### Abstract

Language | English |
---|---|

Pages | 259-321 |

Number of pages | 62 |

Journal | Electronic Journal of Probability |

Volume | 13 |

Publication status | Published - 22 Feb 2008 |

### Fingerprint

### Keywords

- character
- convex cone
- Laplace transform
- LePage series
- Levy measure
- point process
- Poisson process
- random measure
- random set
- semigroup
- stable distribution
- union-stability

### Cite this

*Electronic Journal of Probability*,

*13*, 259-321.

}

*Electronic Journal of Probability*, vol. 13, pp. 259-321.

**Strictly stable distributions on convex cones.** / Davydov, T.; Molchanov, I.; Zuyev, S.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strictly stable distributions on convex cones

AU - Davydov, T.

AU - Molchanov, I.

AU - Zuyev, S.

PY - 2008/2/22

Y1 - 2008/2/22

N2 - 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.

AB - 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.

KW - character

KW - convex cone

KW - Laplace transform

KW - LePage series

KW - Levy measure

KW - point process

KW - Poisson process

KW - random measure

KW - random set

KW - semigroup

KW - stable distribution

KW - union-stability

UR - http://www.math.washington.edu/~ejpecp/include/getdoc.php?id=4273&article=1780&mode=pdf

M3 - Article

VL - 13

SP - 259

EP - 321

JO - Electronic Journal of Probability

T2 - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -