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Simulating the Dickman distribution
 Statist. Probab. Lett
"... Abstract. In this paper, we give a simple algorithm for sampling from the Dickman distribution. It is based on coupling from the past with a suitable dominating Markov chain. ..."
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Abstract. In this paper, we give a simple algorithm for sampling from the Dickman distribution. It is based on coupling from the past with a suitable dominating Markov chain.
Appendix to “Approximating perpetuities”
, 2012
"... An algorithm for perfect simulation from the unique solution of the distributional fixed point equation Y =d UY +U(1−U) is constructed, where Y and U are independent and U is uniformly distributed on [0, 1]. This distribution comes up as a limit distribution in the probabilistic analysis of the Quic ..."
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An algorithm for perfect simulation from the unique solution of the distributional fixed point equation Y =d UY +U(1−U) is constructed, where Y and U are independent and U is uniformly distributed on [0, 1]. This distribution comes up as a limit distribution in the probabilistic analysis of the Quickselect algorithm. Our simulation algorithm is based on coupling from the past with a multigamma coupler. It has four lines of code.
Return to the Poissonian City
"... Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial / final segments of the connecting path formed by travelling off the network in the opp ..."
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Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial / final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination / source. Suppose further that connections are established using “neargeodesics”, constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting neargeodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In earlier work (“Geodesics and flows in a Poissonian city”, Annals of Applied Probability, 21(3), 801–842, 2011) it was shown that a scaled version of this flow had asymptotic distribution given by the 4volume of a region in 4space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two “seminal curves ” defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to zero with increasing computational effort. 1991 Mathematics Subject Classification. 60D05, 90B15. Key words and phrases. improper anisotropic Poisson line process; mark distribution; point process; Poisson line process; Poissonian city network; MeckeSlivynak theorem; seminal curve; spatial network; traffic flow. 1
A Gaussian limit process for optimal FIND algorithms
, 2013
"... We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c · nα are chosen, where 0 < α ≤ 1 2, c> 0 and n is the size of the data set to b ..."
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We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to c · nα are chosen, where 0 < α ≤ 1 2, c> 0 and n is the size of the data set to be split. We consider the complexity of FIND as a process in the rank to be selected and measured by the number of key comparisons required. After normalization we show weak convergence of the complexity to a centered Gaussian process as n → ∞, which depends on α. The proof relies on a contraction argument for probability distributions on càdlàg functions. We also identify the covariance function of the Gaussian limit process and discuss path and tail properties. AMS 2010 subject classifications. Primary 60F17, 68P10; secondary 60G15, 60C05, 68Q25. Key words. FIND algorithm, Quickselect, complexity, key comparisons, functional limit theorem,
Perfect sampling for infinite server and loss systems. arXiv 1312.4088
, 2013
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A statistical view on exchanges in Quickselect
 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO
"... In this paper we study the number of key exchanges required by Hoare’s FIND algorithm (also called Quickselect) when operating on a uniformly distributed random permutation and selecting an independent uniformly distributed rank. After normalization we give a limit theorem where the limit law is a p ..."
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In this paper we study the number of key exchanges required by Hoare’s FIND algorithm (also called Quickselect) when operating on a uniformly distributed random permutation and selecting an independent uniformly distributed rank. After normalization we give a limit theorem where the limit law is a perpetuity characterized by a recursive distributional equation. To make the limit theorem usable for statistical methods and statistical experiments we provide an explicit rate of convergence in the Kolmogorov–Smirnov metric, a numerical table of the limit law’s distribution function and an algorithm for exact simulation from the limit distribution. We also investigate the limit law’s density. This case study provides a program applicable to other cost measures, alternative models for the rank selected and more balanced choices of the pivot element such as medianof2t+1 versions of Quickselect as well as further variations of the algorithm.
CONVERGENCE TO TYPE I DISTRIBUTION OF THE EXTREMES OF SEQUENCES DEFINED BY RANDOM DIFFERENCE EQUATION
"... Abstract. We study the extremes of a sequence of random variables (Rn) defined by the recurrence Rn = MnRn−1 + q, n ≥ 1, where R0 is arbitrary, (Mn) are iid copies of a non– degenerate random variable M, 0 ≤ M ≤ 1, and q> 0 is a constant. We show that under mild and natural conditions on M the su ..."
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Abstract. We study the extremes of a sequence of random variables (Rn) defined by the recurrence Rn = MnRn−1 + q, n ≥ 1, where R0 is arbitrary, (Mn) are iid copies of a non– degenerate random variable M, 0 ≤ M ≤ 1, and q> 0 is a constant. We show that under mild and natural conditions on M the suitably normalized extremes of (Rn) converge in distribution to a double exponential random variable. This partially complements a result of de Haan, Resnick, Rootzén, and de Vries who considered extremes of the sequence (Rn) under the assumption that P(M> 1)> 0. 1.