San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

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) = σs=1tU(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 = ∫rr+Δr((n/2π)½/σexp(−zn/σ²))dz
which, for small Δr, is approximately

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

P(r) = (N/n)((n/2π)½/σexp(−rn/σ²))
or, equivalently
P(r) = N((1/2πn)½/σexp(−rn/σ²))

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