Leadbetter, M.R.On Extremes of Stationary Processes."

..,
•
J;
On Extremes of Stationary Processes
by
M.R. Leadbetter'
•
Abstract.
Certain aspects of extremal theory for (a) stationary sequences and
(b) continuous parameter stationary processes, are discussed in this paper.
A
slightly modified form of a previously used dependence condition, leads to
simple proofs of some key results in extremal theory of stationary sequences.
Dependence conditions of a "weak mixing" type are introduced for continuous
parameter stationary processes and results of classical extreme value theory
.extended to that context .
•
•Research
supported .by Contract N00014-75-C-0809 with the Office of Naval Research.
2
'"
the only work of Watson [16] under m-dependence, and Loynes [14] under strong
mixing assumptions), and the detailed theory (Berman ll]) for stationary normaZ
sequences.
•
The general theory for stationary sequences has been further studied by a
number,of authors under various dependence restrictions.
In particular use has
been made of weak "distributional mixing co'ndi tions" suggested in [11].
Under these conditions it may be shown that not only does Gnedenko's Theorem
still hold, but that in many cases the limiting law is the same as would
occur if the terms of the sequence were independent with the same marginal
,distributions as the original sequence.
Further, recent work of R. Davis ([4])
has made possible a yet more complete and satisfying statement of these results.
One of the purposes of the present paper is to show how a slight modification
of the previously used dependence restrictions, leads to very simple proofs for
these existing results.
This is described in Section 2.
Our main interest here, however, is to study the behavior of extremes of
continuous parameter stationary processes, which we shall do by relating to the
sequence case.
•
A detailed theory is already in existence for stationary nopmaZ
processes (involving many studies, begun by the work of
Cram~r
[2],l3]), where
it is known that the so-called "double exponential" extremal limit applies.
Our object here is to determine circumstances under which extremal theory may
be extended to general continuous parameter stationary processes and in particular, to obtain a form of Gnedenko's Theorem in that context.
These tasks are undertaken in Sections 3-5 where weak mixing conditionsanalogous to those for sequences - are defined and used.
asymptotic form of the mean number
~(x)
As will be seen, the
of upcrossings of a high level x per
unit time plays a central role in the discussion and, in particular, in
determiriing domains of attraction (replacing the tail l-F(x) of the marginal
distribution in the sequence case).
We regard this as one of the most
•
3
•
interesting consequences of the tlleory. and one of potential practical usefulness.
At the same time this automatically restricts attention to cases for
which such means exist (excluding. for example. the more irregularly behaved
stationary normal processes with infinite second spectral moments).
It is
likely that the theory can be extended at the expense of complication (for
example by consideration of so-called "£-upcrossings") but we do not do so here.
Finally we note that we have chosen particular dependence restrictions from
a variety of possible similar conditions.
It is certainly possible that differ-
ent forms of these may prove preferable as the theory develops further.
1.
Two Results from Classical Extreme Value Theory.
lbe field of classical extreme value theory deals substantially with
•
asymptotic distributional questions surrounding Mn
where
~1'~2'.""
= max(~1'~2' ...• ~)
are LLd. random variables (with common d.f. F).
one is first interested in sequences {un} for which P{M
limit as n
+
00.
as n
~
That is,
< U }converges to a
n -
n
In this i.i.d. situation it is almost trivial to show that if
(1.1)
then
P{M
<
n -
(1.2)
u }
n
+
e- T
and conversely.
The main body of the LLd. distributional theory is directed towards the
further study of when such convergence will occur for every x when
un = un (x) = x/an + bn • a n l> 0) and bn being constants.
under what circumstances it is true that
•
(1.3)
P{an (Mn - bn
)< x}
~
Rephrased. we ask
GlX)
at each real x (or at least for continuity points of G).
Clearly such a G is
00.
4
non-decreasing with values between zero and one.
•
It is, in fact, quite readily
shown (cf. [5] or [12]) that if G is a non-degenerate d.f., then it must be
max stable in the sense that for each n=1,2, ... there exist
constants a >O,B
n
n
such that Gn(x) = G(a X + B ) for all x. Further it is not too difficult to
n
n
show (cf. [5], [12] that the "max stable" distributions consist precisely of
the following three general "tyPes" (the so called "extreme value types").
Type I:
G(x) = exp(-e -x ) ,
Type II:
G(x) = exp(-x~ la> 0), x > 0 lzero for x < 0)
Type III:
G(x) = eXPl_(_x)a) (a> 0), x ~ 0
-co
< x <
lIn these x may be replaced by ax
co
+
(1 for x > 0)
b for any a > 0, b)
From these results we see that the only non-degenerate d.f.'s which may
occur as limiting distributions of maxima of i.i.d. sequences as in (1.3), are
the three extreme value types.
