Last update: Fri Mar 15 02:02:37 MDT 2019
@Article{Hildebrand:1993:RPF,
author = "Martin Hildebrand",
title = "Random Processes of the Form {$ X_{n + 1} = a_n X_n +
b_n \pmod p $}",
journal = j-ANN-PROBAB,
volume = "21",
number = "2",
pages = "710--720",
month = apr,
year = "1993",
CODEN = "APBYAE",
ISSN = "0091-1798 (print), 2168-894X (electronic)",
ISSN-L = "0091-1798",
bibdate = "Sun Apr 20 10:44:17 MDT 2014",
bibsource = "http://www.math.utah.edu/pub/tex/bib/annprobab1990.bib;
http://www.math.utah.edu/pub/tex/bib/prng.bib",
URL = "http://www.jstor.org/stable/2244672;
http://projecteuclid.org/euclid.aop/1176989264",
acknowledgement = ack-nhfb,
fjournal = "Annals of Probability",
journal-URL = "http://projecteuclid.org/all/euclid.aop",
remark = "The author proposes a congruential generator whose
multiplier and constant come from two separate
independent random number streams. There is no
discussion of the period of such a generator, but if
$p$ is prime, the period should be the product of the
periods of the three generators, and based on the
effect of shuffling \cite{Bays:1976:IPR,Bays:1990:CIR},
any lattice structure should be well hidden. Hildebrand
shows that convergence to a uniformly distributed
sequence is rapid: under mild, and easily satisfied,
restrictions on $ a_n $ and $ b_n $ only $ O((\log
p)^2) $ steps are required. A test implementation of
such a generator using two 16-bit generators from
\cite{Kao:1996:EAP} with different prime moduli for the
$ a_n $ and $ b_n $, and $ p = 2^{32} - 5 $, passes the
Diehard Battery Test Suite.",
}