Maxima and high level excursions of stationary Gaussian processes
Authors: Berman Simeon M. 
Published: July 2010

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0.1 right-parenthesis limit Underscript t right-arrow 0 Endscripts StartFraction g left-parenthesis c t right-parenthesis Over g left-parenthesis t right-parenthesis EndFraction equals 1 comma f o r e v e r y c greater-than 0 comma">

(
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0.2 right-parenthesis 1 minus r left-parenthesis t right-parenthesis tilde g left-parenthesis StartAbsoluteValue t EndAbsoluteValue right-parenthesis StartAbsoluteValue t EndAbsoluteValue Superscript alpha Baseline comma t right-arrow 0 period">

(
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0.7 right-parenthesis limit Underscript h right-arrow 0 Endscripts log h integral Subscript negative normal infinity Superscript normal infinity Baseline vertical-bar f left-parenthesis lamda plus h right-parenthesis minus f left-parenthesis lamda right-parenthesis vertical-bar d lamda equals 0 comma">

(
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0.8 right-parenthesis exp left-bracket constant integral Subscript 0 Superscript normal infinity Baseline left-parenthesis e Superscript minus lamda xi Baseline minus 1 right-parenthesis d normal upper Psi Subscript alpha Baseline left-parenthesis x right-parenthesis right-bracket comma lamda greater-than 0 period">

(
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.

\]