A law of iterated logarithm for stationary Gaussian processes
Authors: Pathak Pramod K. , Qualls Clifford 
Published: July 2010

In this article the following results are established. Theorem A. Let

{
X
(
t
)
:
0
⩽
t
>
∞
}

\{ X(t):0 \leqslant t > \infty \}

be a stationary Gaussian process with continuous sample functions and

E
[
X
(
t
)
]
≡
0

E[X(t)] \equiv 0

. Suppose that the covariance function

r
(
t
)

r(t)

satisfies the following conditions. (a)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis t right-parenthesis equals 1 minus StartAbsoluteValue t EndAbsoluteValue Superscript alpha Baseline upper H left-parenthesis t right-parenthesis plus o left-parenthesis StartAbsoluteValue t EndAbsoluteValue Superscript alpha Baseline upper H left-parenthesis t right-parenthesis right-parenthesis">

r
(
t
)
=
1
−

|

t

|

α

H
(
t
)
+
o
(

|

t

|

α

H
(
t
)
)

r(t) = 1 - |t{|^\alpha }H(t) + o(|t{|^\alpha }H(t))

as

t
→
0

t \to 0

, where

0
>
α
⩽
2

0 > \alpha \leqslant 2

and

H
H

varies slowly at zero, and (b)

r
(
t
)
=
O
(
1

/

log
⁡
t
)

r(t) = O(1/\log t)

as

t
→
∞

t \to \infty

Then for any nondecreasing positive function

ϕ
(
t
)

\phi (t)

defined on

[
a
,
∞
)

[a,\infty )

with

ϕ
(
∞
)
=
∞
,
P
[
X
(
t
)
>
ϕ
(
t
)

\phi (\infty ) = \infty ,P[X(t) > \phi (t)

i.o. for some sequence

t
n

→
∞
]
=
0
o
r
1

{t_n} \to \infty ] = 0or1

according as the integral
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I left-parenthesis phi right-parenthesis equals integral Subscript a Superscript normal infinity Baseline g left-parenthesis phi left-parenthesis t right-parenthesis right-parenthesis phi left-parenthesis t right-parenthesis Superscript negative 1 Baseline exp left-parenthesis minus phi squared left-parenthesis t right-parenthesis slash 2 right-parenthesis d t">

I
(
ϕ
)
=

∫
a
∞

g
(
ϕ
(
t
)
)
ϕ

(
t
)

−
1

exp
⁡
(
−

ϕ
2

(
t
)

/

2
)
d
t

I(\phi ) = \int _a^\infty {g(\phi (t))\phi {{(t)}^{ - 1}}\exp ( - {\phi ^2}(t)/2)dt}

is finite or infinite, where

g
(
x
)
=
1

/

σ

∼
−
1

(
1

/

x
)

g(x) = 1/_\sigma ^{ \sim - 1}(1/x)

is a regularly varying function with exponent

2

/

α

2/\alpha

and

σ

∼
2

(
t
)
=
2

|

t

|

α

H
(
t
)

_\sigma ^{ \sim 2}(t) = 2|t{|^\alpha }H(t)

.

Theorem C. Let

{

X
n

:
n
⩾
1
}

\{ {X_n}:n \geqslant 1\}

be a stationary Gaussian sequence with zero mean and unit variance. Suppose that its covariance function satisfies, for some

γ
>
0
,
r
(
n
)
=
O
(
1

/

n
γ

)

a
s

n
→
∞

\gamma > 0,r(n) = O(1/{n^\gamma })\;as\;n \to \infty

. Let

{
ϕ
(
n
)
:
n
⩾
1
}

\{ \phi (n):n \geqslant 1\}

be a nondecreasing sequence of positive numbers with

lim

n
→
∞

ϕ
(
n
)
=
∞

{\lim _{n \to \infty }}\phi (n) = \infty

; suppose that

Σ
(
1

/

ϕ
(
n
)
)
exp
⁡
(
−

ϕ
2

(
n
)

/

2
)
=
∞

\Sigma (1/\phi (n))\exp ( - {\phi ^2}(n)/2) = \infty

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts sigma-summation Underscript 1 less-than-or-equal-to k less-than-or-equal-to n Endscripts upper I Subscript k Baseline slash sigma-summation Underscript 1 less-than-or-equal-to k less-than-or-equal-to n Endscripts upper E left-bracket upper I Subscript k Baseline right-bracket equals 1 normal a period normal s period comma">

lim

n
→
∞

∑

1
≤
k
≤
n

I
k

/

∑

1
≤
k
≤
n

E
[

I
k

]
=
1

a
.
s
.

,

\lim _{n \to \infty } \sum _{1 \leq k \leq n} I_k / \sum _{1 \leq k \leq n} E[I_k] = 1\quad \mathrm {a.s.},

\]
where

I
k

I_k

denotes the indicator function of the event

{

X
k

>
ϕ
(
k
)
}

\{ {X_k} > \phi (k)\}

.