Abstract not found

Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables distributed uniformly over the interval

Tausworthe random number generators based on a primitive trinomial allow an easy and fast implementation when their parameters obey certain restrictions. However, such generators, with those restrictions, have bad statistical properties unless we combine them. A generator is called maximally equidistributed if its vectors of successive values have the best possible equidistribution in all dimensions. This paper shows how to find maximally equidistributed combinations in an efficient manner, and gives a list of generators with that property. Such generators have a strong theoretical support and lend themselves to very fast software implementations.

Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed via their equidistribution properties. The huge-period generators proposed so far are not quite optimal in that respect. In this paper, we propose new generators of that form, with better equidistribution and “bit-mixing ” properties for equivalent period length and speed. The state of our new generators evolves in a more chaotic way than for the Mersenne twister. We illustrate how this can reduce the impact of persistent dependencies among successive output values, which can be observed in certain parts of the period of gigantic generators such as the Mersenne twister.

this paper, we shall assume that the sequence is purely periodic, in the sense that the initial state s 0 is always revisited. In other words, the sequence has no transient part. The goal is to make it hard to distinguish between the output of the generator and a typical realization of an i.i.d. uniform sequence over U . In

We give the results of a computer search for maximally-equidistributed combined linear feedback shift register (or Tausworthe) random number generators, whose components are trinomials of degrees slightly less than 32 or 64. These generators are fast and have good statistical properties.

We recall some requirements for "good" random number generators and argue that while the construction of generators and the choice of their parameters must be based on theory, a posteriori empirical testing is also important. We then give examples of tests failed by some popular generators and examples of generators passing these tests.

Thispapersummarizesthecurrentstate-of-the-art onuniformrandomnumbergenerationforstochasticsimulation. Itrecallsthebasicideas,discusses somelinearmethodsandtheirtheoreticalanalysis, andprovidespointerstofurtherdetailsandtorecommendedimplementations. 1 WHATISAGOODRNG? Withoutagoodrandomnumbergenerator(RNG), simulationresultsareoftenmeaningless.Andquestionablegeneratorsarestillallovertheplace, somany experimentsrestonshakyfoundations.Whythis problemwasnotsolvedlongago?Becauseitisnot soeasy.Aso-calledRNGactuallyproducesatotally deterministicandperiodicsequenceofnumbers,once itsinitialstate(orseed)ischosen.Thisisintotal contradictionwiththeassumptionofasequenceofindependentandidenticallydistributed (i.i.d.)random variables,andthereisnocleanwaytocompletely reconcilethesetwooppositeaspects.Therefore,everythingwedointhiscontextisheuristic. Thisbeingsaid, theheuristicargumentsleadtocriteriathat needtheorytobeanalyzed.

: Uniformity tests based on a discrete form of entropy are introduced and studied in the context of empirical testing of uniform random number generators. Comparisons are made with tests based on the Pearson chi-square statistic. Numerical results are provided and several currently used generators fail the tests. The linear congruential and nonlinear inversive generators with power-of-two modulus perform especially badly. CR Categories and Subject Descriptors: G.3 [Probability and Statistics]: Random Number Generation General Terms: Algorithms, statistics Additional Key Words and Phrases: Random number generators; statistical tests; goodness-of-fit; entropy. Authors' Addresses: P. L'Ecuyer and J.-F. Cordeau, D'epartement d'Informatique et de Recherche Op'erationnelle (IRO), Universit'e de Montr'eal, C.P. 6128, Succ. Centre-Ville, Montr'eal, H3C 3J7, Canada; e-mail: lecuyer@iro.umontreal.ca A. Compagner, Faculty of Applied Physics, Delft University of Technology, P.O. Box 5046, 2600...

This paper discusses certain classes of uniform random number generators which have been studied and better understood in the recent few years. Most of the attention is devoted to combined generators. We also mention others and point out some pitfalls. Combination is a good way to obtain fast and reliable generators, but the structural properties of the combined generator should be carefully examined before it could be recommended. Nonlinear generators offer some promise, but still require deeper investigation before specific instances can be safely recommended. 1. INTRODUCTION Random numbers are the nuts and bolts of all stochastic simulations. Simple linear congruential generators (LCGs) (Bratley, Fox, and Schrage 1987; Knuth 1981) are still in widespread use for generating uniform random numbers, mainly because of their simplicity and ease of implementation. However, LCGs have several well-known defects and no longer satisfy the requirements of today's computerintensive simulations...