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  • 1. Biermé, Hermine
    et al.
    Estrade, Anne
    Kaj, Ingemar
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Self-similar Random Fields and Rescaled Random Balls Models2010In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 23, no 4, p. 1110-1141Article in journal (Refereed)
    Abstract [en]

    We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power-law behavior, we prove that the centered and renormalized random balls field admits a limit with self-similarity properties. Our main result states that all self-similar, translation-and rotation-invariant Gaussian fields can be obtained through a unified zooming procedure starting from a random balls model. This approach has to be understood as a microscopic description of macroscopic properties. Under specific assumptions, we also get a Poisson-type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give L-2-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.

  • 2.
    Broman, Erik I.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Stochastic Ordering of Infinite Geometric Galton-Watson Trees2016In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 29, no 3, p. 1069-1082Article in journal (Refereed)
    Abstract [en]

    We consider Galton-Watson trees with Geom(p) offspring distribution. We let T-infinity (p) denote such a tree conditioned on being infinite. We prove that for any 1/2 <= p(1) <= p2 <= 1, there exists a coupling between T-infinity (p(1)) and T-infinity (p(2)) such that P(T-infinity(p(1)) subset of T-infinity(p(2))) = 1.

  • 3.
    Broman, Erik I.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Camia, Federico
    Joosten, Matthijs
    Meester, Ronald
    Dimension (In)equalities and Holder Continuous Curves in Fractal Percolation2013In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 26, no 3, p. 836-854Article in journal (Refereed)
    Abstract [en]

    We relate various concepts of fractal dimension of the limiting set in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of non-trivial Holder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.

  • 4.
    Curien, Nicolas
    et al.
    Ecole Normale Super, Dept Math & Applicat, F-75230 Paris 05, France..
    Konstantopoulos, Takis
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, S-75106 Uppsala, Sweden..
    Iterating Brownian Motions, Ad Libitum2014In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 27, no 2, p. 433-448Article in journal (Refereed)
    Abstract [en]

    Let B-1, B-2, aEuro broken vertical bar be independent one-dimensional Brownian motions parameterized by the whole real line such that B (i) (0)=0 for every ia parts per thousand yen1. We consider the nth iterated Brownian motion W (n) (t)=B (n) (B (n-1)(a <-(B (2)(B (1)(t)))a <-)). Although the sequence of processes (W (n) ) (na parts per thousand yen1) does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W (n) converge to a random probability measure mu (a). We then prove that mu (a) almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W (a) of independent Brownian motions. We also prove that the collection of random variables (W (a)(t),taa"ea-{0}) is exchangeable with directing measure mu(infinity).

  • 5.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
    Martin-Löf, Anders
    Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden..
    A Maxtrimmed St. Petersburg Game2016In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 29, no 1, p. 277-291Article in journal (Refereed)
    Abstract [en]

    Let S-n, n >= 1, describe the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that s(n)/n log(2)n ->(p) 1 as n -> infinity It is also known that almost sure convergence fails. However, Csorgo and Simons (Stat Probab Lett 26: 65-73, 1996) have shown that almost sure convergence holds for trimmed sums, that is, for S-n - max(1 <= k <= n) X-k. Since our actual distribution is discrete there may be ties. Our main focus in this paper is on the "maxtrimmed sum", that is, the sum trimmed by the random number of observations that are equal to the largest one. We prove an analog of Martin-Lof's (J Appl Probab 22: 634-643, 1985) distributional limit theorem for maxtrimmed sums, but also for the simply trimmed ones, as well as for the "total maximum". In a final section, we interpret these findings in terms of sums of (truncated) Poisson random variables.

  • 6.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Stadtmueller, Ulrich
    On the LSL for Random Fields2011In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 24, no 2, p. 422-449Article in journal (Refereed)
    Abstract [en]

    In some earlier work, we have considered extensions of Lai's (Ann. Probab. 2:432-440, 1974) law of the single logarithm for delayed sums to a multi-index setting with the same as well as different expansion rates in the various dimensions. A further generalization concerns window sizes that are regularly varying with index 1 (on the line). In the present paper, we establish multi-index versions of the latter as well as for some mixtures of expansion rates. In order to keep things within reasonable size, we confine ourselves to some special cases for the index set Z(+.)(2).

  • 7.
    Gut, Allan
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
    Stadtmüller, Ulrich
    Cesàro summation for random fields2010In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 23, no 3, p. 715-728Article in journal (Refereed)
    Abstract [en]

    Various methods of summation for divergent series of real numbers have been generalized to analogous results for sums of i.i.d. random variables. The natural extension of results corresponding to CesA ro summation amounts to proving almost sure convergence of the CesA ro means. In the present paper we extend such results as well as weak laws and results on complete convergence to random fields, more specifically to random variables indexed by a"currency sign (+) (2) , the positive two-dimensional integer lattice points.

  • 8.
    Kaj, Ingemar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Biermé, Hermine
    Université Paris Descartes.
    Estrade, Anne
    Université Paris Descartes.
    Self-similar random fields and rescaled random balls models2010In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 23, p. 1110-1141Article in journal (Refereed)
  • 9.
    Kaj, Ingemar
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Sagitov, Serik
    Chalmers University of Technology.
    Limit processes for age-dependent branching particle systems1998In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 11, p. 225-257Article in journal (Refereed)
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