* Research supported by Office of Hava1 lJ.esearch unrer Contract 1',100014-67-1\0321-002, Tsk ~m-042-214
'4EAK CONVERGENCE OF HIGH LEVEL EXCEEOANCES
BY A STATIONARY SEOUENCE
by
M. R. Leadbetter *
Department of Statistios
University of North Carolina at Chapel Hill
Institute of Statistics nimeo Series No. 933
June, 1974
WEAK CONVERGENCE OF HIGH LEVEL EXCEEDANCES
BY A STATIONARY SEQUENCE
by
M. R. Leadbetter*
SUPJt·'lARY
In this paper we consider a stationary sequence
{~ :n
= l, 2
n
satisfying weak dependence restrictions ( (2.1) a.nd (2.2) below).
the point process
level
u
n
!'J
n
nn
For cac:..
n
is defined to consist of the exaeedanaes of a certain
(i.e. the instants
point processes
}
j
for 1I1hich
E;. > U
J
n
).
It is shown that the
converge weakly (as random elements of the natural metric
space to which they belong) to a Poisson process.
generalizations of results of [6, Section 5J.
~his
gives, in particular,
The arguments use results of
[6] and a general convergence theorem for point processes (based on conveTgence of probabilities of no events), which gives a simplifying and clarifying
viewpoint.
1. INTRODUCTION
The asymptotic Poisson nature of the upcrossings of a high level by a
stationary normal process has been known for a considerable time (cf. f2] 3nd
references therein).
This result has been especially useful, in demonstrating
that the maximum of such a normal process in a given time, has the asymptotic
double exponential extreme value distribution.
Conversely it is possible to
show (as in [1]) from this asymptotic distribution of the maximum, that the
* Research supported by Office of naval Research under Contract NOOOl4-67-A0321-002, T5K
N~-042-2l4.
-2upcrossings of n
high level by this stationary normal process, are approxi-
mately Poisson.
Because of these facts, one is led to suspect that the asymptotic Poisson
nature of high level crossings is related closely to normality of the sequence,
or at least to the occurrence of the double exponential asymptotic distribution of the maximum.
This is not the case, however.
In fact the Poisson
property of high level upcrossings occurs in rather general circumstances and
in combination with any of the three "extreme value distributions H according
to the basic distributional properties of the
p~ocess.
Some properties of this kind have been discussed for stationary sequences
As noted there, it is more natural, for a stationary sequence
in [6].
{~n : n
= 1,
for which
~n >
~n-1 <
which
2 .•. }
u
to consider exceedances of a level
u
(i.e. instants
u ) rather than strict upcpossings (i.e. instants
< ~n
n
for
), it being easily seen that these are asymptotically
"equivalent" notions.
It was shown in [6] that, under wide distributional conditions, the
number
val
Ln of exceedances of a suitably chosen level
(1, 2
n)
has a limiting Poisson
un
dist~ibution.
in the time interHe shall show here
that this result is true not only for intervals, but (based on an obvious
normalization) for (almost) arbitrary Borel sets, and that convergence of
joint probabilities of this type also occurs.
These facts will be shown very
simply from the asymptotic distribution of the maximum [6, TheoreM 3.1] by
neans of a general point process convergence theorem.
It will, in fact, be
shown that, after a suitable change of time scale, the point ppocesses
consisting of exceedances of u
n
,converge weakly (as random elements of a
n
-3-
certain metric space) to a Poisson process on the real line.
This provides a
clarifying viewpoint as well as simplification of the arguments an.0. generalization of the results of
~ection
5 of [6].
Ti,e conditions used apply most
meter case consiclered here (as in
the corresponcing (but more
2. ASSUMPTIONS,
---~e
[6~).
com~lex)
NOTATIO~',
assume that
si~?ly an~
AND BAS Ie
--_._---{~n: n = 1, 2
naturally to the discrete para-
A later paper is planned to deal \'lith
situation in continuous tir.e.
RESUL1S
is a stationary sequence with finite
-cimensional distribution functions
xn )
J. eu) = F •
J. (u, u ... u) .
J 1··· n
J 1··· n
F •.
D(un ) , D; (u)
n
Two
depen~ence
t. - ...-t
~.
in [6] are also relevant here.
an,..] write
conditionS referred to as
i
In fact the form of
D(u )
n
given here will be very slightly weaker than that used in (6] (but still sufficient for the proofs and results given there).
sequence of real numbers we shall say that
D(U)
n
{~n}
Specifically
i~
{u }
n
is a
sc,-tisfies the condition
if for any integers
... < i
P
fI,
••• <
.
we have
(2.1)
where
a
sequence
is non-increasing in
n,fI.
k
n
+
00
,
and such that
and where
lim an k
n-+<x>
'n
=0
for some
(l\!ote that if (2.1) ho1c!s for
-4-
some
a
n,t
it is clearly possible to take
Further the condition
(2.2)
lim sup In
n~
I
~ j=2
to be non-increasing in
n
n,N
Di(U)
is said to hold if
> Uk'
~.
n
P{~.
a.
n
1
> U }]
J
nk
= o(l/k) as
k -+
00
t) .