As noted above, we refer to this as Gnedenko's
Theorem.
Further, classical extreme value theory provides necessary and sufficient
conditions on F to ensure that it gives rise to anyone G of the three possible
types (i. e. that F "belongs to the domain of attraction of G").
These condi-
tions mainly involve the behavior of the tail I-FlU) of F as u increases.
example if I-FlU) is regularly varying with exponent -a
as u
4
•
00,
For
then F is
attracted to the type II limit law (cf. [5]).
2.
Dependent Sequences.
Much of the classical theory may be extended to apply to (stationary)
dependent sequences if the dependence is not too strong.
One suitable depen-
dence restriction is that of strong mixing under which Gnedenko's Theorem
certainly still holds ([14]).
However, strong mixing imposes a uniformity
which is not needed in this context, and weaker forms of mixing will, in fact,
•
5
•
suffice.
One convenient such condition is the following which we shall call
the "distributional mixing condition D(u J". Specifically, write F..
.
n
1 1 ... 1
n
1 2
to denote the joint d.f. of!;.1 ... E;.,
F.
.
lU)
= F.
.
lU,u,
•..uJ.
and
1
1 •.. 1
· 1 ... 1
1
1
n
1
n
n
let {u } be a real sequence. Then Dlu J is said to hold if for any choice of
n
n
n, p, p', p
+
p'
->-
0 for some sequence In = 0 (n) .
~
n, i l < i 2 ....
<
ip
<
jl ...
<
jp, jl-i
p
= I.
l2.1)
where
Cl
n, I
n
The following lemma is fundamental in studying the extremes of dep·endent
sequences.
This lemma is a modified version of Lemma 2.5 of [11] to which we
refer for proof (cf. also [12] Lemma 2.3).
Lemma 2.1.
•
Let the stationary sequence lE;n} satisfy U(un), for a given
sequence {un } of constants. Then
for each integer k ([n/k] denoting the integer part of n/k and
The proof of this result - contained in the references cited above involves the standard technique used under mixing conditions of considering
k slightly separated subsets Il •.. I
of (l ... n) and showing that the maxima
k
on these subsets are approximately independent.
o
By means of this result; it is almost trivial to extend Gnedenko's Theorem
to include a wide variety of stationary sequences.
While this is proved in [11]
we sketch a simplified proof, here based on Lemma 2.1 .
•
6
•
n }·be a stationary sequenae, Mn = max(~l""'~n ) and
suppose that for some aonstants {a > a}, {b } we have
Theorem 2.2.
Let
{~
n
n
P{a (M - b ) < x}
n.n n
~
G(x)
where G is a non-degenerate d.f.
If D(un ) holds with un
x, then G is one of the three extreme value forms.
Sketch of proof:
Writing u = x/a
n
n
+
= x/an
b we have pfM < u }
n
n-n
Lemma 2.1, P{M[n/k]'::' Un} ... Gl/k(x) for each fixed k.
~
+
bn , for eaah
G(x) and hence by
This is true with nk
replacing n so that P{M < u k} ~ Gl/k(x) or, rephrasing,
n - n
P{an k(Mn - bn k) -< x} ... Gl/k (x).
l k
Thus for a fixed k, M has the non-degenerate limit G / with normalizing
n
constants ank , b
as well as the limit G (with normalizing constants an,b ).
nk
n
But a well known result of Khintchine (cf [6] and [12]) shows that if two such
•
non-degenerate limits exist, they must be the same apart from a linear transl/k
formation of the argument, Le. G(x) = G (~x + f\) for some ~ > 0, ~.
This shows that G is max stable and hence an extreme value d.f. by the remarks
regarding the classical case in Section 1.
IJ
The other result of the classical theory quoted - the equivalence of (1.1)
and (1.2), also extends to this dependent context, but requires a further
restriction.
One such convenient
condition - which we refer to as D'(u ) n
is the following
(2.2)
(D'(u))
lin which [
n
lim sup
n-+<o
> u , ~.>u}~o
n
J
n
] denotes the integer part).
as
The first part of the following
result has been known for some years ([11]).
A version of the converse part
was obtained recently by R. Davis L4] under a slightly different D'-condition.
Here we give avery simple proof using the present form of (2.2).
•
7
•
Theorem 2.3.
{~n}>
Suppose that U(un ) , D'(un ) hold for the stationary sequence
where {un} is a given sequence of constants.
