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argument that captures Lord Rayleigh's approach is now given. Let us consider a
variable
Q
indexed by the discrete time
j
=
1
,
2
,...,
N
which defines the set of values
{
Q
j
} determined by the equation
Q
j
+
1
=
Q
j
+
U
j
,
(4.1)
where
U
is a random variable taking on the value of +1 or
−
1. The solution to the
iterative equation (
4.1
)isgivenby
N
Q
N
=
U
j
,
(4.2)
j
=
1
which is the sum of
N
identically distributed random variables. The set of values taken
on by the random variable
is called a Bernoulli sequence after
the scientist Daniel Bernoulli. This is the same Bernoulli who introduced the utility
function into science and who first studied how people make decisions under uncer-
tainty. The curly brackets denote a particular realization of such a sequence, each term
of which can be obtained by flipping a coin to decide whether a +1 or a
{
U
j
}={+
1
,
+
1
,
−
1
,...
}
1 is put in
a specific location. A Bernoulli sequence is a completely random process, by which
we mean that no element in the sequence is dependent upon any other element in the
sequence; each toss of the coin is independent of every other toss and therefore so too
is each element in the sequence independent of every other element. Here we visualize
this process by considering a walker on a one-dimensional lattice taking a step in the
indicated direction, to the right for +1 and to the left for
−
1, Figure
4.1
.
Consequently, summing over the elements of a Bernoulli sequence yields a simple
random-walk process. Pearson's question regarding the probability of being at a spe-
cific location after taking
N
steps can now be formally addressed. We emphasize that
the conclusions we are able to draw from this simple random-walk argument are inde-
pendent of the values of the constant step length put into the random sequence of steps
and we use the Bernoulli sequence both because of its conceptual simplicity and on
account of its usefulness later when we develop some rather formidable formalisms.
It is useful to briefly explore the properties of the Bernoulli sequence. Every computer
has a program to generate a random number between 0 and 1 without bias, meaning that
any value on the interval [0, 1] is equally likely. If we take the interval and divide it into
two equal parts, [0, 0.5] and [0.5, 1], then a random number falling on the left interval
is given the value
−
1 and a random number falling on the right interval is given the
value +1; the probability of falling on either side of 0.5 is equal to one half, just as in a
coin toss. In this way a Bernoulli sequence can be generated on the computer. Suppose
that we generate 10,000 data points
−
and we want to determine whether in fact the
computer-generated sequence is a completely random process. Consider the discrete
autocorrelation function
{
U
j
}
Figure 4.1.
Homogeneous lattice sites for a one-dimensional random walk.
