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which, when inserted into (
4.58
) and on rearranging terms, yields
D
D
ξ
ξ
S
Q
(ω)
=
α
=
for
ω
→
0
.
(4.60)
2
ω
2
α
[
ω/
]
)
(
2sin
2
Equation (
4.60
) is the inverse power-law spectrum for the fractional-differenced white
noise in the low-frequency limit. This inverse power-law behavior in the spectrum is
often called 1/
f
noise and the underlying process is described as a 1/
f
phenomenon.
From this analysis we see that the infinite moving-average representation of the
fractional-differenced white-noise process shows that the statistics of
Q
j
are the same
as those of
ξ
j
are
Gaussian, so too are the statistics of the observed process. The spectrum of the ran-
dom force is flat, since it is white noise, and the spectrum of the observed time series
is an inverse power law. From these analytic results we conclude that
Q
j
is analogous
to fractional Brownian motion. The analogy is complete if we set
ξ
j
since (
4.51
) is a linear equation. Thus, because the statistics of
α
=
H
−
1
/
2sothe
spectrum can be written as
D
ξ
S
Q
(ω)
=
for
ω
→
0
.
(4.61)
ω
2
H
−
1
We now consider a random-walk process driven by fluctuations with long-time
memory using
X
j
+
1
=
X
j
+
Q
j
(4.62)
with the solution given by the sum of measurements, each with a long-time memory,
j
X
j
=
Q
k
.
(4.63)
k
=
0
The autocorrelation function for this process is
)
=
X
j
X
j
−
t
∝
t
2
H
C
(
t
,
(4.64)
where we again use
α
=
H
−
1
/
2. The corresponding spectrum is given by
D
ξ
S
X
(ω)
∝
1
,
(4.65)
2
H
+
ω
which can be seen intuitively from the fact that the time series
X
j
is essentially the time
integral of
Q
j
and consequently the spectra are related by
S
Q
(ω)
ω
S
X
(ω)
=
.
(4.66)
2
The long-time memory of financial time-series data has been well established starting
from the application of fractional difference equations by Hosking [
24
] in 1981. One of
the more recent applications of these ideas to financial time-series data was given by
Pesee [
36
], who studied the daily changes in the values of various currencies against
the US Dollar (USD), including the French Franc (FRF), the Deutsche Mark (DM),
the Euro (EUR) and the Japanese Yen (JPY). Each of the time series has a variability
