Abstract
Language | English |
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Pages | 259-321 |
Number of pages | 62 |
Journal | Electronic Journal of Probability |
Volume | 13 |
Publication status | Published - 22 Feb 2008 |
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Keywords
- character
- convex cone
- Laplace transform
- LePage series
- Levy measure
- point process
- Poisson process
- random measure
- random set
- semigroup
- stable distribution
- union-stability
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Strictly stable distributions on convex cones. / Davydov, T.; Molchanov, I.; Zuyev, S.
In: Electronic Journal of Probability, Vol. 13, 22.02.2008, p. 259-321.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 -