* This pesearah was suppopted in part by the Offiae of NavaZ Research
undep Cont'Z'aat N00014-67-A-0002.
and the Unive'Psity of
Lund~
SWeden.
WAVE-LENGTH AND AMPLInJDE FOR A STATIONARY PROCESS
AFTER A HIGH fJlAxIMUM: DECREASING CoVARIANCE FUNCTION*
by
Georg Lindgren**
Depa'Ptment of Statistias
Unive'Psity of Nopth CaroZina at ChapeZ HiZZ
Institute of Statistics MimeD Series No. 745
ApIVU, 1977
WAVE-LENG'TI-I AND PwLITUDE FOR A STATIONARY PRocESS
'e
AFTER A HIGi r1tv<I~tJM: DECREASING CoVARIANCE FUNCTION*
by
Georg Lindgren, Uni versity of North Caroo Una" Chape l, Hi l, l,
and University of Lund" SWeden
I.
INTRODUCTION AND RESULTS.
In a sequence of previous papers, the author
has treated the distribution of the wave-length
of a stationary Gaussian process
the height
u.
{;(t), t€R}
L
U
ou
and the amplitude
,
after a local maximum" with
This paper is a direct continuation of [4] and it deals with
the asymptotic distribution of
L
the covariance function
;
called case iii).
r
of
ou
and
U
u
as
+
00
in the case where
is strictly decreasing for
t > 0,
here
We adopt without further comment all notations and results
from [4], to which the reader is referred, and in which further references
are found.
In this paper, we assume that
and that
r(t)
the dominant term in
t,
near a local maximum with height
r YiV (t»
function
is strictly negative for
and its first four derivatives tend to zero as
for every fixed but large
;1
r V (t)
is negligible.
c(s,t)
u) is
urV(t)
Furthermore, for large
; V(t)
u
t,
of the non-stationary Gaussian process
pears in the definition of
;
u
'(t)
is approximately
+
00.
Then,
(which describes
nu (A 2r V(t) +
while
sand
t
t > 0
the covariance
o(t)
-r"(s-t)
which apwhich we
recognize as the covariance function of the original process derivative
;'(t),
and therefore we expect
large t-values.
o(t)
to behave almost like
for
Thus it seems plausible that the distribution of the time
for the first upcrossing zero of the process
terms of that of the process
conditions on
;'(t)
r
ur'(t) + ;9(t).
;
u
1
(t)
can be expressed in
In addition to the general
given in [4, Section 2], we now require the following
'"
This researoah was supported in parot by the Offiae of Naval, Researah
under Contraat N00014-67-A-0521-0002.
2
condition to be fulfilled.
C3
a)
rV(t)<O
b)
r' (t)
c)
t- (A 4 _r
=
2
J:o
d)
IV
for
t > 0
OCt -Y)
(t»
as
t
= 1,
k
00
-tor some
increases as
t-l(A4_rIV(t»~ dt
For
-+
2, 3, 4
<
00
t
there is a constant
it holds that
C ~ O. ~
tends monotonically to
If
C
=0
then
t,
C as
to > 0
(_l)kr(k)(t)
tkr(k)(t)
such that
00
t
i
decreases to zero and
for some
tive and decreasing for large
y >
-+
_r(k) (t)!r(k-l) (t)
It should be noted that if the condition C3.d
-+
0
is convex, posias
t
-+
00,
_r(k) (t)!r(k-l) (t)
00.
=
O(t-l)
as
t -+
00.
k:: 4, then
is fulfilled for
it is fully fulfilled.
Before we can formulate the results we need some notations.
First de-
fine the function
=
muy (t)
ur' (t) - y{A 2 r'(t)+r"'(t»
and the r.v.
v
=
t
vt(y)
lJ t
=
the number of upcrossing zeros of
~
u
'(s)
in
the number of upcrossing zeros of m (s) + o(s)
uy
::
the number of upcrossing zeros of
uri(s) +
Then our interest centers on
peL
u
~t)
=
1 -
f
f
1 -
y
while we use
P(vt(y)=O)qu*(y) dy
y
::
1 - P{lJ
t
(O,t]
=0)
::
*
P (.6Up[O, t ]muy (s)+o (s) <0) ~ (y) .ely
1 - P(.6Up[O,t]ur'(s)+~'(s)<o)
~'(s)
in
in
(O,t]
(O,t].
3
= I E(Vt(y»
qu*(y) dy
y.
=
E ("t)
•
LJ:=u
y
J: t:
W(B}$ (muy (B)
O{ur' (B) I';;;;;)
IO(B»'I'("uy(~»
q.. u*(y) ds dy
m
.•••.•
..•..•.
..
~ (ur" (B) 11f.;") dB
where (see e.g. [3])
0'2
y2
=
=
=
V(O(s»
)'2(s)
=
V(e' (s»
~ = ~(s)
w
=
=
a 2 (s)
=
c(s,s)
l)2C~s't)1
=
l)sl)t
t
=s
=
Cov(o(s),e'(s»!a(s) yes)
ac(:~t)1 /'a(s)
yes)
t=s
w(s)
=
=
{m '(s) - y(s)~(s)m (s)!a(s)}/y(s) 11-~2(s)
uy
uy
{url-y(A2r"+rIV)!yli=7 - ~{ur'-Y(A2r'+r"')}/ali=7 •
We now split up the discussion into three parts according to the value
C = 0,
°
<
cases is that we define a function,
T
= T(u)
of Um t-+ oo-r"(t)/r'(t) = C;
tive1y) so that the expected values of
C
<
C=
00,
-+
v
and
T
C
= 0,
then the function
constant for large
u
and
t,
-ur'(t)
Common for the three
(= To, TC, Too
00
~T
then we obtain a non-trivial limit of the probability
If
00.
tend to
P(T -T
u
6,
$
respecsay,
x)
and
as
behaves almost like a large
and we might expect a behaviour similar to
the asymptotic Poisson-character of the stream of crossings of a very high
level derived by Cramer and others, cf. [3, Ch.12], although we here have a
very high function.
In fact, Theorem 1.lb contains the first term in the
asymptotic Poisson distribution.
It is possible to establish the full
Poisson distribution by means of the standard proof, generalised as in Section 4 of this paper, under e.g. the following conditions on the function g :
u
4
o
_ __
o
g (t)
-Ln.6
~ t
~
.[)up
s s,
u
T(u)
t
~
T(u)
+
Ig
.[)up
t ~ .T(u)
00,
o~
18 u 2 (s) - gu 2 (t)!
V(t)1 + 0
u
is bounded.
The results by Qualls [8] on multiple level crossings are instructive in this
respect.
1.1.
THEOREM
then, as
u
If condition C3 is fulfilled, if
-r"(t)/r'(t) + 0
as
t +
00,
+ 00
for all
a)
+
b)
I -
x
e-e
for all
x.