(i.e. n
Pl~l >
un}
~
If ll.l) holds
then (1.2) holds li.e. P{Mn -< un }
T)
e
~
-T
) and
conversely.
Proof:
Fix k and write, for each n, n '
= Ln/k].
Using the fact that
n'
{M I > u } = U {~. > u }, standard inequalities for the probability of a union,
n
n
j=l J
n
and stationarity, it is simply shown that
1 - n'(l - F(u » < P{M I u , ~. > u } and lim sup Sn = olk- l ) as k ~
n
j=2
n
J
n
n_
(2.3)
2
If (1.1) holds, 1 - Flu )
n
•
•
1 - T/k _< lim inf P{M
n-tOO
I
~
T/(n'k), so that.n
< U } <
n -
n -
~
00
By taking kth powers throughout, using Lemma 2.1, and letting k
where P{Mn '
~
un}
~
e
-T/k
~
00
we see that
Then (2.3) gives
n-+OO
<
+
Sn
.
by (1.2) and Lemma 2.1 so that we obtain
1 - ·e- T/ k < k- l lim inf nll - F(u
•
o(k- l )
+
l1. 2) holds.
1 - P{M I < u } < n'll - F(u » < 1 - P{M , < u }
n-nn n-n
•
by D'lUn).
in (2.3) gives
lim sup P{M I < U } < 1 - T/k
n - n
n-tOO
Conversely suppose that (1.2) holds.
•
00
1 - e- T/ k
+
n
»
< k- l lim sup n(l - Flu
n~
n
»
o(l/k)
from which by multiplying by k and letting k
~
00,
ll.l) follows.
o
8
•
The importance of this result may be seen from the following corollary,·
which shows in particular that under D, D' conditions the same limit (if any)
occurs in the dependent case as would occur if the sequence were i.i.d.
That
is the maximum for the stationary sequence then has the same asymptotic distribution as it would if the individual terms were independent with the same
Hence the tail of this d.f. may still be used in criteria to
marginal d.f.
In this corollary M will be used to denote
determine domains of attraction.
n
the maximum of n independent random variables, each having the same marginal
d.f. F as the
Corollary:
~ ..
1
Suppose that D(un ), D'lu n ) hold for the stationary
 0 if and only if P{Mn -< un J
+
p.
In particular suppose that for some constants an > 0, b , D(U ), D'(Un )
n
n
hold with un = x/an + bn for each x, and G is a non-degenerate d.f. Then
P{a (M
n
n
-
b ) < x}
n
+
G(x)
+
Glx)
•
if and only if
P{an lMn - bn J
<
x}
and G is, of course, necessarily then one of the extreme value types.
PROOF.
The first part follows at once from the equivalence of l1.l) and (1. 2)
writing (p
= e -T )
in both independent and dependent cases.
follows by identifying p and G(x) where G(x)
G
=0
>
The second part
0 (and using continuity where
" each extreme value d;f. being continuous).
IJ
Finally in this section we note that the conditions D(u ), D'(u ) are
n
n
satisfied when
{~}
is a stationary normal sequence under appropriate restric-
tions on the covariance function.
The simplest of these is that the covariance
•
9
•
·
function {r } should satisfy r
n
n
log n
->
0 as n
-> "',
but even weaker conditions
yet are known cf. (13]).
3.
Continuous Parameter Processes - General Framework.
.
We turn attention now to a stationary process
"time".
{~lt):
t > O} in continuous
It will be assumed, without further comment, that i;(t) has (a.s.)
continuous sample functions, and continuous one-dimensional distributions.
If
E is any set of real numbers we write MlE) = sup{i;(t): t ( E), and N (E) for the
u
number of upcrossings of a level u by
~(t)
in the set E.
It will be convenient
to write M(t), Nu(t) when E = (O,tj.
Our interest lies in asymptotic distributional properties of M(T) when T
becomes large.
•
Since for an integer n
l3.1)
where
~i = max{~lt):
l3.2)
i - I < t < il
i t is reasonable to expect that the properties of M(T) may be approached via
the theory for sequences.
Of course the members of the sequence are not now the
original r. v. 's of the process, but their maxima over fixed intervaZs.
our basic distribution
distribution of
~l
i.e.
Pl~l ~
lienee
x) of the sequence case must be replaced by the
P{~l ~
xl =
Ptmax(~(t):
U~ t
~
1)
<
x}.
