be shown that any non-degenerate stable distribution has a smooth (infinitely differentiable) density function. If denotes the density of X and Y is the sum of independent copies of X: then Y has the density with The asymptotic behavior is described, for , by: where Γ is the Gamma function (except that when and , the tail does not vanish to the left or right, resp., of μ, although the above expression is 0). This "heavy tail" behavior causes the variance of stable distributions to be infinite for all . This property is illustrated in the log–log plots below. When , the distribution is Gaussian (see below), with tails asymptotic to exp(−x2/4c2)/(2c√π). One-sided stable distribution and stable count distribution When and , the distribution is supported by [μ, ∞). This family is called one-sided stable distribution. Its standard distribution (μ=0) is defined as , where . Let , its characteristic function is . Thus the integral form of its PDF is (note: ) The double-sine integral is more effective for very small . Consider the Lévy sum where , then Y has the density where . Set , we arrive at the stable count distribution. Its standard distribution is defined as , where and . The stable count distribution is the conjugate prior of the one-sided stable distribution. Its location-scale family is defined as , where , , and . It is also a one-sided distribution supported by . The location parameter is the cut-off location, while defines its scale. When , is the Lévy distribution which is an inverse gamma distribution. Thus is a shifted gamma distribution of shape 3/2 and scale , , where , . Its mean is and its standard deviation is . It is hypothesized that VIX is distributed like with and (See Section 7 of ). Thus the stable count distribution is the first-order marginal distribution of a volatility process. In this context, is called the "floor volatility". Another approach to derive the stable count distribution is to use the Laplace transform of the one-sided stable distribution, (Section 2.4 of ) , where . Let , and one can decompose the integral on the left hand side as a product distribution of a standard Laplace distribution and a standard stable count distribution,f , where . This is called the "lambda decomposition" (See Section 4 of ) since the right hand side was named as