We prefer to formulate the results in a slightly different way in the
case
0 < C~
TC and
With the choice of
00.
TOO
proposed in Theorem 1.2
in can be shown that
- uri (TC+t)
e
+
00
- uri (T +t)
+
-Ct
{:
for
t < 0
for
t
~
0
respectively, and the theorem reflects this fact.
THEOREM
with
1,2.
If condition C3 is fulfilled, if
0 < C ~
and if
00,
C
- uri (T)
=
I
TC = TC(u) ,
TOO
= Too(u)
u
+
00
+
(0 < C < (0)
(C
then, as
-r"(t)!r'(t)
=
00)
00
+
so that
C as
t
+
00,
5
a)
E(v
(0 < C < 00)
f'. )
T~x
o
.for
1
x
"2 +
21T
u
.~
< Q
(C = 00)
r:;
for
~~
x
~
0
(0 < C < 00)
b)
o
00
P{-r -T sx)
for
x < 0
(C = 00)
-+-
u
1 - P(
o S.6u.p
t S
x
~'(t) < 0)
for
x
~
0
Part b of Theorem 1.2 is included in the following theorem concerning
the joint limit distribution if
THEOREM
1.3.
r.v.
is defined by
T
T
u
and
0 •
u
If the conditions of Theorem 1.2 are fulfilled, and if the
(0 < C < 00)
(C = 00)
then, as
u-+-
(T
-TC
u
'
(T
-T
u
oo
-1
ou-u)
L
-+-
(T, -C
, ou-u)
L
-+-
(T, -~(T»
e
-CT
-~ (T»
(0 < C < 00)
(C
= 00).
The following theorem holds regardless of the value of
THEOREM
1.4.
ou lu
If condition C3 is fulfilled then, as
r o.
u -+- 00
C.
6
The proofs of Theorem l.la and 1.2a are found in Section 3, that of
Theorem l.lb in Section 4, and that of Theorem 1.2b in Section 5.
Section
of Theorem 1. 4.
2. SOME AUXILIARY RESULTS.
In this section, we show that for increasing
(0, T_).
u-values we can disregard what happens in an increasing region
LEW1L\ 2.1,
a)
E(lJ
b)
PROOF,
For any
)
t
+
uo7
as
t0 > 0
ClO
0
0
PC-r <t )
u
~
0
E(V
t
)
+
O.
0
then we extend it to all
E(lJ t
*)
=
0
f
t'
0
o \
ril(t)
~
0
~
~ ~ (ur' (t)]
A2
~
for
o~
~
aCt)
=
(A,.r'(t)
+ A2 r'" (t»!A 2 l3
.,.
bet)
=
(A 2r' (t) + r'" (t»!A2 B•
ua(t)- A2l3(y+u!13)b(t),
t *,
o
and fulfills
'¥(n
t 0 *.
Then, as
uo7
~
Kf
ClO
*
t
t
o
*>
0
are non-negative and strictly increasing in
~
~
,¥(urll(t)] dt
2
Then it is easily proved that there is a
t
t
0
o
~(ur'(t)]dt
+
If;
Let
b)
o<
and
t
t.
o
t 0 * so that
Take
a)
bet)
o*
We first prove that the results hold for one particular
uy (t»
we conclude that
and for all y > -u/13.
'¥(x) ~ l+lxl
we get
muy '(t)
such that
[O,t
o
*l.
-aCt)
Since
and
uy (t)
m
is strictly negative for
Since the function
'¥
is increasing
=
0
7
Thus
(2.1)
c:
oo(t)
~F~1~;J~(nuy(t»qu·(Y)
=
C-u/s
s
fy f t w ,~(muy la)q U *(y)
Jyft r1gl2~
a
+
dt dy
~(muy la)q u *(y)
1m I
uy
dt dy
dt dye
We will now show that the second integral in this sum tends to zero.
first integral is treated similarly.
We first note that
yl~1
The
is bounded.
Furthermore, simple calculations show that
d~ a 2 (t) = -2Sa'(t)b(t),
1muy Iia
Now we can estimate
Write, for short,
but
R(t)/t
2
•
1m uy Ila 2 •
= A4 _rIV(t)
R(t)
increases as
- tR(t)/.(t)
and
t
and recall that
decreases to zero.
-tR(t)/J: .'(s) d. •
R(t)
decreases,
Thus
tR(t)/J: R(s) ds > 1
but
so that
Since
tR(t)/la(t)1
b(t) - K • t,
2
R(t) • b (t)
a 2 (t)
~
is bounded away from zero and infinity for small
we get
R(t)b(t) • ~ ~
-2Ba(t)b(t)
K.'
-0
~
a
2
(t~
~
K'.
a
Since, furthermore,
-
dta(tl/
dt / Ea2(t)
dt
..
-a(t)-ta'(t)
2f3a v (t)b(t)
-a(t) + K
1 • tR(t)
2
s K
s K"
a
t.
8
we get
- ta(t)/0'2(t)
_
Ka "
S
tR(t)b(t)/0'2(t)
s
_
2
• R(t)b (t)
0' (t)
-a(t)
t~(t)
s ~".
Thus it is always possible to find positive constants
o
< t S t
o
*, y
>
Ki
such that for
-u/f>
1muy (t)1
O'(t)
1muy (t)1
t -l{K (y + u/f»
S
R(t) -1 + K uL
)
4
Introducing the explicit expression [4, (2.1)] for
qu *(y),
we get the
following upper bound for the second integral in (2.1):
*
t
t- 1 {K, (y+u/B)R(t)-l + K u} eXP(-K 2 (y+u/B)2/ 2R (t» •
1
t=O
3
4
K Jco
y=-u/B
J
0
• (y+u/f»eXP(-A2By2/2) ~(uIA2/B)-1 dt dy
~(u/A2/B)-1
K
S
5
+ K
JJ
6
y
t
(y;~~~~2
JJ
t
y
(Y+~/a)
2
eXP(-K1 (y+u/f»2/2R(t»
eXp(-K12(y+u/f»2/2R(t».eXP(-A2f>y2/2) dt dy
~(u/A2/S)-1 Jtt=Oo* Jcox=O t~~t)
K
7
S
+ K
B
JJ~
t
x
dt dy
2 2
eXP(-K 1 x /2R(t»
eXP(-K 2x2 /2R(t»
1
dx dt
• eXP(- A2S(X-u/S)2/2) dx dt.
The double integrals in these expressions are bounded by
*
t
0
Jt=O
v'R'[tf d t
and
t
which both are finite.
