Abstract. This is a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.

The ziggurat is an efficient method to generate normal random samples. It is shown that the standard Ziggurat fails a commonly used test. An improved version that passes the test is intro-duced. Flexibility is enhanced by using a plug-in uniform random number generator. An efficient double-precision version of the ziggurat algorithm is developed that has a very high period.

Abstract. We develop exact random variate generators for the polynomially and exponentially tilted unilateral stable distributions. The algorithms, which generalize Kanter’s method, are uniformly fast over all choices of the tilting and stable parameters. The key to the solution is a new distribution which we call Zolotarev’s distribution. We also present a novel double rejection method that is useful whenever densities have an integral representation involving an auxiliary variable.

Ziggurat and Monty Python are two fast and elegant methods proposed by Marsaglia and Tsang to transform uniform random variables to random variables with normal, exponential and other common probability distributions. While the proposed methods are theoretically correct, we show that there are various design flaws in the uniform pseudo random number generators (PRNG's) of their published implementations for both the normal and Gamma distributions [1, 2, 3]. These flaws lead to non-uniformity of the resulting pseudo-random numbers and consequently to noticeable deviations of their outputs from the required distributions. In addition, we show that the underlying uniform PRNG of the published implementation of Matlab's randn, which is also based on the Ziggurat method, is not uniformly distributed with correlations between consecutive pairs. Also, we show that the simple linear initialization of the registers in matlab's randn may lead to non-trivial correlations between output sequences initialized with di#erent (related or even random unrelated) seeds. These, in turn, may lead to erroneous results for stochastic simulations.

This paper proposes an improved gamma random generator. In the past, a lot of gamma random number generators have been proposed, and depending on a shape parameter (say, alpha) they are roughly classified into two cases: (i) alpha lies on the interval (0,1) and (ii) alpha is greater than 1, where alpha=1 can be included in either case. In addition, Cheng and Feast (1980) extended the gamma random number generator in the case where alpha is greater than 1/n, where n denotes an arbitrary positive number. Taking n as a decreasing function of alpha, in this paper we propose a simple gamma random number generator with shape parameter alpha greater than zero. The proposed algorithm is very simple and shows quite good performance.

Abstract. We consider the problem of the exact simulation of random variables Z that satisfy the distributional identity Z L = V Y + (1 − V)Z, where V ∈ [0, 1] and Y are independent, and L = denotes equality in distribution. Equivalently, Z is the limit of a Markov chain driven by that map. We give an algorithm that can be automated under the condition that we have a source capable of generating independent copies of Y, and that V has a density that can be evaluated in a black box format. The method uses a doubling trick for inducing coalescence in coupling from the past. Applications include exact samplers for many Dirichlet means, some two-parameter Poisson–Dirichlet means, and a host of other distributions related to occupation times of Bessel bridges that can be described by stochastic fixed point equations. Keywords and phrases. Random variate generation. Perpetuities. Coupling from the past. Random partitions. Stochastic recurrences. Stochastic fixed point equations. Distribution theory. Markov chain Monte Carlo. Simulation. Expected time analysis. Bessel bridge. Poisson-Dirichlet. Dirichlet means.

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Abstract. We present a method to apply the Malliavin calculus to calculate sensitivities for exponential Levy models built from the Variance Gamma and Normal Inverse Gaussian processes. We also present new sensitivities for these processes. The calculation of the sensitivities is based on a finite dimensional Malliavin calculus and we compare the results with finite difference calculations. This is done using Monte Carlo methods. For European call and digital options we compare the simulation results with exact calculation of sensitivities using Fourier transform methods. The Malliavin method outperforms the finite difference method especially when payoff has serious discontinuities.

Abstract: The Amoroso distribution is the natural unification of the gamma and extreme value families of distributions. Over 50 distinct, named distributions (and twice as many synonyms) occur as special cases or limiting forms. Consequently, this single simple functional form encapsulates and systematizes an extensive menagerie of interesting and common probability distributions.