Digital Signal Processing Reference
In-Depth Information
DEFINITION 4
Aprocess
is said to be asymptotically second-order self-similar if its autoco-
variance and the autocovariance of its aggregate process are related as
{
X
k
}
k
∈Z
r
(
m
)
X
lim
m
(τ )
=
r
X
(τ ).
(6.18)
→∞
It is shown in Reference 2 that Equation 6.6 implies Equation 6.18; in other
words, a long-range dependent process is also asymptotically second-order
self-similar.
The preceding discussion indicates that self-similarity and long-range de-
pendence are different concepts. However, in the case of second-order self-
similarity, self-similarity implies long-range dependence, and vice versa. For
this reason, in the literature and also in this chapter (unless otherwise spec-
ified), the terms
self-similarity
and
long-range dependence
are used in an inter-
changeable fashion.
6.2.2
Heavy-Tailed Distributions and Impulsiveness
DEFINITION 5
A random variable X is heavy-tail distributed with index
α
if
cx
−
α
L
P
(
X
≥
x
)
∼
(
x
)
,
x
→∞
,
(6.19)
for c
>
0
, and
0
<α<
2
, where L
(
x
)
is a slowly varying function such that L
(
x
)
is positive for large x and
lim
x
→∞
L
(
bx
)/
L
(
x
)
=
1
for any positive b.
Intuitively, a heavy-tail distributed random variable can fluctuate far away
from its mean value (defined only when 1
<α<
2), with nonnegligible
probability.
The simplest example of a heavy-tail distribution is the Pareto distribution,*
which is defined in terms of its complementary distribution function (survival
function) as
(
x
k
0
)
−
α
,
x
≥
k
0
,
F
(
x
)
:
=
P
(
X
≥
x
)
=
(6.20)
<
1
,
x
k
0
,
where
k
0
is positive constant and 0
2.
For heavy-tail distributions,
p
th order statistics are finite if and only if
<α<
.Adirect consequence of this property is that for heavy-tail distributed
random variables, the second-order statistics are infinite (the mean is infinite
if
p
≤
α
α<
1).
*Technically, Pareto distribution includes a large class of distributions, namely,
Pareto I, II, III, and
IV,
respectively, according to the increasing complexity in the format of the distribution functions.
Here we concentrate on the Pareto I distribution.
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