San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 Pareto-Lévy Stable Distributions

Stable distributions are such that if random variables x and y come from stable distibutions the x+y will also have a stable distibution.

Paul Lévy found the formula for the characteristic function of all stable distribution. The characterisitc function of a stable distribution must such that the logarithm of the characteristic function Φ(ω) must be of the form:

#### log(Φ(ω)) = iδω - |νω|α(1 - iβF(ω,α,ν)) where F(ω,α,ν)) = sgn(ω)tan(πα/2) if α ≠ 1 = - (2/π)log(|νω|) if α = 1 and where sgn(ω) = +1 if ω >0             = 0 if ω = 0               = -1 if ω < 0

The nature and allowable ranges for the parameters are as follows:

• Alpha: Usually α is called the stability parameter. For the normal distribution α = 2. Alpha has to be in the interval 0<α≤2.
• Beta: Usually β is called the skewness parameter. For the normal distribution and any other symmetric distribution β = 0. Beta can have any real number value.
• Nu: ν is called the scale parameter or the dispersion parameter. For the normal distribution ν is equal to the standard deviation. For non-normal distribution ν has a value but it is not the same as the standard deviation, which for non-normal stable distributions is infinite. Nu can have any positive real number value.
• Delta: δ is called the mean or the measure of centrality. Delta can have any real number value.

For a normal distribution α=2, β=0, ν is equal to the standard deviation and δ is equal to the mean. Thus the log-characteristic function for a normal distribution is of the form:

#### log(Φ(ω) = iδω - |νω|2.

Some cases for particular values of the parameters are shown below:

## The Discovery of the Stable Distributions

Prior to Paul Levy's mathematical analysis empirical investigators were finding cases in which the histograms of some variable, while generally looking like normal distributions, were deviating from the normal distribution in a systematic manner. For example, the economist Wesley Claire Mitchell in 1915 found that the distribution of the percentage changes in stock prices when compared to the best-fitting normal distribution consistently deviated from the normal distribution as shown below:

This sort of distribution means that there would be too many small deviations from the average as well too many very large deviations. What there is too few of is moderate deviations. The extreme large changes were of particular interest because those were the cases of stock market booms and busts. Because a higher proportion of the probability was in the tails of the distribution compared with the case of the normal distribution such distributions were called fat-tailed distributions. They were also given a name based upon Greek, leptokurtic.

The fact that there are an excess of small deviations would tend to lead observers to underestimate the volatility of such variables; at least until a very large deviation comes along.

There are Levy-Pareto stable distributions that are leptokurtic. Furthermore, there is a generalization of the Central Limit Theorem that says that the sum of a large number of independent random variables will have a stable distribution. Thus if some phenomenon such as changes in stock prices or rain from a storm is the result of a large number of independent influences then it would be expected that the distribution would be a stable distribution. If the distribution is a fat-tailed distribution then that fact would account for the unexpected extreme changes in a variables, the sort of occurrences associated with catastrophes.

The following histogram is based upon a sample of 2000 observations of a random variable which has a distribution characterized by the values of the parameters shown. Each time the image is refreshed a new sample of 2000 observations is drawn.