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Random numbers and computers
Kneusel R., Springer International Publishing, New York, NY, 2018. 260 pp. Type: Book (978-3-319776-96-5)
Date Reviewed: Feb 13 2019

[CR has previously published a review of this book (see Review CR146350). The author of the book has written a rebuttal to the review (see Review CR146432), and the reviewer has written a re-rebuttal.]

RE-REBUTTAL

There is no error in the review. The error lies in the way the author has interpreted it. It is not the randomness of pi that is missing from the book, but why it is important, as explicitly stated in my review. The following remark, found in Section 7.1, is grossly misleading:

In this chapter we will, primarily for fun and because they are intellectually interesting, consider other sequences that pass randomness tests but which are generally not suitable for use where pseudorandom numbers are typically used even though a computer program can be used to generate the sequence.

The author is missing two crucial points:

(1) Irrational numbers by virtue of having a non-repeating and non-terminating decimal expansion will not have a cycle that is present in most standard pseudorandom number generators (PRNGs). Thus, in simulation studies with PRNG, the entire simulation work must be over before the cycle breaks or else the simulation results will be seriously affected. This point is already briefly mentioned in my review.
(2) Matsumoto et. al [1] investigated 58 PRNGs and found as many as 40 to be defective. This was caused by improper seed selection and not recursion as claimed in the paper. When an irrational number is proposed as a random number, this seed is the position from which a substring is taken in its decimal expansion. Given that overall randomness of the decimal expansion does not imply mutual randomness for which the substrings also have to be random, it is important to know which irrational numbers can contest with the PRNG over a wide range. This is a serious research problem and not some kind of recreational mathematics. In this context, it is important to mention that Marsaglia [2] rejects Tu and Fischbach’s claim [3] that pi is less random than we thought.

In summary, due to the presence of cycle and due to the possibility of improper seed selection, pseudorandom numbers are actually inferior to irrational numbers so long as the latter exhibits mutual randomness over a sufficiently wide range. Like Matsumoto et. al. [1], I too would recommend using PRNGs only for a nonsystematic choice of seeds or a better scheme of initialization. If neither is ensured, using an irrational number is a better and safer option.

Reviewer:  Soubhik Chakraborty Review #: CR146433
1) Matsumoto, M.; Wada, I.; Kuramoto, A.; Ashihara, H. Common defects in initialization of pseudorandom number generators. ACM Transactions on Modeling and Computer Simulation 17, 4 (2007), Article No. 15 .
2) Marsaglia, G. Refutation of claims such as “Pi is less random than we thought.” http://interstat.statjournals.net/YEAR/2006/articles/0601001.pdf (accessed 1/27/2019).
3) Tu, S.-J.; Fischbach, E. A study on the randomness of the digits of pi. International Journal of Modern Physics C 16, 2(2005), 281–294.
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