Since it is
the tail of this distribution which plays a crucial role (e.g. in (1.1)), one
of our first tasks must be to discuss the asymptotic behavior of P{M(I)
•
•
•
i.e. the tail probability for the maximum over a fixed
interval.
n
be taken up in Section S.
n
~_
1
uJ,
The other
task - to obtain convenient distributional mixing conditions on the
ensure that appropriate D(u ), D'(u ) conditions hold for the
>
~(t)
to
sequence - will
10
•
Before embarking on these tasks it is convenient to give the following
~.
form of Gnedenko's Theorem assuming that the "partial maxima"
priate Dlu ) .- conditions.
n
1
satisfy appro-
A more complete statement will be given in Section
5 where, as noted, conditions on the
original
process·~(t)
will be obtained.
;
Suppose that for some families of constants fay > UJ, {bTl we have
Theorem 3.1.
(3.3)
P{ay(M(T) - b ) .:: xl + G(x), as T +
T
(for a non-degenerate d.f. G), and that the
satisfies ,D(u n ) whenever un
= x/an
+
{~il
00
sequence defined by (3.2)
bn (for each fixed real x),
Then G is one
of the. three extreme value types.
Proof.
the
~
n
~l3.3)
must hold, in particular, as T +
~n-sequence
00
through integral values.
is clearly stationary, the result follows by replacing
Gn
Since
by
in Theorem 2.2, and using (3.1).
Corollary:
II
The result holds in particular if the DlUn ) conditions are replaced
by the assumption that
l~(t)l
is strongly mixing.
For then the sequence
{~il
IJ
is strongly mixing and thus satisfies the D(un)-condition.
4.
•
Maxima over Fixed Intervals.
As noted above, the trail of the distribution of the maximum of
~(t)
in a
fixed interval (e.g. [0,1]), plays a crucial role with respect to the asymptotic
distributional properties of M(T) as T -,
behavior of P{M(h)
>
u} as u+
00
00
Accordingly we consider here the
for a fixed h.
We shall relate this simply to
the mean number of upcrossings of the level u, using the notation N lE), N ltl
u
as defined above.
u
Here we assume the finiteness of first and second moments of
such quantities as N (h).
u
These would usually be checked from known formulae
(cf. llOj) giving these moments in terms of distributions of
~
and its
•
•
11
•
derivative (cf. Eqn. 4.4 below).
In cases where these moments are not finite,
it appears possible to couch the discussion in terms of the so-called
"E-upcrossings" ([15)), ·but we do not attempt to do so here, for the reasons
expressed in the Introduction.
For t > D, write
(4.1)
It(u)
= P{~(D)
-
D may ·be found
then lim I (u) exists, finite, and
00,
q+O
q
EN (D,l)
l4.2)
u
=
lim I (u) .
q+O
q
We remark in passing that It(u) may be simply written in terms of the
distribution of
•
~(D),
~(t)
and that if this is absolutely continuous (4.2) may
be transformed to give
(4.3)
where PtlX,z) is the joint density for
~(D)
and
(~(t)
-
~lD)/t.
This leads
easily - under appropriate. conditions - to the well known expression
~ = ~ z Plu,z)dz
(4.4)
where p is the joint density for
~lD)
example this may be evaluated when
~
and its derivative
~'(O)
(cf. LIDJ).
For
is a standard lzero mean, unit variance)
stationary normal process with covariance function r(T), to give
~
2
=
[l-r"(D))~/2 ne- u /2
In our situation here, u is not usually fixed, and we shall want to let u
•
•
converge to the right hand endpoint lfinite or infinite) of the distribution F
of
~(t).
We shall use "u
->- 00"
to denote this, with the understanding
12
that this is to be replaced by "u
-+
b" if.F has a finite right hand endpoint b .
We shall sometimes (though not in this section) need the further assumption
that q
=
q(u) may be chosen depending on u so that
l4.5)
I
q
(u)
as u
~ ~
-+
•
00
(u) ~ I ( ) (u) as u -+ 00). This assumption may be verified from
q'+o qq u
(4.3), for example in the case of stationary normal processes, under appropriate
(i. e. lim I
conditions.
We also at times require the following condition
(4.6)
EN U (h)[N U (h) - 1]
for a given fixed h.
d o(~)
as u
-+
00,
This is an intuitively appealing "regularity" condition
which asserts that E N2 (h) is appropriately close to E N (h) which will be the
u
u
case if N (h) is zero or one with high probability, i.e. if P{N (h) > I} is
u
u
appropriately small for large u.
(4.6) would be verified in practice by using
(4.4) and the corresponding result giving EN (h)(N (h) - 1) in terms of approu
•
u
priate.joint densities ([10]).
The following result gives the desired tail distribution for the maxima of
~lt)
over fixed intervals.
