Electronic Journal of Probability

Strictly stable distributions on convex cones

Youri Davydov, Ilya Molchanov, and Sergei Zuyev

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Abstract

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.

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 11, 259-321.

Dates
Accepted: 22 February 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819084

Digital Object Identifier
doi:10.1214/EJP.v13-487

Mathematical Reviews number (MathSciNet)
MR2386734

Zentralblatt MATH identifier
1196.60028

Subjects
Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60B99: None of the above, but in this section 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G52: Stable processes 60G55: Point processes

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Davydov, Youri; Molchanov, Ilya; Zuyev, Sergei. Strictly stable distributions on convex cones. Electron. J. Probab. 13 (2008), paper no. 11, 259--321. doi:10.1214/EJP.v13-487. https://projecteuclid.org/euclid.ejp/1464819084


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References

  • C.D. Aliprantis and K.C. Border. Infinite-dimensional analysis. A hitchhiker's guide. Studies in Economic Theory, 4 (1994) Springer-Verlag, Berlin
    Zentralblatt MATH: 0839.46001
  • A. Araujo and E. Gin&eacute.; The central limit theorem for real and Banach valued random variables. (1980) John Wiley & Sons, New York-Chichester-Brisbane.
    Mathematical Reviews (MathSciNet): MR576407
  • F. Baccelli, G. Cohen, Guy, G.J. Olsder and J.-P. Quadrat. Synchronization and Linearity. An algebra for discrete event systems. (1992) John Wiley & Sons, Ltd., Chichester.
    Zentralblatt MATH: 0824.93003
  • Ch. Berg. Positive definite and related functions on semigroups. The analytical and topological theory of semigroups, (1990) 253–278, de Gruyter, Berlin.
  • Ch. Berg, J.P.R. Christensen and P. Ressel. Harmonic analysis on semigroups. Theory of positive definite and related functions. (1984) Springer-Verlag, New York.
    Zentralblatt MATH: 0619.43001
  • T.M. Bisgaard. Bochner's theorem for semigroups: a counterexample. Math. Scand. 87 (2000), 272–286.
    Zentralblatt MATH: 1061.43009
    Digital Object Identifier: doi:10.7146/math.scand.a-14309
  • N. Bourbaki. Éléments de mathématique. Fasc. XXXV. Livre VI: Intégration. (1969) Hermann, Paris.
    Mathematical Reviews (MathSciNet): MR0276436
  • H. Buchwalter. Les fonctions de Lévy existent! Math. Ann. 274 (1986), 31–34.
  • V.V. Buldygin. The convergence of random elements in topological spaces. (1980) (Russian) Naukova Dumka, Kiev.
    Zentralblatt MATH: 0512.60009
  • D.J. Daley and D. Vere-Jones, D. An introduction to the theory of point processes. (1988) Springer-Verlag, New York.
    Zentralblatt MATH: 0657.60069
  • Yu. Davydov and V. Egorov. On convergence of empirical point processes. Statist. Probab. Lett. 76 (2006), 1836–1844.
  • Yu. Davydov, V. Paulauskas and A. Račkauskas. More on $p$-stable convex sets in Banach spaces. J. Theoret. Probab. 13 (2000), 39–64.
  • P. Del Moral and M. Doisy. Maslov idempotent probability calculus. I. Theory Probab. Appl. 43 (1999), 562–576
    Zentralblatt MATH: 0969.60003
    Digital Object Identifier: doi:10.1137/S0040585X97977598
  • N. Dunford and J.T. Schwartz. Linear operators. Part I. General theory. (1988) John Wiley & Sons, Inc., New York.
    Zentralblatt MATH: 0635.47001
  • M. Falk, J. Hüsler and R.-D. Reiss. Laws of small numbers: extremes and rare events. Second, revised and extended edition. (2004) Birkhäuser Verlag, Basel.