•
A discussion of these conditions may be found in [ 6J.
We are particularly concerned with sequences
{un
= Un(T)}
satisfying
(2.3)
The question of definition of such sequences is mentioned in [6].
{Un (I)}
if
is defined satisfying (2.3) for
= U[n/Tl
Un(T)
T
=1
Note that
then we may define
(1) , where'Ix] (here and below) denotes the integer part of x.
It will be useful to give Theorem 3.1 of [ 6] the following slightly more
general form
THEOREM 2.1
{~n}
Let
Di(U
n
(aT))
be a stationary sequence and suppose that
are satisfied for some
to satisfy (2.3).
Then
P{H[
PROOF:
a > 0 • T > 0 , where
an ]
$
u (T)} -+ e
n
-aT
as
n-+ oo •
It follows at once from Theorem 3.1 of [6] that
and hence it is only necessary to show that
D(un(aT)) ,
u (0)
n
is defined
-5-
If
Un(T) > u[an] (aT)
the left hand side of (2.4) is
~ Un (T)} ~ p{
P{u[ a n](aT) < M[ an ]
~
If Un(T) < u[an] (aT)
](aT~ ~ ~.J ~ u.p. r-(T)}
.
[aU]
. _ {u[an.
J=.1"
an{F(un (T)) - F(u[an ] (err))} .
the same result holds with reversed signs.
Hence the
left hand side of (2.4) is in all cases dominated in modulus by
I
= an [aT] (1 + 0(1)) - !'(l + 0(1))
an
= 0(1)
n
as
n
+
00
I
,
as required.
3. CONVERGENCE OF EXCEEDANCES TO A POISSON PROCESS
Throughout this section we assume that
{~n}
is a stationary sequence
and that
D(un ) , !J(u ) hold (u = u (T) satisfying 2.3)), for all T >
n
n
n
For each n = 1, 2, ... , define a discrete parameter proces~ nn(t)
for
t
=
j/n
obtained fror:l
(j = 1, 2,
{~.}
J
,
)
by
nn (j/n)
= ~.
J
simply by time scale changes.
process consisting of the exceedances of
Un(T)
number of such exceedances in the Borel set
B).
by
.
Thus the
Let
nn
N
n
nn
{)
are
be the point
(Nn (B)
denoting the
-6-
The point processes
N
may be properly regarded as random elements in
n
[0,00), or
either the space of integer-valued increasing step functions on
N of integer-valued Borel measures on [0,00).
the space
In either case the
space is metric under the "vague topology'1 (e. g., generated in N by the
~
functions
Jfd~
+
for continuous
f
with bounded support - cf. [3,4]) and
we may consider convergence in distribution of sucn random elements.
will be used to indicate this convergence.)
The following result is a special
case of a theorem of Kallenberg ([4]Theorem 2.3) modified according to a
remark of Kurtz [5].
THEOREM 3. 1
Let
let
~
~n'
n
= 1,
2, ...
be point processes on the positive real line and
~({a})
be a point process without multiple points and such that
a.s. for every fixed real
(i)
a
~
O.
P{~n(B)
= O}
=0
If
= O}
+ P{~(B)
r
for all sets
B of the form
U (ai,b ]
i
1
(ii) lim sup
E~n(a,b] ~ E~(a,b]
(a l
<
hI
<
a2
for all finite
<
b 2 ...
<
ar
<
br )
r
= Nn'
a < b ,
n
~n ~ ~
then
.
rne main task is to verify Condition (i) in our case, where
and
~
=N
is a Poisson process with parameter
been shown for intervals
B = (O,a]
"'n
T.
(i) in fact has already
in 1beorem 2.1.
We verify it for finite
unions of intervals in the following short lemmas.
-7-
LEMMA 3.2
Under the conditions and notation stated at the start of this section, if
o<
a
<
b , B
= (a,b]
= O}
P{N (B)
n
PROOF:
P{N (B)
n
-+
e-(b-a)T
= O} = P{~.J
= P{~.J
by stationarity.