Theorem 4.1.
process
Let h be fixed.
{~(t)},
and that
Suppose that (4.6) holds for the stationary
p{~lO) >
u}
= ol~)
as u
-+
00.
~ ~
as u
-+
00.
(~
= ~(u) = ENu lO,l)).
Then
P{M(h) > u}
(4.7)
Proof.
l4.8)
h
Clearly
P{NU(h) > I}
~
P{M(h) > u}
~ ~ h + o(~)
~
p{F,(O) > u}
+
P{Nulh) > I}
•
13
•
Now i f
by assumption, and since P{N lh) > l} < E N lh) = \1 h.
u
u
Pj = P{N u (h) = j}, we have
ex>
ex>
E N (h) [N (h) - 1] =
u
u
r
j(j-l)p. >
J
j=2
L
jPj = \1 h - P{N u (h) = l}
j=2
so that
P{N (h) > I} > r{N lh) = lJ'::ll h - E N lh) [N lh) - 1]
u
u
u
u
=
by (4.6).
\1 h - 0(11)
Hence from (4.8),
11 h - 0(\1) ~ P{Mlh) > u} < 11 h + 0(\1)
o
from which the result follows.
P{~l >
From this result with h = 1 we see at once that
•
defined by (3.2)).
~.
1
Hence if assumptions on
~(t)
u}
~
\1 (with
~l
can be found under which the
satisfy appropriate D(u ), U'(u ) conditions, then the sequence theory
n
applies to M(n)
n
given by l3.l) provided the function \1lu) is used in lieu of
the tail of the distPibution le.g. in (1.1), or in classical criteria for the
domains of attraction).
As with Gnedenko's Theorem (Theorem 3.1) it will be
convenient to state these results explicitly here under 0 (and 0') assumptions
for the
~.
1
's, completing these results in Section 5 where the dependence condi{~(t)).
tions will be recast in terms of the original process
Theorem 4.2.
Suppose that the conditions of
stationary process
{~lt)}with
•
Suppose that the sequence
l3.2)) satisfies D(u ), D'(u).
n
l4.9)
4.1 are satisfied by the
h = 1, let {un} be a sequence of constants, and
wPite Uy = u[T] foT' eaah T > O.
•
1~eorem
n
P{M(T)
~
Then, foT' a given
T >
-T
Uy} ... e
00
as T
-)0-
0,
{~nJ
(defined by
14
if and onLy if
•
l4.10)
or equivalentLy if and only if
(4.10)'
j.J
= j.J (u
n
n
)
'V
Tin
as n
->
00.
It is clear that (4.10) and (4.10)' are equivalent from the definition
Proof.
of Uy as u[T]'
If (4.9) holds, it holds as T
then follows from Theorem 2.3
that
nP{~l
+
> un)
00
through integral values.
It
and hence from Theorem 4.1
-> T
that (4.10)' holds, as required.
Conversely if (4.10)' holds it follows similarly from Theorems
that P{Mln) u } = P{~l > u }
n
n
'V
j.J
n
n
< u } - P{M(n) < u < M(T)).
n
n
But the second term is dominated by
by Theorem 4.1, and
j.J
n
+
0 by (4.10)', so that
(4.9) follows.
Corollary 1.
the
•
D
Suppose that the oonditions of Theorems 3.1 and 4.1 hold and that
{~n}
sequenoe also satisfies V'(u n ), for all un of the form x/an + b n .
Then G(x) is preoiseLy the extreme vaLue d.f. whioh would be obtained for the
maxima n = max(~1 ... 2n) of an i.i.d. sequenoe 2l,2~ ...
tion satisfies
M
Whose taiL distribu-
(4.11)
Thus the oLassioal oriteria for domain of attraotion may be appLied with
lllX) repLaoing the taU distribtuion of the ;;equenoe terms.
~
•
IS
•
Proof:
By the theorem we have that V
Glx) > O.
.
Since p{<;;. > x}
1
~
n
~
Tin where T = -log G(x), whenever
Vex) by Theorem 4.1, there exist i.i.d. random
variables 2. (having the same marginal distributions as <;;i J satisfying (4.11).
1
.
Hence P{~
> u } ~ vlu ) = V ~ Tin so that P{M < u }
71
n
n
n
n n
+
e
-T
Rephrased, this
states
P{a lM - b ) < x}
n n
n
+
GlX),
0
as required.
Conversely the following result is similarly proved.