    Mathematical Reviews (MathSciNet): MR2104478
  • J. Galambos. The asymptotic theory of extreme order statistics. (1978) John Wiley & Sons, New York-Chichester-Brisbane.
  • E. Giné and M.G. Hahn. Characterization and domains of attraction of $p$-stable random compact sets. Ann. Probab. 13 (1985), 447–468.
    Mathematical Reviews (MathSciNet): MR781416
    Digital Object Identifier: doi:10.1214/aop/1176993002
    Project Euclid: euclid.aop/1176993002
  • E. Giné, M.G. Hahn and J. Zinn. Limit theorems for random sets: an application of probability in Banach space results. Probability in Banach spaces, IV (Oberwolfach, 1982), 112–135, Lecture Notes in Math. 990 (1983) Springer, Berli.
  • U. Grenander. Probabilities on algebraic structures. (1968) Almqvist & Wiksell, Stockholm; John Wiley & Sons Inc., New York-London.
  • W. Hazod. Stable probability measures on groups and on vector spaces. A survey. Probability measures on groups, VIII (Oberwolfach, 1985), 304–352, Lecture Notes in Math. 1210 (1986) Springer, Berlin.
  • W. Hazod and E. Siebert. Stable probability measures on Euclidean spaces and on locally compact groups. Structural properties and limit theorems. (2001) Kluwer Academic Publishers, Dordrecht
    Zentralblatt MATH: 1002.60002
  • E. Hewitt and K.A. Ross. Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations. (1963) Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Gˆttingen-Heidelberg.
    Zentralblatt MATH: 0115.10603
  • H. Heyer. Structural aspects in the theory of probability. A primer in probabilities on algebraic-topological structures. (2004) World Scientific Publishing Co., Inc., River Edge, NJ.
    Mathematical Reviews (MathSciNet): MR2111057
    Zentralblatt MATH: 1074.60001
  • G. Högnás and A. Mukherjea. Probability measures on semigroups. Convolution products, random walks, and random matrices. (1995) Plenum Press, New York.
  • L. Hörmander. Sur la fonction d'appui des ensembles convexes dans un espace localement convexe. Ark. Mat. 3 (1955), 181–186.
  • J. Jonasson. Infinite divisibility of random objects in locally compact positive convex cones. J. Multivariate Anal. 65 (1998), 129–138.
    Zentralblatt MATH: 1138.60300
    Digital Object Identifier: doi:10.1006/jmva.1997.1732
  • V.V. Kalashnikov and S.T. Rachev. Mathematical methods for construction of queueing models. (1990) Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
  • K. Keimel. Lokal kompakte Kegelhalbgruppen und deren Einbettung in topologische Vektorräume. Math. Z. 99 (1967), 405–428.
  • K. Keimel and W. Roth. Ordered cones and approximation. Lecture Notes in Mathematics, 1517 (1992) Springer-Verlag, Berlin.
    Zentralblatt MATH: 0752.41033
  • J.L. Kelley. General topology. (1955) D. Van Nostrand Company, Inc., Toronto-New York-London.
    Zentralblatt MATH: 0066.16604
  • J.L. Kelley and I. Namioka. Linear topological spaces. (1963) D. Van Nostrand Co., Inc., Princeton, N.J.
    Mathematical Reviews (MathSciNet): MR0166578
    Zentralblatt MATH: 0318.46001
  • S. Kotz and S. Nadarajah. Extreme value distributions. Theory and applications. (2000) Imperial College Press, London.
    Zentralblatt MATH: 0960.62051
  • R. LePage, M. Woodroofe and J. Zinn. Convergence to a stable distribution via order statistics. Ann. Probab. 9 (1981), 624–632.
    Zentralblatt MATH: 0465.60031
    Digital Object Identifier: doi:10.1214/aop/1176994367
    Project Euclid: euclid.aop/1176994367
