The Normal Distribution and Ergodicity

The normal distribution, which is represented by the familiar bell curve, is probably the most important probability distribution in statistics. The reason for this is the Central Limit Theorem, which holds that the sum of a large number of identically distributed but independent random outcomes will tend toward having a normal distribution as the number of included results increases, regardless of their original probability distribution.

For the purposes of finance, an important aspect of the Central Limit Theorem is that normal distributions will be ergodic, which is to say that they will exhibit the fractal quality of having indistinguishable characteristics regardless of at what scale they are viewed: a variable that moves with instantaneous, normally distributed perturbations will create a time path that looks the same whether you are looking at movements over one minute, one day, one month, or several years. The reason for this is that, as with every other sort of variable, sums of normally distributed variables tend to be normally distributed variables; the exception is that normally distributed variables also start that way.

To a very broad degree in finance, measurement time sequences are arbitrary. There is no reason to think that the "proper" span of time over which to consider results is a minute, a day, a month, or a year. There are some cosmological conditions -- such as the passing of night and day and the changing of the seasons -- and some social conditions -- such as regularly scheduled weekends and holidays during which financial and economic activity is restricted -- that will create regular cycles that might be taken into account for the purpose of measuring financial results, but beyond this there is no reason to think that any span of measurement is better than any other.

The usefulness of the normal distribution's ergodic property is so great that, in my estimation, it is often worthwhile to use the distribution even when it is known not to match experience for the purpose to which it is put.