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  • Articles  (130)
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  • 11
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    Stratonovich’s signatures of Brownian motion determine Brownian sample paths (2012)
    Springer
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    Publication Date: 2012-10-08
    Description:    The signature of Brownian motion in \mathbb R d over a running time interval [0, T ] is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension d ³ 2 , almost all Brownian motion sample paths (running up to time T ) are determined by their signature over [0, T ] . Content Type Journal Article Pages 1-15 DOI 10.1007/s00440-012-0454-z Authors Yves Le Jan, Départment de Mathématiques, Université Paris-Sud 11, 91405 Orsay, France Zhongmin Qian, Mathematical Institute, University of Oxford, Oxford,  OX1 3LB England Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
    Print ISSN: 0178-8051
    Electronic ISSN: 1432-2064
    Topics: Mathematics
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  • 12
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    The Bernstein–Orlicz norm and deviation inequalities (2012)
    Springer
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    Publication Date: 2012-10-08
    Description:    We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case. Content Type Journal Article Pages 1-26 DOI 10.1007/s00440-012-0455-y Authors Sara van de Geer, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland Johannes Lederer, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
    Print ISSN: 0178-8051
    Electronic ISSN: 1432-2064
    Topics: Mathematics
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  • 13
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    Matchings on infinite graphs (2012)
    Springer
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    Publication Date: 2012-10-04
    Description:    Elek and Lippner (Proc. Am. Math. Soc. 138(8), 2939–2947, 2010 ) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton–Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erdős-Rényi random graphs. Content Type Journal Article Pages 1-26 DOI 10.1007/s00440-012-0453-0 Authors Charles Bordenave, Institut de Mathématiques, Université de Toulouse, CNRS, Toulouse, France Marc Lelarge, INRIA-Ecole Normale Supérieure, Paris, France Justin Salez, Université Paris Diderot-LPMA, Paris, France Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
    Print ISSN: 0178-8051
    Electronic ISSN: 1432-2064
    Topics: Mathematics
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  • 14
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    Limits of spiked random matrices I (2012)
    Springer
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    Publication Date: 2012-10-04
    Description:    Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general β analogue, proving a conjecture of Baik et al. (Ann Probab 33:1643–1697, 2005 ). We also treat shifted mean Gaussian orthogonal and β ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrödinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which β appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at β  = 2,4, yielding in particular a new and simple proof of the Painlevé representations for these Tracy–Widom distributions. Content Type Journal Article Pages 1-31 DOI 10.1007/s00440-012-0443-2 Authors Alex Bloemendal, Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Bálint Virág, Departments of Mathematics and Statistics, University of Toronto, Toronto, ON M5S 2E4, Canada Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
    Print ISSN: 0178-8051
    Electronic ISSN: 1432-2064
    Topics: Mathematics
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  • 15
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    Asymptotic expansions in the CLT in free probability (2012)
    Springer
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    Publication Date: 2012-09-29
    Description:    We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. Content Type Journal Article Pages 1-50 DOI 10.1007/s00440-012-0451-2 Authors G. P. Chistyakov, Faculty of Mathematics, University of Bielefeld, Postfach 100131, 33501 Bielefeld, Germany F. Götze, Faculty of Mathematics, University of Bielefeld, Postfach 100131, 33501 Bielefeld, Germany Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
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    Topics: Mathematics
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  • 16
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    BK-type inequalities and generalized random-cluster representations (2012)
    Springer
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    Publication Date: 2012-09-27
    Description:    Recently, van den Berg and Jonasson gave the first substantial extension of the BK inequality for non-product measures: they proved that, for k -out-of- n measures, the probability that two increasing events occur disjointly is at most the product of the two individual probabilities. We show several other extensions and modifications of the BK inequality. In particular, we prove that the antiferromagnetic Ising Curie–Weiss model satisfies the BK inequality for all increasing events. We prove that this also holds for the Curie–Weiss model with three-body interactions under the so-called negative lattice condition. For the ferromagnetic Ising model we show that the probability that two events occur ‘cluster-disjointly’ is at most the product of the two individual probabilities, and we give a more abstract form of this result for arbitrary Gibbs measures. The above cases are derived from a general abstract theorem whose proof is based on an extension of the Fortuin–Kasteleyn random-cluster representation for all probability distributions and on a ‘folding procedure’ which generalizes an argument of Reimer. Content Type Journal Article Pages 1-25 DOI 10.1007/s00440-012-0452-1 Authors J. van den Berg, CWI and VU University Amsterdam, Amsterdam, The Netherlands A. Gandolfi, Dipartimento di Matematica U. Dini, Univ. di Firenze, Firenze, Italy Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