  • W. Linde. Probability in Banach spaces–-stable and infinitely divisible distributions. Second edition. (1986) John Wiley & Sons, Ltd., Chichester.
    Zentralblatt MATH: 0665.60005
  • V.P. Maslov and S. N. Samborskiĭ (Editors). Idempotent analysis. (1992) American Mathematical Society, Providence, RI.
  • G. Matheron. Random sets and integral geometry. (1975) John Wiley\thinspace &\thinspace Sons, New York-London-Sydney.
    Mathematical Reviews (MathSciNet): MR0385969
  • K. Matthes, J. Kerstan and J. Mecke. Infinitely divisible point processes. (1978) John Wiley & Sons, Chichester-New York-Brisbane.
    Zentralblatt MATH: 0383.60001
  • I. Molchanov. Theory of random sets. (2005) Springer-Verlag London, Ltd., London.
    Zentralblatt MATH: 1109.60001
  • I.S. Molchanov. Limit theorems for unions of random closed sets. Lecture Notes in Mathematics, 1561 (1993). Springer-Verlag, Berlin.
    Zentralblatt MATH: 0790.60015
  • I.S. Molchanov. On strong laws of large numbers for random upper semicontinuous functions. J. Math. Anal. Appl. 235 (1999), 349–355.
    Zentralblatt MATH: 0959.60003
    Digital Object Identifier: doi:10.1006/jmaa.1999.6403
  • K.R. Parthasarathy. Probability measures on metric spaces. (1967) Academic Press, Inc., New York-London.
    Zentralblatt MATH: 0153.19101
  • S.T. Rachev. Probability metrics and the stability of stochastic models. (1981) John Wiley & Sons, Ltd., Chichester.
  • H. Ratschek and G. Schrˆder. Representation of semigroups as systems of compact convex sets. Proc. Amer. Math. Soc. 65 (1977), 24–28.
  • S.I. Resnick. Extreme values, regular variation, and point processes. (1987) Springer-Verlag, New York.
    Mathematical Reviews (MathSciNet): MR900810
    Zentralblatt MATH: 0633.60001
  • S.I. Resnick and R. Roy. Superextremal processes, max-stability and dynamic continuous choice. Ann. Appl. Probab. 4 (1994), 791–811.
    Zentralblatt MATH: 0809.60064
    Digital Object Identifier: doi:10.1214/aoap/1177004972
    Project Euclid: euclid.aoap/1177004972
  • J. Rosiński. On series representations of infinitely divisible random vectors. Ann. Probab. 18 (1990), 405–430.
  • I.Z. Ruzsa. Infinite divisibility. Adv. in Math. 69 (1988), 115–132.
  • I.Z. Ruzsa. Infinite divisibility. II. J. Theoret. Probab. 1 (1988), 327–339.
    Zentralblatt MATH: 0646.60017
    Digital Object Identifier: doi:10.1016/0001-8708(88)90063-1
  • G. Samorodnitsky and M.S. Taqqu. Stable non-Gaussian random processes. Stochastic models with infinite variance. (1994) Chapman & Hall, New York.
    Zentralblatt MATH: 0925.60027
  • R. Schneider. Convex bodies: the Brunn-Minkowski theory. (1993) Cambridge University Press, Cambridge.
    Zentralblatt MATH: 0798.52001
  • A.N. Shiryayev. Probability. (1984) Springer-Verlag, New York.
    Mathematical Reviews (MathSciNet): MR737192
  • M.F. Smith. The Pontrjagin duality theorem in linear spaces. Ann. of Math. (2) 56 (1952), 248–253.
    Zentralblatt MATH: 0047.10701
    Digital Object Identifier: doi:10.2307/1969798
  • N.N. Vakhania, V.I. Tarieladze and S.A. Chobanyan. Probability distributions on Banach spaces. (1987) D. Reidel Publishing Co., Dordrecht.
  • V.M. Zolotarev. One-dimensional stable distributions. (1986) American Mathematical Society, Providence, RI.
    Zentralblatt MATH: 0589.60015
  • V.M. Zolotarev. Modern theory of summation of random variables. (1997) VSP, Utrecht.
    Zentralblatt MATH: 0907.60001

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