16,059 research outputs found

    On the contraction method with degenerate limit equation

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    A class of random recursive sequences (Y_n) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X\stackrel{L}{=}X. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the ``unique'' solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000017

    Long and short paths in uniform random recursive dags

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    In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.Comment: 16 page

    Width and mode of the profile for some random trees of logarithmic height

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    We propose a new, direct, correlation-free approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width and almost sure convergence. It is widely applicable to random trees of logarithmic height, including recursive trees, binary search trees, quad trees, plane-oriented ordered trees and other varieties of increasing trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000187 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Constrained exchangeable partitions

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    For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown

    Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop

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    We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual distance in the map. For large maps, this yields explicit formulas for the bulk-boundary and boundary-boundary correlators in the various encountered scaling regimes: a small boundary, a dense boundary and a critical boundary regime. The critical boundary regime is characterized by a one-parameter family of scaling functions interpolating between the Brownian map and the Brownian Continuum Random Tree. We discuss the cases of both generic and self-avoiding boundaries, which are shown to share the same universal scaling limit. We finally address the question of the bulk-loop distance statistics in the context of planar quadrangulations equipped with a self-avoiding loop. Here again, a new family of scaling functions describing critical loops is discovered.Comment: 55 pages, 14 figures, final version with minor correction

    An algebraic approach to Polya processes

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    P\'olya processes are natural generalization of P\'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference operator on polynomial functions. Especially, it provides new results for {\it large} processes (a P\'olya process is called {\it small} when 1 is simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part ≤1/2\leq 1/2; otherwise, it is called large)

    Topological Properties of Epidemic Aftershock Processes

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    Earthquakes in seismological catalogs and acoustic emission events in lab experiments can be statistically described as point events in linear Hawkes processes, where the spatiotemporal rate is a linear superposition of background intensity and aftershock clusters triggered by preceding activity. Traditionally, statistical seismology interpreted these models as the outcome of epidemic branching processes, where one-to-one causal links can be established between mainshocks and aftershocks. Declustering techniques are used to infer the underlying triggering trees and relate their topological properties with epidemic branching models. Here, we review how the standard Epidemic Type Aftershock Sequence (ETAS) model extends from the Galton-Watson branching processes and bridges two extreme cases: Poisson and scale-free power law trees. We report the statistical laws expected in triggering trees regarding some topological properties. We find that the statistics of such topological properties depend exclusively on two parameters of the standard ETAS model: the average branching ratio nb and the ratio between exponents α and b characterizing the production of aftershocks and the distribution of magnitudes, respectively. In particular, the classification of clusters into bursts and swarms proposed by Zaliapin and Ben-Zion (2013b, https://doi.org/10.1002/jgrb.50178) appears naturally in the aftershock sequences of the standard ETAS model depending on nb and α/b. On the other hand swarms can also appear by false causal connections between independent events in nontectonic seismogenic episodes. From these results, one can use the memory-less Galton-Watson as a null model for empirical triggering processes and assess the validity of the ETAS hypothesis to reproduce the statistics of natural and artificial catalogs
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