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and can assist in formulating new experiments that can discriminate between different
hypotheses. Simple random walks were, in fact, among the most quoted models of spec-
ulation in the stock market [
32
] until fairly recently, when the fractal market hypothesis
was introduced; see Peters [
37
] or Mandelbrot [
29
] for a more complete discussion
and the relatively new discipline of econophysics [
30
]. This example is taken up again
subsequently, but for the moment we follow the random-walk strategy to gain an under-
standing of how to construct a stochastic process as we increase the complexity of the
phenomenon being modeled.
In this simple random-walk model we assume that the walker takes one step in each
unit of time, so that the “time” variable can be defined by
t
t
is the
time interval between steps and is usually taken to be unity between elements in the
Bernoulli time sequence. The time becomes continuous as the time increment becomes
a differential,
=
N
t
, where
, in such
a way that their product remains constant, that being the continuous time
t
. Physically,
time can be considered a continuous variable even though it is treated as discrete in the
walk as long as the sampling time of the experiment is sufficiently small and the sample
size is sufficiently large.
We refer to (
4.2
) as a diffusion trajectory since it signifies the location of the walker
after each step. This nomenclature stems from the use of single-particle trajectories to
denote the instantaneous position of a diffusing particle. The mean-square value of the
diffusion trajectory is given by
t
→
0, and the number of steps becomes very large,
N
→∞
Q
2
N
N
N
N
N
U
j
U
k
=
=
1
δ
jk
=
N
.
(4.5)
j
=
1
k
=
1
j
=
1
k
=
The angle brackets denote an average over an ensemble of realizations of Bernoulli
sequences and, since the elements in the Bernoulli sequence are independent of one
another, the average vanishes except when the indices are equal, as formally denoted
by the Kronecker delta
k
. Consequently the random
diffusion variable, as measured by its root-mean-square (rms) value, increases as the
square root of the number of steps taken by the walker. This increas
e in
the rms value
of the random variable suggests that the normalized quantity
Q
N
/
δ
jk
=
1if
j
=
k
;
δ
jk
=
0if
j
=
√
N
will remain of
order unity as
N
becomes arbitrarily large. Interpreting the number of steps as the time
(
4.5
) shows that the second moment increases linearly with time, giving rise to a value
α
=
=
.
5 for the scaling index introduced in the first chapter.
This analysis can be extended beyond the Bernoulli sequence by introducing the
probability of making a transition between two sites arbitrarily far apart. For this more
general random walk the equation for the probability of being at a lattice site
s
after
n
H
0
+
1 steps is given by
P
n
q
,
q
)
P
n
+
1
(
q
)
=
p
(
q
−
(4.6)
q
where the probability of stepping from site
q
q
)
to site
q
is
p
(
q
−
.Inthesimple
case of the Bernoulli sequence
q
is only one lattice space to the right or to the left of
