called the skewness parameter, is a measure of asymmetry. Notice that in this context the usual skewness is not well defined, as for the distribution does not admit 2nd or higher moments, and the usual skewness definition is the 3rd central moment. The reason this gives a stable distribution is that the characteristic function for the sum of two independent random variables equals the product of the two corresponding characteristic functions. Adding two random variables from a stable distribution gives something with the same values of and , but possibly different values of μ and c. Not every function is the characteristic function of a legitimate probability distribution (that is, one whose cumulative distribution function is real and goes from 0 to 1 without decreasing), but the characteristic functions given above will be legitimate so long as the parameters are in their ranges. The value of the characteristic function at some value t is the complex conjugate of its value at −t as it should be so that the probability distribution function will be real. In the simplest case , the characteristic function is just a stretched exponential function; the distribution is symmetric about μ and is referred to as a (Lévy) symmetric alpha-stable distribution, often abbreviated SαS. When and , the distribution is supported by [μ, ∞). The parameter c > 0 is a scale factor which is a measure of the width of the distribution while is the exponent or index of the distribution and specifies the asymptotic behavior of the distribution. Parametrizations The above definition is only one of the parametrizations in use for stable distributions; it is the most common but is not continuous in the parameters at . A continuous parametrization is where: The ranges of and are the same as before, γ (like c) should be positive, and δ (like μ) should be real. In either parametrization one can make a linear transformation of the random variable to get a random variable whose density is . In the first parametrization, this is done by defining the new variable: For the second parametrization, we simply use no matter what is. In the first parametrization, if the mean exists (that is, ) then it is equal to μ, whereas in the second parametrization when the mean exists it is equal to The distribution A stable distribution is therefore specified by the above four parameters. It can