*
t
J
0
t=O
Since
R(t) dtrespectively
t
~(U/A2/S) +
that the total bound tends to zero.
co
dominated convergence gives
9
We now extend the results to general
t
Since -i.n 6(t * t) Ir'(t)1 > 0,
a)
o '
for some
o
.6Up(t * t )lr"(t)1 <
0
0
'
00,
we get,
dt
+
0
Ki, K2 > 0
t
t
f o* <j>(ur'(t)/!r;) 'Y(ur"(t)/n:;) dt
f o* ~(Klu)(I+K2u)
s
0
0
Taking -i.n.6
b)
-i.n.6 o(t)
.6UP
and .6UP
0+" .6UP o(t) <
> 0,
1~(t)I(I-~2(t»~
<
over
00,
t
~
t o* we get that
y_ = -i.n.6 y(t) > 0,
y+ = .6UP yet) <
since that assumption implies that the distribution of
non-singular and that, as
t
0_"
+
00,
~(t),
This follows from the ergodicity of
00.
O.
t
t
€ ..
•
(o(t), o'(t»
is
00
(2.2)
respectively.
Imuy' (t) I
Since
and y we get
KI 'u + K2 ' Iyl
S
'Y(n uy (t»
Imuy(t) I
KI"u + K2"lyl
S I + In uy (t)1 S Klu + K21yl.
and
If we further notice that .6Up(t * t
o '
0
for all
S
)IA 2r'(t)+r
tli
u
(t)I!-i.n.n(t * t )lri(t)1
0
'
0
is finite, we get
for all y with
fY
> -ul S
Iyl S K u.
4
Separating
to
w ~(m la)
ft=t
*
uy
o
t
S K
5
'Y(n
uy
+ K7
0
4
f
ft
o
o
lyl~K4U t=t o*
and the lemma is proved.
~
K u and
4
S Ku
4
we get
) q *(y) dt dy
u
flylSK ft=t * ~(K6u)
U
Iyl
(u+lyl) qu*(Y) dt dy
(u+lyl) qu*(Y) dt dy
+
0
as
u
+
00
0
The following lemma plays an important role in the future development.
10
lEMMA 2;28
There exist a constant
that, for all interval
where
III
PROOF;
Since muy '(t)
1+
u
and y,
such
dt
K(III+1)
$
is the length of the interval.
is monotonic' for large
Let
independent of
sufficiently remote from the origin
I
~(muy (t)/a(t» ~(n uy (t»
fI
K,
and
{l -
Y..
u (:>. 2 +rIV(t)/r"(t»}
and
iV(t)/r"(t)
muy '(t) can change sign at most once in 1.
denote those parts of I in which muy ' (t) ~ 0 and < 0
I
respectively.
= ur"(t)
t,
With the same notation
a+, a
etc. as in the proof of Lemma
2.1, we get
~(m
uy (t)/a+) ~(Elmu(t)l/a)
y - dt
~(m
uy (t)/o+) {1 + Elmuy (t)lla}
- dt
since
~(x)
fI +
$
$
$
and
fI
fI
+
+
x~(x)
~(muy (t)/a+)
{m
uy
t
~
to
to
f x~(x/a+) dx
$
T_
(t)l/a) dt
uy-
K+(II+I+1).
large enough to assure that
and define a function
+ Elm
'(t)/y 11-8 2 + 1 + Elm (t)lla} dt
uy-
A combination of the results yields the lemma.
Now take
K-II- I
are bounded.
~(muy (t)/a+) ~(muy V(t)/y - 11-E 2
K_II+I + K
$
= T_(u)
~
00
0
-r'(t)
decreases for
such. that (for
= 0)
(if
C
(if
C> 0).
(2.3)
0 > 0)
11
REMARK 0
It is only in the case
-ur' (T_)
satisfies (2.3) then~ as
If T
P(T ST)
b)
over
= ...U,mt-+oor" (t) Ir' (t) = 0
that we require
to tend to infinity in a specified"way.
lEMMA 2.3,
PROOF.
C
u
S
-
E(V
T
)
u
+
00
o.
+
By Lemma 2.1, it suffices to show that the corresponding integrals
(t
o
~T)
a)
tend to zero.
-
If
-r"(t)/r'(t)
+
f
H ur' (t) II>;)
'¥ (ur" ( t) / ~) dt
Tt
then
0
o
~
( - Hur' (t)/II:;) {l + ur"(t)/n:;l dt
o
S
T_~(ur'(T_)/II;)
k fur' (T )
+ A4-~
- ~(x/~) dx
ur' (t )
o
T
S
~ exp(-(2+0)A2logT_/2A2) + (A2/A4)-~ ~(ur'(T_)/~)
V21T
by the definition of T •
If
-r"(t)/r'(t)
then ..[nn(t
+ C > 0
T
)ur"(t)
+
00
and we can use
0' -
the simple estimate
Write for a moment
b)
for
Iyl
'¥(ur"(t)/~) S Kur"(t).
y
u
= T-
and expand
Yu :
S
m (t)
uy
=
ur'(t) {l - R (t)}
uy
nuy (t) = ur"(t)A4-~
{l - S
uy
(t)}
where the residuals
R (t)
uy
=
.Y
u (A 2 + r"'(t) /r' (t»
S
=
1-
uy
(t)
~
y/1-1l2
+
1
• .Y.
y/l-J.l2
u
m (t)
uy
and
nuy (t)
+
0
12
will be shown to converge to zero uniformly foro to :s; t ::;; T_ 1i
as
u
cording to if
C <
or
00
= o(r'(t»
~(t)
or o(r
(sic!).
C > 0,
IV
(t»
as
00,
respectively:
r'''(t)rIV(t)!r''(t);
.6UP
t
yU
r'(t)2!r"(t),
yu!u,
.6UP
:s;t:S;T YurIV(t)!ur"(t).
t
t
-+ ""
ac-
Then the assertion is proved if
we can show that the following quantities tend to zero as
-+
S
-+ "",
First we note that
u
Iyl
t
r'(t)rIV(t)!r"(t),
-+
and
00
r'(t)r"'(t)!r"(t),
:S;t:S;T Yur"'(t)!ur'(t),
o
-
That the first four quotients tend to zero is
o
a consequence of the assumed convexity of
three expressions, we note that
u we have y
so that for large
(_l)k r(k)(t).
r'(t)
= O(t- Y)
= T-
:s; u1!y.
u
convergence is obvious in the case
C<
Thus
If
00.
implies
C
=
For the last
-ur'(T_):S; MuT -Y
y!u
u
00
-+ 0,
and the
we have for
t:S; T
that
y
r'" (t)
ur' (t)
y r
u
T_tr'" (T_)
u
ur' (T_)
ur"(t)
t
(2.4)
(2.5)
m (t)
uy
o(t)
=
ur I (t)
uy (t)
=
ur"(t)
l1
uniformly in
Now let
(1 -E)(l
o
-+
(1 + 0 (1»
II;
(1 + 0(1»
~
S
0 > 0
be as in (2.3) and take
L
If
T_,
to
T_rIY (T_)
ur" (T_)-
00
to:S; t
+~o) >
IV (t)
Iyl
S
t
S
S
Yu'
T_,
1muy (t) I/o (t) ~ (1- e:) Iur I (t) I!~,
Iyl
S
E > 0 so that
Yu'
111 uy (t) I
S
we can then assume that
(1 + e:) ur" (t) !IC".
