Stable distributions

See also: Distributions introduction

A distribution is said to be stable if summing independent random variables from that distribution results in a random variable from that same distribution. These distributions are called stable because their shape is unchanged (stable) when they are being summed.

There are three stable distributions for which there is a closed form (meaning a simple algebraic equation) of the probability density function: the Normal distribution, the Cauchy distribution and the Levy distribution.

One can prove that the general form of adding random variables from a stable distribution together has the form:

with

 

where a is called the index of stability.

Normal

For the Normal distribution, this index of stability is a = 2. Because the parametrisation used for stable laws is different than the one used for the Normal distribution, the general formula doesn't hold. Summing n random variables with mean m and standard deviation s results in a random variable with mean n*m and standard deviation SQRT(n)*s.

Cauchy

For the Cauchy distribution, the index of stability is a = 1. This means that summing n Cauchy(a,b) distributions results in a n*Cauchy(a,b) distribution because the general formula is now:

Levy

The Levy distribution has index of stability a = 1/2, meaning that the general formula becomes

 

and thus the sum of n Levy(c,a) distributions results in a n^2*Levy(c,a) distribution.

Heavy tailed distributions

For stable distributions with index of stability a smaller than 2 (i.e. Cauchy or Levy distribution) the upper tail densities are asymptotically power laws. This means that the functional form looks something like:

 

Because this is of the same functional form as the Pareto distribution, the tails are also sometimes called Pareto tails.

A consequence for these distributions of having heavy tails, is that the moments doesn't exist. This is the reason why on the Cauchy equations page and the Levy equations page the moments are not specified.

 

 

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