The Lévy distribution, named for the French mathematician Paul Lévy, is important in the study of Brownian motion, and is one of only three stable distributions whose probability density function can be expressed in a simple, closed form.
The Standard Lévy Distribution
Definition
So the standard Lévy distribution is a continuous distribution on .
Distribution Functions
We assume that has the standard Lévy distribution. The distribution function of has a simple expression in terms of the standard normal distribution function , not surprising given the definition. Recall that is considered a special function in mathematics.
has distribution function given by
Details:
For ,
Similarly, the quantile function of has a simple expression in terms of the standard normal quantile function .
has quantile function given by
The quartiles of are
- , the first quartile.
- , the median.
- , the third quartile.
Details:
The quantile function can be obtained from the distribution function in [2] by solving for .
Open the quantile app and select the Lévy distribution. Keep the default parameter values for the standard Lévy distribution. Note the shape and location of the distribution function. Compute a few values of the quantile function.
Finally, the probability density function of has a simple closed expression.
has probability density function given by
- increases and then decreasing with mode at .
- is concave upward, then downward, then upward again, with inflection points at and at .
Details:
The formula for follows from differentiating the CDF in [2]:
But , the standard normal PDF. Substitution and simplification then gives the results. Parts (a) and (b) also follow from standard calculus:
Open the Special Distribtion Simulator and select the Lévy distribution. Keep the default parameter values for the standard Lévy distribution. Note the shape of the probability density function. Run the simulation 1000 times and compare the empirical density function to the probability density function.
Moments
We assume again that has the standard Lévy distribution. After exploring the graphs of the probability density function in [5] and distribution function in [2], you probably noticed that the Lévy distribution has a very heavy tail. The 99th percentile is about 6400, for example. The following result is not surprising.
Details:
Note that is increasing. Hence
Of course, the higher-order moments are infinite as well, and the variance, skewness, and kurtosis do not exist. The moment generating function is infinite at every positive value, and so is of no use. On the other hand, the characteristic function of the standard Lévy distribution is very useful. For the following result, recall that the sign function is given by for , for , and .
has characteristic function given by
Related Distributions
The most important relationship is the one in definition [1]: If has the standard normal distribution then has the standard Lévy distribution. The following result is bascially the converse.
If has the standard Lévy distribution, then has the standard half-normal distribution.
Details:
From definition [1], we can take where has the standard normal distribution. Then , and has the standard half-normal distribution.
The General Lévy Distribution
Like so many other standard distributions
, the standard Lévy distribution is generalized by adding location and scale parameters.
Definition
Suppose that has the standard Lévy distribution, and and . Then has the Lévy distribution with location parameter and scale parameter .
Note that has a continuous distribution on the interval .
Distribution Functions
Suppose that has the Lévy distribution with location parameter and scale parameter . As before, the distribution function of has a simple expression in terms of the standard normal distribution function .
has distribution function given by
Details:
Recall that where is the standard Lévy CDF in [2].
Similarly, the quantile function of has a simple expression in terms of the standard normal quantile function .
has quantile function given by
The quartiles of are
- , the first quartile.
- , the median.
- , the third quartile.
Details:
Recall that , where is the standard Lévy quantile function in [3].
Open the quantile app and select the Lévy distribution. Vary the parameter values and note the shape of the graph of the distribution function function. For various values of the parameters, compute a few values of the distribution function and the quantile function.
Finally, the probability density function of has a simple closed expression.
has probability density function given by
- increases and then decreases with mode at .
- is concave upward, then downward, then upward again with inflection points at .
Details:
Recall that where is the standard Lévy PDF in [5], so the formula for follow from the definition of and simple algebra. Parts (a) and (b) follow from the corresponding results for .
Open the special distribtion simulator and select the Lévy distribution. Vary the parameters and note the shape and location of the probability density function. For various parameter values, run the simulation 1000 times and compare the empirical density function to the probability density function.
Moments
Assume again that has the Lévy distribution with location parameter and scale parameter . Of course, since the standard Lévy distribution has infinite mean, so does the general Lévy distribution.
Also as before, the variance, skewness, and kurtosis of are undefined. On the other hand, the characteristic function of is very important.
has characteristic function given by
Details:
This follows from the standard characteristic function in [8] since . Note that since .
Related Distributions
Since the Lévy distribution is a location-scale family, it is trivially closed under location-scale transformations.
Suppose that has the Lévy distribution with location parameter and scale parameter , and that and . Then has the Lévy distribution with location parameter and scale parameter .
Details:
From definition [10], we can take where has the standard Lévy distribution. Hence has the Lévy distribution with location parameter and scale parameter .
Of more interest is the fact that the Lévy distribution is closed under convolution (corresponding to sums of independent variables).
Suppose that and are independent, and that, has the Lévy distribution with location parameter and scale parameter for . Then has the Lévy distribution with location parameter and scale parameter .
Details:
The characteristic function of is
for . Hence the characteristic function of is
where is the location parameter and is the scale parameter.
As a corollary, the Lévy distribution is a stable distribution with index :
Suppose that and that is a sequence of independent random variables, each having the Lévy distribution with location parameter and scale parameter . Then has the Lévy distribution with location parameter and scale parameter .
Stability is one of the reasons for the importance of the Lévy distribution. From the characteristic function, it follows that the skewness parameter is .