..
Using
13
Lemma 2.2 if
Iyl > y
+
we get
u
I
Iyl>yu
K T q *(y) dy.
- u
For the first integral, we use the same technique as in part a), while for
T JI I
q *(y)dy - 2T (l-~(y » - 2T ~(T )/T
- y >Yu u
u
-
the second we use that
+
0
For later use, we extend part of the proof to a new lemma.
Le~
T
2.4.
C co
T , T
= To ,
fulfills (2.3), and
muy (t)
a (t)
=
ur' (t)
T'luy(t)
=
urn (t)
where
Iyl
If T
~
0(1)
+
T
Yu'
If
PROOF.
v>;
~
=T
y
u
(1
~ t
C < co,
~
(1 +
T if
then
as in Theorem 1.1 and 1.2 respectively,
co
then
+ 0(1»
0 (1»
uniformly in
0
+
C
Iyl
~
y,
u
T
~
t
~
T+x
if
~
MuT
to show that
-2
-ur'(T) ~ MuT- Y implies that
by condition C3.d which implies that
Trill (T)/ur' (T) + O.
T2r VII (T)2/(_r' (T)/rli(T»
co
and in
=T ~
M,u1 / Y,
= co.
and we can proceed as in the proof of (2.4) and (2.5).
ur"(T)
C<
y
If
y
u
u
C
= co,
/u + O.
then
It remains
If we square the quotient, we obtain
which tends to zero since
rill (T)r"(T)/r' (T)
does. 0
14
3.
1.lA AND 1.2A
PROOF OF THEOREM
As
u +
Write
I
y
u
o C CIO
(=T ,T ,T ).'
= T = T(u)
00
T+X
f
PROOF.
$(m
W
t=O
uy
~(n
/0)
uy
) q *(t) dt dy
+
u
O.
The lemma is a direct consequence of Lemma 2.3, Lemma 2.2 and the
fact that
TIlyl>T qu*(y)dy + 0
PROOF OF THEOREM
1.1A.
as
T+
0
00.
In the light of Lemma 2.3 and Lemma 3.1 it only re-
mains to prove that
fIyl:s;y fTT+X
u
W
-
We do this by proving
note that for
~(n
$(m /0)
uy
t >T
I1yl:s;yu
) q *(y) dt dy
uy
I~_
and
6
+
the derivative
urll(t)
'r"(T)/(-r'(T_) = K/.e.ogT_ r"(T_)/(-rt(T_)
Lemma 2.4, it then holds that
Iyl :s; y,
u
and thus
<
I Iy 1-yu
T+x
I T:
~(n
uy )
nuy (t)
+ ~(O)
+
0
~
+
-ur'(T)
0
by condition C3.d.
+ 0
uniformly in
By
T :s; t :s; T+x,
uniformly.
Here the inner integral is taken over an interval of
fixed length and the integrand is bounded by
ur"(t)
6.
+
u
K $(K ur'(t»
l
2
uniformly, a little reflexion will show that
qu*(y).
-urt (t)
~
Since
-ur' (T+x)
must tend to infinity if the integral
I~ Hurt (t)/n:;)~(ur"(t)/~)dt
of the integrand is
is to be bounded.
q *(y) • 0(1)
uniformly in
u
Therefore the said bound
(T, T+x)
and the total
integral tends to zero.
II y I :s;yu I T:
T_
~(ur"/If;) + ~(O)
We have seen that
uniformly as
$(muy (t)/0(t»/$(urt(t)/~2)
+
u
1
+
w(t)
00.
+
IA 4 /A 2
and
~(n uy) -
It remains to prove that
uniformly, i.e. to prove that
15
2
m
uy
(t)
The difference is
cr-2(t){u2rV(t)2(A2-cr2(t»/A2 + y2(A 2r'(t)+r Vif (t»2
=
- 2uyr'(t)(A 2r V (t)+r n '(t»}
Since
A2 - cr 2 (t)
=
and T_ S Yu
=
O(r'(t)2),
T S u
l/y,
A S K u2r v (T_)4
1
•
2
O(t-Y),
riVf(t)
=
rV(t)
-2
log T_
+
=
O(rV(t»,
0
log T
B S K y 2 rV (T )2 S K (y /u)2
3 u
4 u
-
Ici S K6uyuri(T_)
say.
we have
K T_
2
S
A + B - C,
log T_
S K (yu/u)
7
K u- 2 (1- l/y) log u
S
-
5
+
+ 0
o.
Thus all the three factors in the considered integral are uniformly asymptotica11y close to the corresponding factors in the integral
/A 4/A 2 ~(ur'/II;) ~(O) dt
6.
which in turn is approximately
By this, the
0
theorem is proved.
For future use, we formulate the obtained results as a lemma.
LEM'1A 3.2.
If
-r"(t)/r' (t) + 0
as
t +
and if
00
T
and T are defined
as in (2.3) and Theorem 1.1 respectively then
Y(t)/1-~2(t)/cr(t) +
IA 4 /A 2
~(m
uy (t)/cr(t»/~(urV(t)/~)
~(n
•
uy
(t»
uniformly in
"" ljI(urn(t)/~4)
T_
S
t
S
T,
Iyl
+
1
+ IjI(O)
S
T as
= 1/12-:;
u
+
00
For the proof of Theorem 1.2a, we need another lemma.
16
As
u
+
for every
00,
-uri (TC+t) lexp(-Ct) + 1
t
> 0,
o
t
00
uniformly in
-t o
S
o
uniformly in
t
0
> 0
I
Itl
uniformly in
00
o
t
S
S t
o
-t 0 '
-ur' (T +t)
PROOF. If -r"(t)/r' (t)
C<
+
then for every
> 0
00,
~
such that
-(C-e:)r'(t) S r"(t) S -(C+e:)r'(t)
arguments,
,
this can be shown to imply that
-rl(t) S -r'(s)exp(-(C-e:)(t-s»
for
~
t
for
e: > 0
there is a
t > t*.
By continuity
-r'(s)exp(-(C+e:)(t-s»
S ~ t*,
t*
S
which gives the lemma in
this case.
If
tion of
•
-r"(t)/r'(t)
00
+
the result follows if one considers the defini-
00
0
T.
PROOF OF THEOREM
1.2A~
Let
T_
+
so that
00
T_-T
+
-00,
-ur'(T_)
+
00,
so that the convergences of the foregoing lemma hold uniformly in
and
T -T
00
S
t
S
t
-t o "
~
respectively.