[(b - a)n]
for any fixed
h
~
as
u
n
~ u
n
n -+
00
~ b}
a < jln
[an]
•
~
[bn]}
[(b
+ h -
< j
But it is easily seen that
~
[bn] - [an]
>
0 , when
n
~ [(b -
a)n]
+ 1 ~
is large enough, and hence then
By Theorem 2.1, the outside terms have the respective limits
e-T(b-a) , and since
LEt4~1A
h
3.3
r
a
r
<
B
= 1U(a.,b.l
1
1-
b , then
r
lim P{N (B)
n-+oo
n
= O} = e -Tm(B)
r
where
m(B)
= L(b.
1
1
- a.)
1
e
-T
(b+h-a)
"
is arbitrary we have the conclusion of the lemma.
Under the same conditions if
<
a)n]
is the Lebesgue measure of
B.
-8-
PROOF:
If
B. = (a. ,b.] , N (B.) = 0
J
J
n
J
~
is equivalent to M(E.)
J
J
Ej = ([na ] + 1 , [na ] + 2 , ••• [nb ])
j
j
j
and
r1(E
j
) =
max(~k
u
where
n
: k
€
E ) .
j
Hence
= O} = P{
pIN (B)
n
r
= II pIN (B.) = O}
j=l
n J
+
r
n (M(E.)
j=l
J
~( ~ (l.1(E.)
j=l
~
J
~
un )} - II P{M(E.)
j=l
J
r
The first term converges, as
n-+
e
II
to
oo
u )} =
n
r
-T(b.-a.)
J
J
Unl]
~
= e
-Tm(B)
.
by Lemma
j=l
3.2.
On the other hand, by Lemma 2.3 of [6], the modulus of the remaining
difference of terms does not exceed
But by D(un ) , a
k -+ 0
n, 'n
(r - l)a
n1nA
A=
] where
min
l~j~r-l
as
J
J+
is non-
Since a n, R.
for some
R., and eventually rnA]> k n ' we have an[nA] -+ 0
increasing in
(b. I-a.).
n -+
00
,
from which the result follows.
We may, finally, now obtain the main result.
THEORE~l
3.4
(u
n
T > 0 , for the stationary sequence
n = I, 2,
. ..
{~.}
Let
J
(2.3) ) for all
nn(j/n) =
~j
, j = 1, 2, ...
, and let nn be the point process consisting of the exceed-
ances of un (T)
Then
by Nn
process with parameter
PROOF:
= u n Cr) satisfying
nn ~
n
n
-+
00
,
where
1'1
is a Poisson
T .
(i) of Theorem 3.1 holds by
PIN (8)
N as
Le~~
3.3 since for such
= O} -+ e-Tffi(B) = P{N(B) = O}
B,
-9-
(ii) is immediate since
- n (b - a) (Tin) .-
T
(b - a)
= EN(a,b] .
Hence the result holds.
COROLLARY
Under the conditions of the theorem if
boundary has Lebesgue measure zero
{
} +e
PN(B)=r
n
B is any Borel set whose
(m(aB) = 0)
-TID (B)
[tm(B)]
then, for any r = 0,1, 2...
r
Ir!
The joint distribution of any finite number Nn (B 1) ... Nn(Bk ) corresponding
to disjoint B. (with maB. = 0 for each i) converges to the product of
1
1
the corresponding Poisson probabilities
This follows at once since the random variables
distribution to
converge in
(N (B )
!'In (B ) ) ~ (N(B l ) ... N(Bk ) ) , if
n 1
k
(N(aB) = 0 a.s. if m(aS) = 0 since N is a Poisson
N(B)
lin ~ N (cf. [4]).
Nn(B)
(and
process).
Thus we have a "full" weak Poisson limit for the exceedances of un (T)
under rather general dependence restrictions.
It is easily seen (even by con-
sidering i.i.d. sequences) that this Poisson behaviour can occur together with
any of the possible asymptotic extreme value distributions for the maxinum.
-10-
REFERENCES
[1]
Berman. S. H.
"Asymptotic independence of the numbers of high and low
level crossings of stationary Gaussian processes". Ann.. Math.
42,
[2]
~27
Cram~r
---945 (1971).
H. and Leadhetter . M~R. lIStationary and related stochastic pro-
cesses", John Niley
[3]
StatiBt~;
&Sons.
New York (1967).
Jagers. P. "Aspects of random measures and point processes", Adv. in
Pr>ob. & Y'eZated topics. ~ Marcel Decker, New York.
[4]
Kallenberg. O.
ilCharacterization and convergence of random measures and
point processes" Z. WahY'o veY'W. Geb ••
[5]
Kurtz. T.
'!:J..... 9 - 21 (1973).
"Point processes and completely monotone set functions Yl •
Technical Report, Dept. of Math .• Univ. of Wisconsin (1973).
[6]
Leadbetter, M. R.
"On extreme values in stationary sequences". Z. rJahJ:'.
verw. Gab •• 28. 289 - 303 (1974).
';