NONUNIFORM RANDOM VARIATE GENERATION
Luc Devroye
School of Computer Science
McGill University
Abstract.
This chapter provides a survey of the main methods in nonuniform 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 ex
pected time complexity of various algorithms, before addressing modern topics such as indirectly
specified distributions, random processes, and Markov chain methods.
Authors’ address: School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A
2K6. The authors’ research was sponsored by NSERC Grant A3456 and FCAR Grant 90ER0291.
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1. The main paradigms
The purpose of this chapter is to review the main methods for generating random vari
ables, vectors and processes. Classical workhorses such as the inversion method, the rejection
method and table methods are reviewed in section 1.
In section 2, we discuss the expected
time complexity of various algorithms, and give a few examples of the design of generators that
are uniformly fast over entire families of distributions. In section 3, we develop a few universal
generators, such as generators for all log concave distributions on the real line. Section 4 deals
with random variate generation when distributions are indirectly specified, e.g, via Fourier coef
ficients, characteristic functions, the moments, the moment generating function, distributional
identities, infinite series or Kolmogorov measures. Random processes are briefly touched upon
in section 5. Finally, the latest developments in Markov chain methods are discussed in section
6. Some of this work grew from Devroye (1986a), and we are carefully documenting work that
was done since 1986. More recent references can be found in the book by H¨
ormann, Leydold
and Derflinger (2004).
Nonuniform random variate generation is concerned with the generation of random
variables with certain distributions. Such random variables are often discrete, taking values in
a countable set, or absolutely continuous, and thus described by a density. The methods used
for generating them depend upon the computational model one is working with, and upon the
demands on the part of the output.
For example, in a
ram
(random access memory) model, one accepts that real numbers
can be stored and operated upon (compared, added, multiplied, and so forth) in one time unit.
Furthermore, this model assumes that a source capable of producing an i.i.d. (independent
identically distributed) sequence of uniform [0
,
1] random variables is available. This model is of
course unrealistic, but designing random variate generators based on it has several advantages:
first of all, it allows one to disconnect the theory of nonuniform random variate generation from
that of uniform random variate generation, and secondly, it permits one to plan for the future,
as more powerful computers will be developed that permit ever better approximations of the
model. Algorithms designed under finite approximation limitations will have to be redesigned
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 Fall '10
 Tarter
 Normal Distribution, Probability theory, Devroye

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