0
Using this
function in Lemma 2.3, we only need to prove that
tends to the respective expressions as
-r"(t)/r'(t)
r
C<
+
x
=
•
'W" <p
K
6
00:
u
+
00.
1muy (t)lla(t)
By Lemma 2.4 we have
(m fa) 'II (n ) ds
uy
uy
fur' (T+x)
ur' (T_)
<P(k s)ds
1
A little ref1exion also shows that
S
K
S
K
7
S
r
<p
(K
1
~ K ujr'(t)l,
1
ur' (t» ur" (t) dt
independent of
u, y.
m 2(t)/a 2 (t)-eXP(-2C(t-T»/A
uy
2
and
and
17
nuy (t)·-cexp(-t+T)
can be made uniformly small in
T s t
S
T+x.
Dominated
convergence then gives the assertion.
-r"(t)/r'(t)
E(V
T+x )
If
-+
00:
If
Lemma 2.3 and Lemfua 3.3 give
that
O.
-+
= 0,
x
then
=
If
x < 0
Iyl s y
u
q>«ur' (T)-y(A2r' (T)+r"'(T»/a(T».
= T,
we have
yrm(T)
-+ 0,
yrV(T)
the argument of the q>-function tend to zero.
so that all terms in
-+ 0
Thus
1
2'
•
A reverse inequality follows again from Lemma 2.4.
I
T w
T
~(muy fa) ~(n uy )
~
< (1+<)
(1+8)
f:
V
ur (t)]
~ ( (1-e:)
~
~ (1+e:) ur" (t)] dt
~
vT;
((I-e:)
ur' (t)] ur" (t) dt
15;
T_
(1+e:)2 {q> (1-8) ur v (T) ) _ q> [ (I-e:) ur I (T - ) ] }
=
I>::
1-e:
Thus Um .6up E(V ) S ~
T
u
1 (l+e:) 2
2
2
1- e:
and this part of the theorem is proved.
It remains to prove that, if
T x
{
I
.
fIylsy T+
-+
I>::
2
•
e: > 0 and
dt
21fT
--~
For
w <p<mu fa)
y
~
(n
x > 0,
uy
)
then
dt} q *(y) dy
U
-+
~ \I~
IS" .
21T
Lemma 2.2 shows that the inner integral is bounded by a constant
K,
u
large
18
independent of
u
and
y.
The dominated convergence theorem applied to the
outer integral gives that result if, for fixed
T
dt
+
x/2~.
To see this convergence, let
s
be fixed,
I
~(muy ;0; ~(n uy )
X
+
T
muy(t)
to
and
nuy(t)
= 1/2~.
~(O) ~(O)
n
uy
(t)
both tend to zero.
4,
I,m.
(t) + o(t)
<
0
for all
(4.1)
e -e
+
= Te o •
+
O.
We introduce the event
y
T
Then
0
and prove that for fixed
Let
= T+s.
Thus the integrand tends pointwise
ur'(T+x) - ur'(T)
=
PRooF OF THEOREM
uy
t
K + ur"(t),
3
::;;
ur"(t) dt
Er : {m
and
K ur"(t) + K ,
1
2
S
The theorem is proved.
•
0 < s < x,
Also this convergence is dominated since
~(muy/o) 'P(Tl uy )
fT+X
T
y
t
€
E
r
where
I S. R:
I}
Since the expected number of crossings in
(T, T+x]
tends to
zero, we then have , by (4. 1)
lim P(L U
u+ oo
> T+x)
= lim
u+
•
oo
I
=
lim
u+ oo
f
y
P(E(O T+x]) qu*(Y) dy
P(E(O T]) qu*(y) dy
y'
by dominated convergence •
,
=
I
y
e-e lim qu*(y) dy
= e-e
19
To prove (4.1), we start along well-known lines and divide the interval
[T_, T]
into
~:
subintervals, each of length
=
T
Let a
Let
n
be a small number and split the intervals into two parts
= tk+j(l-~)~/nk'
tk,j
= O,l, ••• ,nk
j
be a subdivision of the
Ik-in-
terva1s and write for short
Then (4.1) is a consequence of the following four lemmas.
4.1.
I..EM'v1A
Um
If
a -+ 0
Um
u
-+
00
Um
00
If
•
~
= log
p( ~ Fk]
-
k=l
If
n
k
t
~
P(F )
k
k=l
= log
t
k
,
n
Um
u -+
CIO
II P(F )
k
k=l
= e -e
~
•
-+ 0
as
u
-+ co,
then
= o.
and
k
then
p(k=l
~ Ek]
and
then
-+ co,
00
p(k=l
~ EkJ - p( ~ Fk)
k=l
LEMMA 404.
u
-+
-+ co
~ = log t k
If
LEMMA 4.3.
u -+
u
'
U:Mfv1A 4.2.
u
= Um
P(E(O T])
u -+ co
as
~
=
a -+ 0,
-+
0
sufficiently slowly as
u
-+
0,
0
and
~ -+ 0
sufficiently slowly as
then
20
Before we proceed to the proof of these lemmas, we state and prove the
following auxiliary results.
lEMMA 4151
For
1(2-o)£.ogt
if
u
a>
~
0,
Y
fixed and
~
1muy (t)l/a(t)
T
~ t ~ T it holds
1(2+o)£.ogt
is large enough.
PROOF, By Lemma 2.4, we have that 1muy (t)l/a(t)
T ~ t ~ T.
Since
-uri(t) ~ -ur'(T_)
have the right hand inequality.
Since
= 1(2+o)A 2£'OgT_
-t·rll(t)/r'(t)
following inequality holds in the interval
-ur' (t) ~ -uri (T) + ro~ T
.6/lP ur"(t)
0
~ -uri(t)/~2 for
~ 1(2+o)A2£'Ogt
we
is bounded, the
(T - T/£.ogT, T):
~ -ur' (T) + K/I£.ogT •
Therefore
I
T
T-T/£.ogT
T
K·-"-'-;o..-..
~
£.og T
e.x.p {-
2~2
(-ur' (T) ..-K' /1£'09T)2}
e.x.p {-
2~2
u r' (T)2 +K
from which it follows that
tegral has a finite limit).
as asserted.
LEMMA 4061
•
o(t)/a(t)
~(ur'(t») dt
52 '
2
iI
}
-url(T) ~ 1(2-o)A 2 £'OgT
(since the left hand in-
Thus
0
The covariance function
fulfills
c
of the normalized process 5(t) =
21
•
a)
c(t, t+h)
b)
is bounded as
There is a
y > 1
~
h
for
h,
o
~
there exists
> 0
o
~
t
4.1.
