Alpha stable distributions are either gaussian or have power law like tails, asymptotically. Sometimes tails will be described as Pareto-like. The definition of alpha stable distributions comes from properties of sums of numbers and there is a sort of generalised central limit theorem that is central two the definition: one refers to the maximumdomain of attraction.Extreme value theory is based on the properties of maxima or minima of distributions: there is a version of the central limit theorem that refers to maxima instead of sum or average, and there is a maximum domain of attraction argument in classifying how the extreme tails of a given distribution converge to one of three extreme value distributions.Surely there must be a connection here? A given alpha stable distribution must have a fairly direct connection to a given extreme value distribution right?

Intuitively infinite variance really means that the longer your data sample the higher the variance. The Cauchy (Lorentz) distribution has infinite variance but has a well defined distribution of its maxima - I think it becomes the Gumbel distribution in the deep tails though.

Yes - I reckoned one of you folks here would know the answer - or maybe there isn't a straightforward connection...

QuoteOriginally posted by: erstwhileAlpha stable distributions are either gaussian or have power law like tails, asymptotically. Sometimes tails will be described as Pareto-like. The definition of alpha stable distributions comes from properties of sums of numbers and there is a sort of generalised central limit theorem that is central two the definition: one refers to the maximumdomain of attraction.Extreme value theory is based on the properties of maxima or minima of distributions: there is a version of the central limit theorem that refers to maxima instead of sum or average, and there is a maximum domain of attraction argument in classifying how the extreme tails of a given distribution converge to one of three extreme value distributions.Surely there must be a connection here? A given alpha stable distribution must have a fairly direct connection to a given extreme value distribution right?Interesting question.Just thinking out loud, and looking up some definitions, you can see some *very* close connections (pre-rescaling limit). (http://en.wikipedia.org/wiki/Extreme_value_theory)Extreme value distributions are distributions of the minimum or maximum of lists{X_1,X_2,...,X_n} of n IID draws of some rv, call it X.Start with the distribution of the minimum M_n for a rv with positive support:M_n = min {X_1,X_2,...,X_n} This has complementary distribution function, if you think about it a little,(*) P(M_n >= x ) = P(X_1 >= x, X_2 >= x, ..., X_n >= x) = F(x)^n,where F(x) = P(X >= x) = complementary distribution function of X.On the other hand, consider the Laplace transform of the sum S_n of n IID draws of some other r.v. Y with positive support.S_n = Y_1 + Y_2 + ... + Y_n This sum has LT with respect to x(**) E[e^(- x S_n)] = G(x)^n where G(x) = int(0,infty) e^(-x y) p_Y(y) dy is the LT of the density of Y.Look at (*) vs (**). Can G(x) be intrepreted as a complementary distribution function?Yes! We have G(0) = 1, G(infty) = 0, G(x) >= 0, and G(x) is monotone decreasing in x.These are the properties that characterize complementary distribution functions F(x) ![Again, this is all for a r.v. supported on (0,infty)]. So (*) and (**) associate two distributions with positive support. In other words, the LT of the position of the random walk S_n, generated by draws of Y, can be interpreted as the complementary distribution function of the minimum of n draws of a corresponding rv, X. The correspondence X <==> Y is that X has a complementary distribution function equal to the LT of the density of Y. So, say given a distribution of X you could get the density of the associated Y by a Laplace inversion.This is for arbitrary distributions and arbitrary n = 1, 2, ...As for the rescaling limit n -> infty, alpha-stable, etc, I'm not sure; will stop here.

Thanks.Some alpha stable <==> extreme value connections follow pretty immediately from what I wrote, as follows.Remember my stuff requires positive support.Let Y be distributed one-sided alpha-stable, which has positive support. The LT of the density (http://arxiv.org/pdf/1007.0193v3.pdf) has a simple formula: G(x) = int(0,infty) e^(-x y) p_Y(y) dy = exp(-x^alpha), for 0 < alpha < 1So, this alpha-stable Y is associated to another r.v. X, by what I wrote, namely the one with complementary distribution function P(X >= x) = exp(-x^alpha). That means the density of X is (1) p_X(x) = alpha x^(alpha-1) exp(-x^alpha), 0 < alpha < 1Now let Z be a Frechet distibuted variate.The Frechet distribution (http://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution)is one of the possible extreme value densities, in the rescaling limit, for a maximum.It has the density (2) p_Frechet(z) = beta z^(-1 - beta) exp{-z^(-beta)}, 0 < beta < infty.Notations: I am using parameter beta here instead of conventional Frechet alpha to distinguish (at least initially) from the alpha of (1). The density variable z is used for the realizations of Z. Since the Frechet is associated with maxima, we flip the IID list of maxima to minima by reciprocals:max(Z_1,Z_2, ...., Z_n) = [min (X_1,X_2,....,X_n)]^(-1), where X_i = 1/Z_i. So, if Z is Frechet distributed, then we should consider the distribution of X = 1/Z to get minima. Call the distribution of X the associated-Frechet (I just made that name up, so don't know if that is a convention, but it fits).From (2), the density of the associated-Frechet will be(3) p_AssocFrechet(x) = beta x^{beta-1} exp(-x^beta), 0 < beta < inftyCompare (1) and (3). You can see they are the same function,now identifying beta = alpha, at least in the parameter overlap range 0 < alpha < 1. So, to summarize, here's how it works for this example: The one-sided alpha-stable distribution is indeed connected to an extreme value distribution for a minimum, namely the associated-Frechet distribution, verifying erstwhile's intuition.Here's the connection, again, in words:First, an "associated-Frechet variate with parameter alpha" is definedto be the reciprocal of the Frechet variate with parameter alpha.Then, the connection is that the Laplace transform of the density of the one-sided alpha-stable variate with parameter alpha is identical to the complementary distribution function of an associated-Frechet variate with the same parameter alpha. The next step would be to try to directly relate some other cases, but not today ...

Alan that was impressive, thanks - I need to let this connection sink in now...