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dimension. These are called fractal time series and when they are also irregular, which
is to say random, we call them random fractal time series.
Let us consider how the fractal idea can be married to the notion of probability. We do
this by constructing a fractal random process for which the moments, such as the arith-
metic mean, may or might not exist. It does seem that it ought to be possible to repeat
a given experiment a large number of times, take the measurements of a given quantity,
and obtain the data sequence
Q
1
,
sum the measured values,
j
=
1
Q
j
,
divide by the number of data points
N
, and thereby determine their average value
Q
,
Q
2
,...,
Q
N
,
N
1
N
Q
=
Q
j
.
(2.32)
j
=
1
This procedure gives us the arithmetic mean of the measured process. In statistical the-
ory, this one realization of the measured values is one sample taken from the entire
population of all the possible experimental values. The mean determined from this sam-
ple of data is called the sample mean and the mean of the entire population is called
the population mean. The sample mean can always be determined using the available
data to carry out the averaging. The population mean, on the other hand, is a theoreti-
cal concept, and the sample mean provides an estimate for it. If the population mean is
finite, then as we collect more and more data the sample mean converges to a fixed value
that is identified as the best estimate of the population mean. This need not be the case,
however. If, for example, as we collect and analyze more data the value of the sample
mean keeps increasing, we must conclude that the population mean of the process does
not exist, which is to say that the mean is not finite. This is exactly what happens for
some fractal processes. Bassingthwaighte
et al
.[
7
] point out that, for a fractal process,
as more data are analyzed,
... rather than converge to a sensible value, the mean continues to increase toward ever larger
values or decrease toward ever smaller values.
The nineteenth-century social philosopher Quetelet proposed that the mean alone can
represent the ideals of a society in politics, aesthetics and morals. His view involved,
in the words of Porter [
60
], “a will to mediocrity.” Like Cannon in medicine [
12
],
Quetelet's notion involved a tendency for the enlightened to resist the influences of
external circumstances and to seek a return to a normal and balanced state. The idea that
a property of society could be defined, but not have a finite average, would have been
anathema to him as well as to most other nineteenth-century scientists. We mention the
few exceptions who saw beyond the normal distribution in due course.
For a concrete example of such a fractal statistical process consider the following
game of chance. A player tosses a coin and the bank (a second player with a great deal
of money) agrees to pay the first player $2 if a head occurs on the first toss, $4 if the first
head occurs on the second toss, $8 if the player obtains the first head on the third toss,
and so on, doubling the prize every time a winning first head is delayed an additional
toss. What we want to know is the player's expected winnings in this game.