    Print ISSN: 0178-8051
    Electronic ISSN: 1432-2064
    Topics: Mathematics
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  • 17
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    A contour line of the continuum Gaussian free field (2012)
    Springer
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    Publication Date: 2012-09-20
    Description:    Consider an instance h of the Gaussian free field on a simply connected planar domain D with boundary conditions - l on one boundary arc and l on the complementary arc, where l is the special constant Ö   p /8   . We argue that even though h is defined only as a random distribution, and not as a function, it has a well-defined zero level line g connecting the endpoints of these arcs, and the law of g is SLE (4) . We construct g in two ways: as the limit of the chordal zero contour lines of the projections of h onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. We also show that, as a function of h , g is “local” (it does not change when h is modified away from g ) and derive some general properties of local sets. Content Type Journal Article Pages 1-34 DOI 10.1007/s00440-012-0449-9 Authors Oded Schramm, Department of Mathematics, MIT, 2-180, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Scott Sheffield, Department of Mathematics, MIT, 2-180, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
    Print ISSN: 0178-8051
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    Topics: Mathematics
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  • 18
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    The asymptotic distribution of a single eigenvalue gap of a Wigner matrix (2012)
    Springer
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    Publication Date: 2012-09-17
    Description:    We show that the distribution of (a suitable rescaling of) a single eigenvalue gap l i +1 ( M n ) - l i ( M n ) of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin–Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin–Mehta law required either an averaging in the eigenvalue index parameter i , or fixing the energy level u instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function N ( - ¥ , x ) ( ~ M   n   ) (where ~ M   n   is a suitably rescaled version of M n ) with the event that there is no spectrum in an interval [ x , x + s ] , in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel. Content Type Journal Article Pages 1-26 DOI 10.1007/s00440-012-0450-3 Authors Terence Tao, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
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    Topics: Mathematics
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  • 19
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    Asymptotic behaviour of first passage time distributions for Lévy processes (2012)
    Springer
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    Publication Date: 2012-09-03
    Description:    Let X be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function c . As an analogue of the random walk results in Vatutin and Wachtel (Probab Theory Relat Fields 143(1–2):177–217, 2009 ) and Doney (Probab Theory Relat Fields 152(3–4):559–588, 2012 ), we study the local behaviour of the distribution of the lifetime z under the characteristic measure n of excursions away from 0 of the process X reflected in its past infimum, and of the first passage time of X below 0, $$T_{0}=\inf \{t〉0:X_{t}
    Print ISSN: 0178-8051
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    Topics: Mathematics
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  • 20
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    Functional limit theorems for random regular graphs (2012)
    Springer
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    Publication Date: 2012-09-03
    Description:    Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2 d on n vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as n grows to infinity, either when d is kept fixed or grows slowly with n . In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein’s method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn–Szemerédi argument for estimating the second largest eigenvalue for all values of d and n . Content Type Journal Article Pages 1-55 DOI 10.1007/s00440-012-0447-y Authors Ioana Dumitriu, Department of Mathematics, University of Washington, Seattle, WA 98195, USA Tobias Johnson, Department of Mathematics, University of Washington, Seattle, WA 98195, USA Soumik Pal, Department of Mathematics, University of Washington, Seattle, WA 98195, USA Elliot Paquette, Department of Mathematics, University of Washington, Seattle, WA 98195, USA Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051
    Print ISSN: 0178-8051
    Electronic ISSN: 1432-2064
    Topics: Mathematics
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