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The Probability Distribution of
Trends for Cumulative Sums of Random
Disturbances

Let U(t) for 0≤t≤N be a set of independent, normally-distributed random
variables with means of zero and variances of σ&su2;. The distribution of

T(t) = σ_{s=1}^{t}U(s)

is also a normally distributed variable with a mean of zero but a variance of
tσ². The growth rate r over an interval of t is

r = [T(t)−T(0)]/t

is also normal and of mean zero but with a variance of σ²/t.

The Distribution of Growth Rates

Suppose the interval of size N is subdivided into m intervals of size n; i.e., nm=N.
Let p be the probability that the growth rate over a subinterval is in the
range r to r+Δr. Then the probability of the growth rate not being in that
range is (1−p). The probability of none of the m subintervals not having a
growth rate in that range is (1−p)^{m}. Therefore the probability of
at least one of the subintervals having a growth rate in that range is
[1−(1−p)^{m}]. Let the probability density funcion for this
probability distribution be denoted as P(r).

Since the probability distribution of the growth rate over an interval of size m
is a normal distribution of mean zero and variance σ²/n the probability
that the growth rate is between r and r+Δr is

p = ∫_{r}^{r+Δr}((n/2π)^{½}/σexp(−zn/σ²))dz
which, for small Δr, is approximately
((n/2π)^{½}/σexp(−rn/σ²))Δr

For an infinitesimal range in the growth rate the
probability of at least one subinterval having a growth rate between r and r+dr, [1−(1−p)^{m}], reduces to mp. Therefore