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and the second moment is realized by the integral
Q
2
q
2
p
=
(
q
)
dq
,
(2.76)
R
resulting in the time-independent variance
q
2
2
2
σ
=
−
q
.
(2.77)
Notice that we distinguish the average determined by the pdf from the average deter-
mined from the discrete values of the dynamic variable given by (
2.32
), where the sum
is over the discrete time index. If the measurements are taken at equally spaced time
points that are sufficiently close together we could use the continuous time average
given by
t
1
t
t
)
dt
.
Q
=
Q
(
(2.78)
0
The ergodic hypothesis requires that
Q
after sufficiently long times.
If the random variable is determined by an ensemble of trajectories, then the statistics
depend on time, and the probability density function also depends on time,
p
Q
=
(
q
,
t
)
.
Consequently the variance is also time-dependent:
q
2
t
2
2
σ(
t
)
=
;
−
q
;
t
.
(2.79)
The divergence of the variance with increasing amounts of data (for time series this
means longer times) is a consequence of the scaling property of fractals. In this con-
text scaling means that the small irregularities at small scales are reproduced as larger
irregularities at larger scales. These increasingly larger fluctuations become apparent as
additional data are collected. Hence, as additional data from a fractal object or process
are analyzed, these ever larger irregularities increase the measured value of the variance.
If the limit continues to increase, the variance continues to increase; that is, the variance
becomes infinite. We lose the variance as a reliable measure of the data, but scaling
implies that there is a fractal dimension that can replace the variance as a new measure;
the variance is replaced with a scaling index.
The increased variability indicated by the divergence of the variance has been
observed in the price of cotton [
45
], the measured density of microspheres deposited
in tissue to determine the volume of blood flow for each gram of heart tissue [
6
] and
the growth in the variance of the mean firing rate of primary auditory nerve fibers with
increasing length of time series [
81
,
82
]. Many other examples of the application of this
concept in physiologic contexts may be found in reviews [
7
,
95
]. We shall discuss a
select subset of these examples in detail later.
It is sometimes confusing, so it should be stated explicitly that fractal random
processes are indicated by two distinct kinds of inverse power laws, namely inverse
power-law spectra and inverse power-law pdfs. The inverse power-law spectrum, or
equivalently the corresponding autocorrelation function, indicates the existence of long-
term memory, far beyond that of the familiar exponential. In probability the inverse
power law indicates the contribution of terms that are quite remote from the mean, even
