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fractal dimensions. Consider an autocorrelation function in terms of a dimensionless
variable of the following form:
1
C
(
t
)
=
(2.46)
t
α
)
β/α
(
1
+
(see Gneiting and Schlather [
21
] for a complete discussion of this autocorrelation
function and the implications of its form). Any combination of parameters 0
<α
≤
2
and
0 is allowed, in which case (
2.46
) is referred to as the Cauchy class of
autocorrelation functions. Now consider two asymptotic limits.
In the short-time limit the autocorrelation function can be expanded in a Taylor series
to obtain the power-law form
β>
−
β
|
α
as
t
C
(
t
)
≈
1
α
|
t
→
0
,
(2.47)
for the range of parameter values given above. The autocorrelation functions in this
case are realizations of a random process in an
E
-dimensional Euclidean space that has
a fractal dimension given by
D
=
E
+
1
−
α/
2
(2.48)
with probability unity [
21
]. In the one-dimensional case (
E
=
1) the power spectrum
corresponding to (
2.47
)istheinversepowerlaw
1
|
ω
|
α
+
1
as
S
(ω)
≈
ω
→∞
.
(2.49)
Note that the
t
→
0 limit of the autocorrelation function corresponds to the asymptotic
limit as
of the spectrum since the two are Fourier-transform pairs. Conse-
quently, the inverse power-law spectrum obtained in this way can be expressed as a
straight-line segment on bi-logarithmic graph paper. The line segment corresponds to
the spectrum and has a slope related to the fractal dimension in (
2.48
)by
ω
→∞
α
+
1
=
5
−
2
D
.
(2.50)
At the long-time extreme, the autocorrelation function (
2.46
) collapses to
1
C
(
t
)
≈
as
t
→∞
,
(2.51)
|
β
|
t
where the inverse power-law indicates a long-time memory when
β>
0. In this case
we introduce the Hurst exponent
2
H
=
2
−
β,
where 2
≥
β>
0
(2.52)
and the index
H
was introduced by Mandelbrot [
44
] to honor the civil engineer Hurst
who first investigated processes with such long-term memory. Here again the Fourier
transform of the autocorrelation function yields the power spectrum
1
(ω)
∝
|
ω
|
β
−
1
S
=
1
,
(2.53)
2
H
−
|
ω
|
