Abstract. This chapter provides 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.

Applying the ratio-of-uniforms method for generating random variates results in very efficient, fast and easy to implement algorithms. However parameters for every particular type of density must be precalculated analytically. In this paper we show, that the ratio-of-uniforms method is also useful for the design of a black-box algorithm suitable for a large class of distributions, including all with log-concave densities. Using polygonal envelopes and squeezes results in an algorithm that is extremely fast. In opposition to any other ratio-of-uniforms algorithm the expected number of uniform random numbers is less than two. Furthermore we show that this method is in some sense equivalent to transformed density rejection.

this paper we show that transformed density rejection is well suited to construct universal algorithms suitable for correlation induction which is important for variance reduction in simulation. 2 Transformed density rejection

this paper we introduce an new approach for universal bounding curves based on the ratio-of-uniforms method. The new algorithm is even simpler and can be applied to a larger class of distributions, including all log-concave distributions. As for Devroye's algorithm the expected number of uniform random numbers does not depend on the particular distribution. In opposition to other black-box algorithms hardly any setup step is required. Thus it is superior in the changing parameter chase.

In this paper we present some variants of transformed density rejection (TDR) that provide more exibility (including the possibility to halve the expected number of uniform random numbers) at the expense of slightly higher memory requirements. Using a synchronized rst stream of uniform variates and a second auxiliary stream (as suggested by Schmeiser and Kachitvichyanukul (1990)) TDR is well suited for correlation induction. Thus high positive and negative correlation between two streams of random variates with same or dierent distributions can be induced.

This paper presents an overview of the most powerful universal methods. These are based on acceptance/rejection techniques where hat and squeezes are constructed automatically. Although originally motivated to sample from non-standard distributions these methods have advantages that make them attractive even for sampling from standard distributions and thus are an alternative to special generators tailored for particular distributions. Most important are: the marginal generation time is fast and does not depend on the distribution. They can be used for variance reduction techniques, and they produce random numbers of predictable quality. These algorithms are implemented in a library, called UNURAN, which is available by anonymous ftp.

Abstract. We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions for a particular generator. The algorithms can be implemented in a few lines of high level language code. 1.

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This short note discusses performance bounds for "Ahrens" algorithm, that can generate random variates from continuous distributions with monotonically decreasing density. This rejection algorithm uses constant hat-functions and constant squeezes over many small intervals. The choice of these intervals is important. Ahrens has demonstrated that the equal area rule that uses strips of constant area leads to a very simple algorithm.

There exists a vast literature on nonuniform random variate generators. Most of these generators are especially designed for a particular distribution. However in pratice only a few of these are available to practioners. Moreover for problems as (e.g.) sampling from the truncated normal distribution or sampling from fairly uncommon distributions there are often no algorithms available. In the last decade so called universal methods have been developed for these cases. The resulting algorithms are fast and have properties that make them attractive even for standard distributions.