) pp. 51-73.
[24]
Qualls, C., &Watanabe, H. Asymptotic properties of Gaussian processes.
Ann. Math. statist' J 43 (1972) pp. 580-596.
[25]
Watson, G.S. Extreme values in samples from m-dependent stationary
stochastic processes. Ann. ~1ath. Statist. s ~ (1954) pp. 798-300.
[26)
Welsch, R.E. A weak convergence theorem for order statistics from
strong-mixing processes. Ann. Math. Statist.~ 42 (1971) pp. 17371946.
[27]
Mittal, Y., &Ylvisaker, D. Limit distributions for the maxima of
stationary Gaussian processes. To appear in J.·Stooh. Proo.
Tech. P-ept. Dept.
Limit theorens for the maximum term of a stationary
Ann. Frob., (1974) pp. 540-543.