Suppose that the conditions of Theorem 4.1 hoZd and that there are
Corollary 2.
i.i.d. random variabZes
e1 22 ...
such that P{an (Mn
-
b ) < x}
n-
->
for some
G(X)
A
•
non-degenerate GJ constants a
p{2 l > x} ~ vex) as x
n
> D,b , M
n
n
= max(e l ,22 ···en ), and such that
If D(u n ), D'lUn ) are satisfied by the sequence {<;;n}
for aU un of the form xlan +b n then P{a,.lM(T) - bTl .::. x} + G(x) (where
+
00.
o
'J
It may seem curious that it is the tail I-FLu) of the distribution of one
I
of the random variables which is central in the sequence case, whereas the mean
number of upcrossings of a level per unit time plays the same role in the
continuous context.
However, it is worth noting that for an i.i.d. sequence,
the probability of an upcrossing of a level u in unit time is
F(u) [1 - F(u)]
~
1 - Flu) for large u.
P{~.
1-
1
<
u
< ~.} =
1
Since only zero or one such upcrossing
may occur in unit time, this is also the mean number of such upcrossings per
,
unit time.
Thus the tail 1 - Flu) of the distribution is, in the discrete case,
also the asymptotic form of the mean number of upcrossings ofu per unit time,
•
•
and so naturally corresponds to V(u) in the continuous case.
16
5.
Dependence Restrictions for set).
In the sequence case we used the conditions D(u ), D'(u ) to restrict depenn
dence.
We now define continuous analogs (0 lu J, D'(u )) of these conditions
c
which will apply to the process
the
n
~n-sequence· in
~(t),
n
c
n
•
and may replace the D, D' requirements for
•
the previous theorems.
The condition D (u ) is the following analog of D(u ), where again {u J is
c n
n
n
any real sequence.
In this I (u) is defined by (4.1) and if F
(Xl ... x)
t 1" .t
n
'1
n
denotes a finite dimensional d.f. of ~lt), F "' lU) is written for
tl
tn
F
t (u ... u).
t l' .. n
Specifically.
.{~}, ~ -> 0
as n
{~(t)J
c
such that I
-> "',
integers n, i l '"
jl ... <j , < n/q , jl - i >
n
p P -
~/q
qn
n
=
n
n
n
n
and, for any
we have (writing
~ = '1)
(5.1)
IF.
1
where an
..
.
(u) - F.
1
,'1 n
1 '1"
1
1 '1 •.. p q,Jlq···J p
.
~
, n
-> 0
for some sequence
~
n
.
In discussing
D(un ) for the
.
~
n
.
.1
p
q
(u) F.
.
(u)
n
Jlq ... Jp,q n
I
<
a
= o(n).
-sequence, it will be convenient to first
obtain the following lemma, which enables us to approximate the maximum of
a fixed interval by the maximum of discrete values
in that interval.
•
0
n,"-
~(jq)
~
in
for points jq (j=O,I,2 ... )
In this lemma, and subsequently, N(q) (E) will denote the number
u
of upcrossings of the level u by ·the sequence F,(jqi in a set E, i.e. the number
of
in~egers
j such that jq
E
E for which
~(jq)
 "',
if I is an intervaL of
fixed Length h,
(i)
E N(q) (I) = ~ h
u
+
Ol~)
•
17
•
(ii)
p{~(U)
If also
= o(~),
> u}
(~(jq)
U
o < p{
then we have
0(1l}
< u)} - p{r4(I) < u} =
jqd
as u
+
00
Proof.
as u
+
(uniformly in all such fixed length intervals).
The number, m=m , say, of points jq ( I, clearly satisfies m 'V h/q
li)
00
u
so that, by stationarity,
= m p{~(U)
E N(q)(I)
u

+
P{N (I) > 0, N(q) (I) = U}
u
if a denotes the left hand endpoint of I.
.p{~(O)
u
But the first term is equal to
> u} = 0(1l) and the second is dominated by
P{N lI) - N(q) lI) >
u
u
O}
< EiN (I) - N(q) ll) }(N
-
u
u
. that (by li»), (5.2) does not
exceed
~ h -
u
[Il h + 0(1l)] + 0(1l)
- Nu(q) being non-negative) so
= o(~)
0
as required.
By using this lemma we may now obtain conditions under which D(u ) holds for
n
the
by
•
•
sequence{~}.
We write
n
~(tJ
~
n
= ~(u ) for the mean number of upcrossings of un
n
per unit time.
Lemma 5.2.
Let {un} be a sequence of constants and suppose that
the process
~(t) .
Then the sequence
Suppose also that niln is bounded, and
{~
n
p{~(U)
oc (u n ) holds for
> un} = Ol~).
