### RANDOM NUMBER GENERATION (LUC DEVROYE)

 L. Devroye, J. Fill and R. Neininger, Perfect Simulation from the Quicksort Limit Distribution, Electronic Communications in Probability, vol. 5, pp. 95-99, 2000. PDF file. Technical Report #603, Department of Mathematical Sciences, The Johns Hopkins University. We show how to generate a random variable from the quicksort limit distribution (for which only a distributional identity is known). L. Devroye and R. Neininger, Density approximation and exact simulation of random variables that are solutions of fixed-point equations, Advances in Applied Probability, 2002, to appear. (PDF version). An algorithm is developed for the exact simulation from distributions that are defined as fixed-points of maps between spaces of probability measures. The fixed-points of the class of maps under consideration include examples of limit distributions of random variables studied in the probabilistic analysis of algorithms. The sampling algorithm relies on a modified rejection method. L. Devroye, Simulating perpetuities Methodologies and Computing in Applied Probability, vol. 3, pp. 97-115, 2001. (PDF version). A perpetuity is a random variable that can be represented as $1+ W_1 + W_1 W_2 + W_1 W_2 W_3 + \cdots$, where the $W_i$'s are i.i.d.\ random variables. We study exact random variate generation for perpetuities and discuss the expected complexity. For the Vervaat family, in which $W_1 \inlaw U^{1/\beta}$, $\beta > 0$, $U$ uniform $[0,1]$, all the details of a novel rejection method are worked out. There exists an implementation of our algorithm that only uses uniform random numbers, additions, multiplications and comparisons. L. Devroye, Random variate generation in one line of code, 1996 Winter Simulation Conference Proceedings, J.M. Charnes, D.J. Morrice, D.T. Brunner and J.J. Swain eds, ACM, pp. 265-272, 1996. (PDF version). For many distributions, it is possible to generate a random variate in one assignment statement using only simple mathematical functions and ordinary arithmetic operations. L. Devroye, Simulating Bessel random variables, 2002. To appear in Statistics and Probability Letters. (PDF version). This paper develops a uniformly fast algorithm for generating random variates from the Bessel distribution. No Bessel function or Bessel ratio is needed. L. Devroye, On random variate generation when only moments or Fourier coefficients are known, Mathematics and Computers in Simulation, vol. 31, pp. 71-89, 1989. We consider algorithms for generating random variates having a density, when only its Fourier coefficients or moments are known. We also study the expected time per random variate. L. Devroye, Simulating theta random variates, Statistics and Probability Letters, vol. 31, pp. 275-2791, 1997. We develop an exact simple random variate generator for the theta distribution, which occurs as the limit distribution of the height of nearly all models of uniform random trees. Even though the density is only known as an infinite sum of functions, our algorithm does not require any summation. L. Devroye, Random variate generation for the digamma and trigamma distributions, Journal of Statistical Computation and Simulation, vol. 43, pp. 197-216, 1992. We derive uniformly fast random variate generators for Sibuya's digamma and trigamma families. Some of these generators are based upon the close resemblance between these distributions and selected generalized hypergeometric distributions. The generators can also be used for the discrete stable distribution, the Yule distribution, Mizutani's distribution and the Waring distribution. L. Devroye, On random variate generation for the generalized hyperbolic secant distributions, Statistics and Computing, vol. 3, pp. 125-134, 1993. We give random variate generators for the generalized hyperbolic secant distribution and related families such as Morris's skewed generalized hyperbolic secant family and a family introduced by Laha and Lukacs. The rejection method generators are uniformly fast over the parameter space and are based upon a complex function representation of the distributions due to Harkness and Harkness. L. Devroye, A triptych of discrete distributions related to the stable law, Statistics and Probability Letters, vol. 18, pp. 349-351, 1993. We derive useful distributional representations for the discrete stable distribution of Steutel and Van Harn, the discrete Linnik distribution introduced by Pakes, and a discrete distribution of Sibuya. These representations may be used to obtain simple uniformly fast random variate generators. L. Devroye, Algorithms for generating discrete random variables with a given generating function or a given moment sequence, SIAM Journal on Scientific and Statistical Computing, vol. 12, pp. 107-126, 1991. We present and analyze various algorithms for generating positive integer-valued random variables when the distribution is described either through the generating function or via the sequence of moments. L. Devroye, Random variate generation for multivariate unimodal densities, ACM Transactions on Modeling and Computer Simulation, vol. 7, pp. 447-477, 1997. A probability density on a finite-dimensional Euclidean space is ortho-unimodal with a given mode if within each orthant (quadrant) defined by the mode, the density is a monotone function of each of its arguments individually. Up to a linear transformation, most of the commonly used random vectors possess ortho-unimodal densities. To generate a random vector from a given ortho-unimodal density, several general-purpose algorithms are presented; and an experimental performance evaluation illustrates the potential efficiency increases that can be achieved by these algorithms versus naive rejection. L. Devroye, The branching process method in random variate generation, Communications in Statistics---Simulation, vol. 21, pp. 1-14, 1992. The generalized Lagrange probability distributions include the Borel-Tanner distribution, Haight's distribution, the Poisson-Poisson distribution and Consul's distribution, to name a few. We introduce two universally applicable random variate generators for this family of distributions. In the branching process method, we produce the generation sizes in a Galton-Watson branching process. In the uniform bounding method, we employ the rejection method based upon a simple probability inequality that is valid for all members in a given subfamily.

### RANDOM NUMBER GENERATION JOURNALS

 ACM Transactions on Mathematical Software ACM Transactions on Modeling and Computer Simulation INFORMS College on Simulation: go here for the free full version of the Winter Simulation Conference (WSC) Proceedings.

### NON-UNIFORM RANDOM NUMBER GENERATION LINKS

 winrand 1.0/95 (CRAND) CRAND is the highest quality package (in my opinion) for non-uniform random variate generation, developed and implemented by Ernst Stadlober. In C++. RANLIB Random number generation package by Brown, Movato and Russell. Here are the files. Download the C or FORTRAN implementations. Regress+ Mac-based freeware package for fitting models to data. It includes as an essential component a battery of non-uniform random variate generators (currently for 29 distributions, soon for 50). Developed by Michael P. McLaughlin. Compendium of probability distributions An on-line list of distributions compiled by Mike McLaughlin. Non-Uniform Random Variate Generation Boris Shukhman has programmed most algorithms from my book in C, but the code is not suitable for general consumption yet. Also, the algorithms are not as fast as those developed by Ernst Stadlober, Wolfgang Hörmann, Jo Ahrens and Ulrich Dieter. Alan Miller's programs Alan Miller from the CSIRO Mathematical & Information Sciences programmed most algorithms from Non-Uniform Random Variate Generation. Newran02A A C++ library for generating sequences of random numbers from a wide variety of distributions. Gaussian random number generator A fast time-correlated gaussian generator. FORTRAN.