PROOF OF LEMMA
process
m
uy
P(v
P(V
T
~
J
(t)
less than the
y
Write
+ o(t)
v
o.
c(t, t+h).
S
1)
S
E(V )
S
K
E(v
t
(y»
T
J
=
J
J
€
J.
~
0
w
Then
as
u
~~.
Furthermore, by Lemma 3.2
~(muy /0) ~(n uy )
J ~(ur'(t)J
dt
II;
dt
if
0
~
a
~
o.
0
J
With
the number of upcrossing zeros of
Vk '
=
m (t) + o(t)
uy
the number of "upcrossing zeros" of the sequence
muy (tk ,J.) + o(tk , j)
for
j
= 0, 1, •.• , n
k
we have
0
•
S
p(
n
S
~ Fk]
k=l
I
k=l
- p(
P(v k ~ 1,
~
E )
k=l k
v
k
9
S
= 0)
~ E~
P(
k=l
n
S
I
k=l
n
Fk)
S
In
which exists according
y
for the number of upcrossing zeros of the
J
for
(y)~l)
PROOF OF LEf'iW\ 4.2.
M so that
0
to condition C3.b.
•
h
h
Part a) follows from the explicit representation of
part b), we only have to take
where
~ ~.
t
0,
such that for every
S Mh- Y
Ic(t, t+h)!
PROOF,
·
3
2 2
4
:r + hor'(t) 2
o~(t) + O(h ·r'(t) +h °r'(t) )
A4
~.
= 1 -
l~(t)
where
h2
I
k=l
E(v k - vk ') •
P(E~ Fk )
22
To make this difference small, we take a small
enough to apply Lemma 3.2.
·
fI k
w
(Ito)
~(m
uy
and choose
u
large
/0)
~(n
uy
) dt
J J~: ~ [u~t) J
I
.(0) dt
2
k
L
=
0
Then
k
n
E >
P(m
(t k · 1)
uY,J-
j=l
+ o(tk . j _1 )
•
<
0
<
.»
.) + o(t
k ,J
u yk, J
m (t
=
(4.2)
Below we will show that
=
P(muy (t k ,j_1) + o(tk2j _1 ) < 0 < mUy(tk2j) + o(t k2j »
~~ cp(muy(tk,j)/a(tk,j»
is uniformly greater than
1 -
E
1¥(O) • (t k ,j-tk ,j_1)
in all
I
if
k
u
is large.
If we use
the remainder of (4.2) as an approximating Riemann-sum (remember that
ur'(t)
is monotonic) we get
and
:;;
{(l-E) - (l-E)3}
I fI
k=l
j'"
k
A2
~(ur' (t»)
~2
J
1¥(O) dt
:;;
4E'
from which the lemma follows.
•
It remains to show that
press the index k
and write
is uniformly not less than
1 -
E.
Sup-
e
23
.Ilk
=
OJ
=
l;j
= (8j
and let
o(tk,j)/a(tk,j)'
-1
8j _ l )/.Ilk ,
be the density function of the r.v. (8. l,l;.).
fj(o,o)
probability in
.Ilk
-
J-
Qkj
J
Then the
can be written as
-0 j-l P(Xj-.Ilk°.ll
< 0j_l < x _ )
j l
k
<5
j
(4.3)
Here we used that
Since
8j _ l
X - x _
j
and
j
l;j
l
< 0
for large
u.
have a joint normal distribution with mean zero
and the covariance matrix
Dj
=
=
we have
f.(x. - .Ilkzy, z)
J
where
J
=
1
2Trldet
D-:J
eXp (-A/2detD j )
24
Here
maxO$Y$l A
= Ay=O = d22 x j 2
2
- 2d 12 x j z + Z
0 $ Z $ Xj/~k
as soon as
and we get the following lower bound for the integral (4.3).
xj/t.k
Jz=o
(4.4)
=
det
dz.
If c = c(tk,j_l,tk,j) then, by Lemma 4.6a, we have
-2 -2
-1
2
Dj = ~k (I-c) = A4/A 2 -t.
rY(tk,j_l) (1+0(1»+0(1)
k
tk,j_l +
o
00.
If
nk = log t k
~k
then
= ~/nk
~k +
as
- ~/log t k _ l
0,
and
$
ReMARKD
t.
We observe that the results hold even if
+
0
sufficiently
slowly.
if
~
+ O.
This implies that the integral in (4.4) tends to
A4/A 2 •
These considerations give that the lower bound in (4.3) is at least
(I-E)
\I~~
PROOF OF
</>(Xj)'¥(O)
LEMMA 4.3.
Le.
Qkj
Write, for
~
I-E.
i
= O,l, ••• ,nk ,
~i
K
We will estimate the difference
0
= -m
(t
j
k
= O,l, ••• ,n
l
.)/cr(tk ,~.).
uy,~
P(nF ) - ITP(F )
k
k
by a method originally used
25
by Berman [1] and improved by Cramer and Leadbetter [3], et ale
The present
version is due to Qualls and Watanabe [6].
IP( ~FkJ - .~ P(Fk)I ~
k=l
k=l
(4.5)
· J~
where
X~j;
.(101'
(x
0
p:
= _-..;;;.1__ e.xp (-.tB (A) )
211'11-A 2 c 2
(x, y; A'C)
=
~J
AC ij ) dA
with the correlation coefficient
B(A)
I"cHI
is the density of a standardized bivariate normal r. v.
ep( 0,0; p')
<t>
I I
l~k<!.~n
2
-
- 2ACxy +
2-2
~,i,Xl,j ~
Since, by Lemma 4.5,
B(A) ~ (2-0) log T
2
Y )/(1 - A c ).
1(2-o)logT, we have that
(2-2A'C)/(1-A 2c 2 ) ~ (2-0) (l+lcl)-l log T
0
and we get
the following upper bound for the integral in (4.5):
(211')-1 (1_c 2)-.t e.xp{-
.
-2)-.t
(1 -c
Here
1+0' > 1
~ E
-.t ,
for all
c.
-
Using
n
(l-a)~,
k
~
Il
l~k
T}.
and we can take
0
so small that
By Lemma 4.6b, there is an M such that
Ic(t,t+h)I ~ Mh- Y for all
~lk~1
2-~ log
1+1 cl
h ~ a~.
Since
Itk,i-tL,jl ~ Itk-t ' - (l-a)~ ~
l
we therefore have
log T, we get the following upper bound for the double sum:
nknl M i ~ -Y
~n
a
~ M iI~-Y
a
I I
Ik-:tl-Y
l~k<£.~n
E -.t
(log T)2
exp (- (1+0' )logT)
Ik-ll-Y T- l - o'
26
M "b.-Y
$
a
-M m
$
a
n
L
~l
(log T)2 T- l - o'
00
L v-Y
~l
b. -Y Tb. -1 (log'!') 2 ct-l-o'
=M
vtY
a
b.- l - y (log
Thus the double sum in (4.5) is bounded by a function of
zero as
T
+
b.
