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PERFECT SIMULATION OF VERVAAT PERPETUITIES
, 908
"... Abstract. We use coupling into and from the past to sample perfectly in a simple and provably fast fashion from the Vervaat family of perpetuities. The family includes the Dickman distribution, which arises both in number theory and in the analysis of the Quickselect algorithm, which was the motivat ..."
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Abstract. We use coupling into and from the past to sample perfectly in a simple and provably fast fashion from the Vervaat family of perpetuities. The family includes the Dickman distribution, which arises both in number theory and in the analysis of the Quickselect algorithm, which was the motivation for our work.
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.
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 suita ..."
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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.
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 be spli ..."
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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,