Strictly stable distributions on convex cones

By Y. Davydov, Ilya S. Molchanov, Sergei A. Zuyev. 2005

78 citations recorded by the source corpus.

Using the LePage representation, a symmetric alpha-stable random element in Banach space B with alpha from (0,2) can be represented as a sum of points of a Poisson process in B. This point process is union-stable, i.e. the union of its two independent copies coincides in distribution with the rescaled original point process. This shows that the classical definition of stable random elements is closely related to the union-stability property of point processes. These concepts make sense in any convex cone, i.e. in a semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. We prove that random samples (or binomial point processes) in rather general cones converge in distribution in the vague topology to the union-stable Poisson point process. This convergence holds also in a stronger topology, which implies that the sums of points converge in distribution to the sum of points of the union-stable point process. Since the latter corresponds to a stable law, this yields a limit theorem for normalised sums of random elements with alpha-stable limit for alpha from (0,1). By using the technique of harmonic analysis on semigroups we characterise distributions of alpha-stable random elements and show how possible values of the characteristic exponent alpha relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. It is shown that several conditions imply that a stable random element admits the LePage representation. The approach developed in the paper not only makes it possible to handle stable distributions in rather general cones (like spaces of sets or measures), but also provides an alternative way to prove classical limit theorems and deduce the LePage representation for strictly stable random vectors in Banach spaces.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/caf2f9d4227474f97827b7fb70ef7a3857ab8b1b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Thomas Madaule
2. Eliran Subag
3. Ofer Zeitouni
4. P. Maillard
5. Alice Guionnet
6. F. Augeri

In Finance

1. P. Maillard
2. Ofer Zeitouni
3. Eliran Subag
4. Thomas Madaule
5. Zhan Shi
6. Joseph Najnudel
7. L. Arguin
8. Jian Ding

In Mathematical Physics

1. Thomas Madaule
2. P. Maillard
3. Ofer Zeitouni
4. Eliran Subag
5. Zhan Shi
6. Joseph Najnudel
7. Jian Ding
8. L. Arguin

Citations in the corpus

• Convergence in Law for the Branching Random Walk Seen from Its Tip (2011, background): “2 of Maillard [22] (see also [12]), to prove that μ∞ is superposable.”
• Convex geometry of max-stable distributions (2006, unknown): “ar X iv :m at h/ 06 03 42 3v 3 [ m at h. PR ] 2 Keywords: copula; max-stable random vector; norm; cross-polytope; spectral measure; support function; zonoid”
• A note on stable point processes occurring in branching Brownian motion (2011, methodology): “Note that if Y is a simple point process (i.e. every atom has unit mass), then the set of its points is a random closed subset of (0 , ∞ ) and the stability property is then also known as the union-stability for random closed sets (see e.g. [20, Ch. 4.1]).”
• Freezing and Decorated Poisson Point Processes (2014, methodology): “This can be proved using a representation of the Laplace functional for infinitely divisible processes (see Maillard [44]; implicitely, this also appears in [21]).”