The bound may become large when
00.
T) 2 T-
09
•
T that tends to
and
a
are small but
can be kept small by regulating the rate with which they tend to zero.
Therefore the lemma is proved.
PROOF OF LE~
=
4.4.
0
have to prove that if
\-le
=
m (t)/o(t)~
uy
o(t)/cr(t),
for
i
= O,lt ••• ,~}
then
n
- L
k=l
log P(Fk )
+
6.
The idea in the proof is to approximate 6(t)
ent sums of
°
where
stationary~
I
k
by two differ-
separable stochastic processes:
=
';1-Pk1 0l+(t) + Pk 02(t)
(t)
=
.;l-Pk 2 01 - (t) + Pk 02(t)
°+
and
1
in each
are "smooth" simple Gaussian processes with mean zero
01
and the covariance functions
=
and
02
CO.6 { (1±E:)
~
\ 1.. 2
0
h}
respectively
is a process of "Markov" type with the covariance function
Ihl
for some constant
K~
< l/K
depending only on the upper bound of
~(t)
in
27
Lemma 4.6a.
The weights
Pk
shows that, for ,very
€ > 0
large
0
u,
t€I ,
k
and
<
are taken to be
there is a
-r'(t ).
k
A little reflexion
(l-a)A > 0 such that for all
h < (l-a)A it holds
<
2
2
(1-Pk ) cl-(h) + Pk c2 (h)
c(t,
2
> (l-Pk )
Write
A~
= €/ulr'(tk)1
and
(1-Pk2)~ (-ur'(tk)/~ + 2A~)
=
+
~ =
2
(l-Pk )
Then, for large
-4"
,
(-ur (tk+l)/~ - 2A~).
u and for
-~
-m(tk)
S -A 2 ur'(tk)+A~.
We now use the ordering property of the probabilities of extreme crossings derived by Slepian [7, Theorem 1] which says that, of two normalized
Gaussian processes, the one with the uniformly larger covariance function
has the larger probability of staying below a certain positive boundary.
Thus
P(F )
k
=
P(I(tk,i)
~
+
P(O (tk,i)
~
P( I l-Pk2 01 +(tk,i)
o
~
-m(tk,i) ,
<
-m(tk,i) ,
P(Pk 0 2 (tk,i)
P(6 +(h)
1
o
<
P (02 (h)
<
<
i
= 0, 1,
1,
i = 0,_ 1,
b.~,
~,
= 0,
-m(tk+l ) - A~,
<
+
<
i
·.. , ~)
• •• t
i
n )
k
= 0,
·.. , ~)
0 < h < (l-a)A)·
A~/Pk'
o<
h < (l-a)A)
=
P +
1 • P2 '
and
P(F )
k
S
P(o-(tk,i)
<
1, ••• , n )·
k
-m(tk,i) ,
i
= 0,
1, ••• , nk)
say,
28
~ p(/1- Pk 2 °l+(tk,i)
~)
i = 0, 1, ... , ~)
+ 1 - P(PkoZ(tk,i)
S
P(Ol-(h) < ~-,
Ol-
+ P{
has at least two
intervals
+ 1 - P(oz(h) <
The probabilities
0 < h < (l-a)A)
(tk,i' t k ,i+1)'
6~/Pk'
PI+ and PI-
Similarly
since
= (l- Pk2 )
we get, for large
P
u,
+ ; : 1 _ (1- a) 6
1
21T
S
1 _ (1-a)6
21T
~--crossings
the bounds
i
= 0,
1,
in one of
• a • ,
0 < h < (1-a)6)
are known, see Slepian [7]:
n -1
k
29
Since the process
01
is a simple
CO~-process
direct integrations show
that
(l-a)~/~
P {there is a sub-interval of length
which
01
has at least two crossings of the
~
COn6to~oP(at
~
COn6to~.(~/nk) ~ exp(-~
(0,
least two crossings in
3 -
_2
2
2
o(l)oexp(-u r'(t ) /2A 2 ).
k
=
)
~/~»
In total, we get
P
1
+
Q
1
>
-
1 _ (l-a)A (1-3<>211. 4
2'11'
\ /..2
For the probability
which implies that if
1 - P
2
P '
2
exp(
2, (t )2/ 2 >.. )
k
2 •
-u r
we use a result due to Pickands [5, Lemma 2.5]
~~/Pk ~
00
and
a
~
0
=
P(¢UPO~h~(l-a)~ °2(h) > ~~/Pk)
-
COn6to~(~~/Pk)
Here
2
(~~/Pk)
-1
then
2
2
exp(-~~ / 2Pk ).
obviously tends to infinity and if we use Lemma
4.5 it is easily proved that
We sum up the obtained estimates:
which give
+R
n
+R
n
30
(l-a)(1+3€)
n
3
2
L
R S COn6t o
6 2exp(-u r'(t )
k
n
k=l
where
e + Rn
2
IA Z)
as
+ 0
6 + 0
and
u
+ ~.
Similarly,
n
- L
k=l
~
logP(Fk)
(I-a) (1-3€)
3
n
k~l
6
21r
r::;
2
2
~~ exp(-u r' (tk ) /2A Z )
(I-a) (1-3€)3 e + 0(1).
Since
is arbitrary and
€
0
the asserted result.
5s
PROOF OF THEOREM
or
C=
co.
a is permitted to tend slowly to zero, we have
1.2B.
Let
T be TC or
co
T
according to if
C<
co
The crucial point in the proof is the observation that the
translated process
To (t)=o (T+t)
converges weakly to
;'(t).
More" precisely, let
P be probability
measures for the processes
{TO(t), lelst}
tively on the space
of continuous functions with the topology for
o
{C,C}
and
PT and
{;'(t), ItlSt}
0
respec-
uniform convergence.
LE~1MA
PROOFs
5.1.
The process
tends to
of
P => P as
T
-r"(s-t)
T
To
as
{To}
Le.
PT
converges weakly to
has the covariance function
T
P tend to those of
T
that the sequence
+ co,
c(T+s, T+t)
which "
so that all finite dimensional distributions
+ 00,
P.
P.
To establish that
P => P, we have to prove
T
is "tight", see e.g. Billingsley [2, eh. 2]. Suf-
ficient conditions for this are
31
b)
there are T , h , K such that s for
V(To(t+h) _ To(t»
~
Ihl : ; ho
T ,
KhZ.
