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- Holmgren, Cecilia, 1984-, et al.
(author)
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Fringe trees, Crump-Mode-Jagers branching processes and m-ary search trees
- 2017
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In: Probability Surveys. - 1549-5787. ; 14, s. 53-154
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Journal article (peer-reviewed)abstract
- This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump-Mode-Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) m-ary search trees, as well as some other classes of random trees.We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in detail. In particular, we consider fringe trees of m-ary search trees in detail; this seems to be new.Various applications are given, including degree distribution, protected nodes and maximal clades for various types of random trees. Again, we emphasise results for m-ary search trees, and give for example new results on protected nodes in m-ary search trees.A separate section surveys results on the height of the random trees due to Devroye (1986), Biggins (1995, 1997) and others.This survey contains well-known basic results together with some additional general results as well as many new examples and applications for various classes of random trees.
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| 2. |
- Janson, Svante
(author)
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Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
- 2007
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In: Probability Surveys. - 1549-5787. ; 4, s. 80-145
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Journal article (peer-reviewed)abstract
- This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright’s constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area.The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources.
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| 5. |
- Lindskog, Filip, et al.
(author)
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Regularly varying measures on metric spaces : Hidden regular variation and hidden jumps
- 2014
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In: Probability Surveys. - : Institute of Mathematical Statistics. - 1549-5787. ; 11:2014, s. 270-314
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Journal article (peer-reviewed)abstract
- We develop a framework for regularly varying measures on complete separable metric spaces S with a closed cone C removed, extending material in [15, 24]. Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular-variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in ℝ∞+ with marginal distributions having regularly varying tails and to càdlàg Lévy processes whose Lévy measures have regularly varying tails. In both cases, an infinite number of regular-variation properties coexist distinguished by different scaling functions and state spaces.
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| 6. |
- Malyarenko, Anatoliy, 1957-, et al.
(author)
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Tensor- and spinor-valued random fields with applications to continuum physics and cosmology
- 2023
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In: Probability Surveys. - The Hague, The Netherlands : Institute of Mathematical Statistics. - 1549-5787. ; 20:none, s. 1-86
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Journal article (peer-reviewed)abstract
- In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes the physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points.The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp., rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp., its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random crosssection models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic.For the convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.
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