Let
X
(
t
)
,
t
≧
0
X(t),t \geqq 0
, be a stationary Gaussian process with mean 0, variance 1 and covariance function
r
(
t
)
r(t)
. The sample functions are assumed to be continuous on every interval. Let
r
(
t
)
r(t)
be continuous and nonperiodic. Suppose that there exists
α
,
0
>
α
≦
2
\alpha , 0 > \alpha \leqq 2
, and a continuous, increasing function
g
(
t
)
,
t
≧
0
g(t),t \geqq 0
, satisfying
\[
(
0.1
)
lim
t
→
0
g
(
c
t
)
g
(
t
)
=
1
,
f
o
r
e
v
e
r
y
c
>
0
,
(0.1)\quad \lim \limits _{t \to 0} \frac {{g(ct)}}{{g(t)}} = 1,\quad for\;every\;c > 0,
\]
such that
\[
(
0.2
)
1
−
r
(
t
)
∼
g
(
|
t
|
)
|
t
|
α
,
t
→
0.
(0.2)\quad 1 - r(t) \sim g(|t|)|t{|^\alpha },\quad t \to 0.
\]
For
u
>
0
u > 0
, let
v
v
be defined (in terms of
u
u
) as the unique solution of
\[
(
0.3
)
u
2
g
(
1
/
v
)
v
−
α
=
1.
(0.3)\quad {u^2}g(1/v){v^{ - \alpha }} = 1.
\]
Let
I
A
{I_A}
be the indicator of the event
A
A
; then
\[
∫
0
T
I
[
X
(
s
)
>
u
]
d
s
\int _0^T {{I_{[X(s) > u]}}ds}
\]
represents the time spent above
u
u
by
X
(
s
)
,
0
≦
s
≦
T
X(s),0 \leqq s \leqq T
. It is shown that the conditional distribution of
\[
(
0.4
)
v
∫
0
T
I
[
X
(
s
)
>
u
]
d
s
,
(0.4)\quad v\int _0^T {{I_{[X(s) > u]}}ds,}
\]
given that it is positive, converges for fixed
T
T
and
u
→
∞
u \to \infty
to a limiting distribution
Ψ
α
{\Psi _\alpha }
, which depends only on
α
\alpha
but not on
T
T
or
g
g
. Let
F
(
λ
)
F(\lambda )
be the spectral distribution function corresponding to
r
(
t
)
r(t)
. Let
F
(
p
)
(
λ
)
{F^{(p)}}(\lambda )
be the iterated
p
p
-fold convolution of
F
(
λ
)
F(\lambda )
. If, in addition to (0.2), it is assumed that
\[
(
0.5
)
F
(
p
)
i
s
a
b
s
o
l
u
t
e
l
y
c
o
n
t
i
n
u
o
u
s
f
o
r
s
o
m
e
p
>
0
,
(0.5)\quad {F^{(p)}}\;is\;absolutely\;continuous\;for\;some\;p > 0,
\]
then
max
(
X
(
s
)
:
0
≦
s
≦
t
)
\max (X(s):0 \leqq s \leqq t)
, properly normalized, has, for
t
→
∞
t \to \infty
, the limiting extreme value distribution
exp
(
−
e
−
x
)
\exp ( - {e^{ - x}})
. If, in addition to (0.2), it is assumed that
\[
(
0.6
)
F
(
λ
)
i
s
a
b
s
o
l
u
t
e
l
y
c
o
n
t
i
n
u
o
u
s
w
i
t
h
t
h
e
d
e
r
i
v
a
t
i
v
e
f
(
λ
)
,
(0.6)\quad F(\lambda )\;is\; absolutely \;continuous\; with\; the\; derivative\; f(\lambda ),
\]
and
\[
(
0.7
)
lim
h
→
0
log
h
∫
−
∞
∞
|
f
(
λ
+
h
)
−
f
(
λ
)
|
d
λ
=
0
,
(0.7)\quad \lim \limits _{h \to 0} \log h\int _{ - \infty }^\infty {|f(\lambda } + h) - f(\lambda )|d\lambda = 0,
\]
then (0.4) has, for
u
→
∞
u \to \infty
and
T
→
∞
T \to \infty
, a limiting distribution whose Laplace-Stieltjes transform is
\[
(
0.8
)
exp
[
constant
∫
0
∞
(
e
−
λ
ξ
−
1
)
d
Ψ
α
(
x
)
]
,
λ
>
0.
(0.8)\quad \exp [{\text {constant}}\int _0^\infty {} ({e^{ - \lambda \xi }} - 1)d{\Psi _\alpha }(x)],\quad \lambda > 0.
\]