<
= V(o(T» = c(T,T)
Here a) is obvious since V(To(O»
AZ < 00
T
0 0 0
as
T
+ 00.
which tends to
For b), we have
=
V(To(t+h) - To(t»
c(T+t+h, T+t+h)
+ c(T+t, T+t) - Zc(T+t+h, T+t)
=
Kl(r'(t+h)-r'(t»Z + KZ(r'(t+h)
-r' (t» (r VVV (t+h)-r'Yi (t»
+ K (r'''(t+h)-r"' (t»Z + K (r"(t+h)-r"(t»Z
3
for small h
::;;
KhZ
Let
E
and all T.
PROOF OF THEOREM 1.280
(l+e:)exp(-Ct)
+
me: (t)
=
{:
t < 0
t ~ 0
(l-E)exp(-Ct)
m - (t)
e:
>
0 be arbitrarily small and define
(C < 00)
(C
= 00)
(C < 00)
=
{tbe:
Also write, for fixed
Tm (t)
u
4
t < 0
t ~ 0
(C
= 00).
y
= muy (T+t).
Lemma 3.3 then gives that for sufficiently large
u
It I ::;; t o •
32
It is also always possible to find a
1 -
such that
t0
p(~'
+
(t) < m (t)
e:
for
t < -t )
0
<
1-P(~t(t)<
m -(t)
e:
for
t
<
< e:
-t )
0
and for large T
for
-T < t < -t)
o
< e:o
Then
P(m
uy
(t)+o(t) < 0
for
0 < t < T+x)
= p(Tmu (t)+To(t) < 0
S
p(To(t) < m + (t)
e:
-T < t < x)
for
-t
for
< t < x)
0
which, by Lemma Sol, tends to
for
S
P(~'(t)
-t
< me: + (t)
o
<
t
for
< x)
t < x) + e:o
Therefore
(5.1)
lim ~up P(r u
u+
>
T+x)
P(~'(t) < me: +(t)
S
oo
for
Similarly, we get the lower bound
p(To(t) < - Tm (t)
U
for
-t
0
- (l_p(To(t) < - Tm (t)
u
;;::
p(To(t) < m - (t)
e:
< t
for
for
<
x)
-T < t < t 0
-t 0 < t < x) - e:
which has the limit
P(~I (t) < m -(t)
e:
for
»
-t o < t < x) - e:
t < x)o
33
~
P(~'(t)
< m
e:
for
(t)
t <
x) - 2e:
and
lim
( 5.2)
~6 P(T
u+
oo
u
> T+x)
~
P(~'(t) < m -(t)
e:
for
t < x).
Since the right hand sides in (5.1) and (5.2) can be made arbitrarily close
to the required probability by taking
0
convergence.
6
D
PROOFS OF THEOREM
1a3
PROOF OF THEOREM
1.3.
ou-u = -~ u (1'),
u
we have
AND
1.4.
We already know that
= (r,
- 6(T+T»
where
o
0
< C <
p
00
(1)
small we have proved the asserted
e:
!
or
0
C
and
=
00.
-
ur(T+T)
Since
~
-1
C
T6(t)
T
= l' u -T
-ur (T+'T) +
-
exp(-CT)
or
0
p
0
= 6(T+t) = T6(0)
.!;
1'.
(1) -
Since
T6(":["» ,
according to if
+ It0 T6(s)ds we can ex-
tend Lemma 5.1 and use the tightness criterion on each of the conditional
processes
~ (0) +
= x).
T
IT 6(0)
(0(0)
ro ~'(s) ds = ~ (":[") ,
PROOF OF THEOREM
We conclude that
behaves like
and get the theore11l;. 0
1.4;
oulu = (u-~ u (1' u»/u =
1
1 - r(1' ) + u- n (A 2r(1' )+r"(1' »
u
u
u
u
- u- 1 6(1' ).
u
Since
l'
u
tends to infinity, all the variables
tend to zero in probability.
We estimate
r(1'),
u
u- 16(1')
u
by
r"(1')
u
and
u
-1
nu
34
=
ThEm, for all
u
-1
t, x . > 0
u
u
o(t)dtl
1 - P('u>t u ) - P(.6u.PO<t<t lo(t)l>xu)'
u
t,
Then
luri(t )1 ~ Kut -y -+ 0
then
peL >t ) -+ 0,
U
U
y > 1
T
1/y
K u +K ,
1
2
peL >t ) -+ 0
u
Now let
:S;
and
1 < y' < y
such that
and define
-rll(t)/r'(t) -+ 0,
M > 0,
we observe that for
00,
so that for large
This in turn implies that
e: > 0,
we have
u we have
tu-T -+
00
and consequently that
even in this case,
t
-+
so that
00
Um P (.6UPO<t<t
u-+ oo
0 > 0
and we conclude that if
-r"(t)/r'(t) -+ C > 0,
~n6u+oo lur~(T-e:)1 ~
u
Take
K> 0
and a constant
u
lim
S
u
u u
Ir'(t)1 :S; Kt- Y for large
If
Io(t).I
we have
By assumption, there is a
U
u-1 'u .6UPO<t<,
:s;
u-1 t x )
:S;
~
'u
IJo
20 2 (t):S; (2+0)A 2
10 (t) I<xu)
= Um
u-+
u
The probability of no crossings in
oo
(t , t )
-
u
which tends to zero if
u -+
00,
Therefore
t ~ t
P(.6UP t <t<t
and
lo(t)l<x ).
u
u
is bounded by the expected num-
ber of crossings which, in turn, is not more than
:S; K t x exp(-x 2/(2+o)A 2 )
u
4 u u
for
35
P (.6UPO<t<t
u
-+
lo(t)l>x)
u
Since, furthermore,
u
-1
t
x
U U
O.
-+ 0,
the theorem is proved.
o
REFERENCES
[1]
Berman, S.M.:
quences.
Limit theorems for the maximum term in stationary se-
Ann. math. Statistics 35, 502-516 (1964).
Convergence of pztobabiUty measures.
[2]
Billingsley, P.:
Wiley 1968.
New York:
[3]
Cramer, H., Leadbetter, M.R.: Stationary and related stochastic
processes. New York: Wiley 1967.
[4]
Lindgren, G.: Wave-length and amplitude for a stationary process after
a high maximum. Institute of Statistics Mimeo Series No. 742,
University of North Carolina Statistics Department (1971).
[5]
Pickands, J.:
process.
Upcrossing probabilities for a stationary Gaussian
Trans. Amer. Math. Soc. 145, 51-73 (1969).
[6]
Qualls, C., Watanabe, H.: An asymptotic 0-1 behavior of Gaussian
processes. Institute of Statistics Mimeo Series No. 725,
University of North Carolina Statistics Department (1970).
[7]
Slepian, D.:
The one-sided barrier problem for Gaussian noise.
Bell. Sys. Tech. J. 41, 463-501 (1962).
[8]
Qualls, C.: On the joint distribution of crossings of high multiple
levels by a stationary Gaussian process. Arkiv for matematik 8,
No. 15, 129-137 (1969).