Digital Signal Processing Reference
In-Depth Information
6.2
Preliminaries—Self-Similarity, Long-Range Dependence,
and Impulsiveness
In this section, we formulate the concepts of long-range dependence and
self-similarity, and point out how they are related. In a communication con-
text, self-similarity is closely related to heavy-tail distributions. We provide
the definition of heavy-tailed distributions, and discuss two special classes,
namely, the Pareto distributions and the
α
-stable distributions.
6.2.1
Long-Range Dependence and Self-Similarity
6.2.1.1
Long-Range Dependence
{
}
k
Let
be a discrete-time second-order stationary stochastic process with
finite second-order statistics, mean
X
k
∈Z
µ
=
σ
2
=
{
(
−
µ)
2
}
E
X
k
, and variance
E
X
k
.
The autocovariance of
{
X
k
}
is denoted as
1
σ
r
(τ )
=
2
E
{
(
X
k
−
µ)(
X
k
+
τ
−
µ)
}
.
(6.1)
DEFINITION 1
{
X
k
}
k
∈Z
is a long-range dependent process with Hurst parameter H, if for
τ
∈ Z
,
(τ )
k
2
H
−
2
r
lim
k
=
c
r
,
1
/
2
<
H
<
1
(6.2)
→∞
where
0
<
c
r
<
∞
is a constant.
An equivalent definition
1
of long-range dependence (LRD) is based on the
spectrum of the process (provided it exists),
f
(λ)
.Aprocess is long-range
dependent if its spectrum satisfies
1
−
2
H
lim
λ
→
f
(λ)/
|
λ
|
=
c
f
,
1
/
2
<
H
<
1
,
(6.3)
0
where
c
f
is a positive constant. Equation 6.2 implies that for long-range de-
pendent processes, it holds that:
∞
r
(τ )
=∞
,
(6.4)
τ
=−∞
1
2
for
1. Thus, although at high lags the autocovariance is small, its
cumulative effect is large, giving rise to a behavior that is distinctly different
<
H
<
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