} of maxima of I, over the intervals [n-l,nJ, aatisfies D(u n ).
18
Proof.
Fix n, i
> £. Let q=Q"Jl be chosen so
l < iZ···<i
.
p < jl···<j p , < n, jl-j pthat (5.1) holds. Then, writing Ir=[ir-l,irL .Js=[js-l,js]' we have, by Lenuna 5.1,
o
p
n
< p{
{i;(jq) <
U
r=l
n
,jq
<
E
-
U
n
,jq
•
J }}
E
s
p'
lZ;.
1
which tends to zero as n
+
and also to the groups of I
00
r
< u)
r -
()
n n s=l
since nv
and J
s
n
is bounded.
By applying this as stated
intervals separately, we see at once that the
difference, R , say, between
n
(5.3)
Ip.d, (Z;.
1
r=l
< u)
r
n n
-
A (Z;.
s=l
pp'
<u)} - p{ f\ (Z;i < u )} P{A (Z;. < u )}!
Js - n
Js - n
r=l
r - n
s=l
and
•
p'
P
(5.4)
li;(jq) -< un ,jq
Ip{ (\
r=l
E
I r ) n (\ ll,(jq) <
s=l
U ,
n
jq
E
J )}
s
p
p{ ()
r=l
tends to zero as n
+
and the largest in any I
Un,£_l where un £
, n
+
But since the smallest jq in any J
00.
r
s
is at least jl-l,
is at most i , (5.1) shows that (5.4) does not exceed
p
0 for some sequence £
n
U~,£ = un,£_l + Rn and u*n, £*
n
+
0 for £* = £
n
n
= o(n).
+
Hence (5.3) does not exceed
1, so that the sequence {Z; }
n
o
satisfies D(u ) as asserted.
n
The continuous version of Gendenko's Theorem (Theorem 3.1) may now-be
restated in terms of conditions on the process set).
Theorem 5.3.
Suppose that faY' some famiLies of constants {aT> O}, {bTl we have
P{~ (M(T) -
bTl < x} -, G(x)
•
19
•
fop a non-degenepate d.f. G and
tha~fop
real x), {l;lt)} satisfies 0
and p{F,LO) > u} ; olll), lim sup n II
\l ; Il(u), II
n
; Il(u).
n
c
(ll )
n
each sequence
u
n
{x/a n
;
n
(all
+ b}
n
< 00, I"her'e
n
Then G is one of the -three extpeme value types.
o
This follows at once fmm Theorem 3.1 and Lemma 5.2.
The condition 0 (u ) (and the other conditions of this Theorem) are not too
c
n
difficult to verify when F,(t) is a normal process.
The appropriate 0' condition
for the continuous case - which we shall call D'(u ) _ is simpler to state and
c
n
more interesting than 0c (un ), though evidently also more difficult to verify.
oe(un ),
Like
D'(u
e n ) is a natural analog of the corresponding sequence condition (D'(u
_n )).
Specifically, -we say that {F,(t)} satisfies 0' (u ), for a given sequence
c n
(5.5)
•
lim sup E N ([n/kJ) (N
U
n
n~
u
([n/k)) - 1) ; o(k
-1
) as k
->- ro.
n
We note that while this seems formally quite different from the stated D'(u )
n
condition for sequences, the difference
is somewhat illusory, since the sequence
condition may also be recast in terms of a second factorial moment (of numbers
of exceedances).
oc ,
Theorem 4.2 may now be restated in terms of the assumptions
0'c as follows.
Theorem 5.4.
Let {un} be a sequcnce of constanta,
~
; u[T)'
Suppose that the
stationary ppocess {F,(t)} satisfies Dc(u n ), D~(un)' that (4.6) holds and that
P{F,(O) > u} ; 0(11) as u ->- ro. Then (4.9) holds (P{M(T) ~ ~} ->- e- T ) if and only
if (4.10) holds
Proof:
•
•
(5.6)
(~
;
Il(~) ~
TIT).
For fixed k, writing n' ; [n/k) and arguing as in Theorem 4.1 we obtain
n'll
n
- EN
u
(n')(N
n
u
n
(n') - 1) < P{M(n') > u } < P(F,(O) > u }
nn
+ n'll
If (4.9) holds since the central term of (5.6) does not exceed P{M(n)
>
n
u }
n
20
which converges to l-e -T , it follows from the left hand inequality in (5.6) that
n'
~n
( and hence n
of (5.5) is finite.
~n)
is bounded when k is chosen so that the left hand side
If (4.10) holds it is trivially true that n
Hence under (4.9) or (4.10), by Lemma 5.2 the sequence
over the intervals [n-l,n], satisfy D(u).
P{M(n) < u } - pk{M(n') < u }
-
for each k.
2.3.
n
n
is bounded.
-
n
0
+
•
tSn} of maxima of set)
Hence also by Lemma 2.1 (with
n
(5.7)
~
as
n+
for
~.
1
oo
The rest of the proof now follows the same arguments as in Theorem
For example if (4.10) holds we may let n
+
00
in (5.6) and use (5.5) to
obtain
1 - T/k _< lim inf P{M(n') < u } < lim sup P{M (n') < u }
-
n-700
n
0-)<0
n
-
n
< 1 - T/k + o(l/k)
from which (4.9) follows by taking the kth power, using (5.7) and letting k
+
00.
•
[I
Similarly (4.9) implies (4.10).
REFERENCES
[1]
Berman, S.M., "Limit theorems for the maximum term in a stationary sequence."
Ann. Math. Statist.
[2]
~,
502-516, 1964.
Cramer, H., "A limit theorem for the maximum values of certain stochastic
processes", Terr. ver. i. prim . .!Q., 137-139, 1965.
[3]
Cramer, H. and Leadbetter, M.R., "Stationary and related stochastic processes",
John Wiley and Sons, New York, 1967.
[4]. Davis, R., "Maxima and Minima of Stationary sequences", To appear in Ann.
Prob.
[5J
De Haan L., "Sample extremes - an elementary introduction", Stat. Nederlandica
30, 1976, 161-172.
•
21
•
[6]
Feller, W., "An Introduction to probability theory and its applications"
~,
[7]
John Wiley and Sons, New York, 1966.
Fisher, R.A. and Tippett, L.H.C., "Limiting forms of the frequency distributions of the largest or smallest members of a sample", Proc. Camb .
•
Phil. Soc. 24, 1928, 180-190.
[8]
Frechet, M., "Sur la loi de probabilite de l'ecart maximum".
Polonaise de Math (Cracow)
~,
Ann de la Soc.
1927, p.93.
[9] . Gnedenko, B.V., "Sur la distribution limite du terme maximum d'une serie
aleatoire", Ann
[10]
Math~,
1943, p. 423-453.
Leadbetter, M.R., "A note on the moments of the number of axis-crossings by
a stochastic process", Bull. Amer. Math. Soc.
[11]
•
1967, p. 129-132.
, "On extreme values in stationary sequences", Z. Wahr. verw.
Geb.
[12]
~,
~,
1974, p. 289-303.
_________, Lindgren, G., Rootzen. H., "Extremal and related properties
of stationary stochastic processes" in preparation; to appear in Univ.
of Umea Mimeo Series.
'J
L13j
------_.,
"Conditions for the convergence in distribution of maxima
of stationary normal processes" to appear in J. Stoch. Proc.
[14]
Loynes, R.M., "Extreme values in uniformly mixing stationary stochastic
processes", Ann. Math. Statist. 36, 1965, p. 993-999.
[15]
Pickands, J., "Upcrossing probabilities for stationary Gaussian processes",
Trans. Amer. Math. Soc. 145, 1969, p. 51-73.
[16]
Watson, G.S., "Extreme values in samples from m-dependent stationary
stochastic processes", Ann. Math. Statist.
•
~,
1954, p. 798-800 .
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•
1.
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RECIPIENT'S CAT "LOG NUMBER
••
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On Extremes of Stationary Processes
••
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PERf"ORMING
ape ... Klz .... TlaN
Mimeo Series No. 1194
CONTRACT OR GRANT NUMBER(s)
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-'-"
10.
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Department of Statistics
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11.
..
PERIOD COVERED
PERrORMING O~G. REPORT NUMBER
M.R. Leadbetter
,.
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TECHNICAL
<.
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•
2. GOVT ACCESSION NO.
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•••
SUPPLEMENTARY NOTES
19.
KEY WORDS (Continue on revellUI IIlrle U n.c ••••ry "Id Iden"'v by block number)
Extreme values, stochastic processes, maxima
'0.
•
ABSTRACT (Conl/nue on r."'.,.• • Id. If nec ••• at)" ."d Iden,lfy by block nUtl'lber)
Certain aspects of extremal theory for (a) stationary sequences and (b)
continuous parameter stationary processes, are discussed in this paper. A slight1;
modified form of a previously used dependence condition, leads to simple proofs
of some key results in extremal theory of stationary sequences. Dependence condi
tions of a "weak mixing" type are introduced for continuous parameter stationary
processes and results of classical extreme value theory extended to that context.
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