Period Length of the random numbers generated by rand() and randn()

Hello,

from what I understand, (uniform) random number generators will produce numbers that at some point will repeat.

Is this always the case? Is it possible to read somewhere how long this period length is for the uniform number generator rand()?

And does randn() use rand()? Is there a way to see how randn() is coded?

What I actually would like to know is the Period Length of randn(), if the output of this is actually repeating after a while.

Thank you very much for your help! :)

Greetings

Sofie

Answer 1

[Steven Lord]

See the "Choosing a Random Number Generator" section on this documentation page for a brief description of each of the available random number generators in MATLAB and their approximate periods.

We do not distribute the source code for either rand or randn.

[Sofie Brammer]

Perfect, that is what I was searching for.

Thank you very much!

Answer 2

[Jonas]

don't confuse yourself, random number do not have a period length, but what you actually mean may be the rate e.g. after which a number may appear again. this is not deterministic, but you can measure e.g. the mean number of lements after which a specific number appears again:

short example with numbers 1 to 100

numberOfPossibilities=100;

data=randi(numberOfPossibilities,1e6,1);

for nr=1:numberOfPossibilities
    where=find(data==nr);
    spacings=diff(where);
    meanSpacing(nr,1)=mean(spacings);
end
disp(meanSpacing')

  Columns 1 through 19

   99.8758  100.6833  101.6778  101.5189   99.8014  100.6575  100.1841  101.7151  100.2505  100.5372  100.0663   98.9614  101.1416   98.2634   97.5752  101.8207   99.1696   99.2484   99.4164

  Columns 20 through 38

  100.1717  100.1275  100.4499   99.7139   98.7456  101.6561   98.9993   99.8174   99.9619  100.3358   98.5253   97.4752   99.9273   99.0303  101.2657  100.9726  101.1186   97.5004  101.2726

  Columns 39 through 57

  100.8305   99.5587   98.4710   99.4178   99.3804   98.8412   99.9486  100.9847   99.5266  100.0800  101.2774  100.0746   99.4222  100.0274   98.1896   98.3103  100.2012   99.7047   99.8475

  Columns 58 through 76

   99.8735  100.9039  101.2885  101.5181   99.3311  101.0973  100.2707   99.2883  100.9896   98.5731  101.1047  101.6660  100.6660  100.0169   99.9413  101.3303  101.2446  100.1078   99.8385

  Columns 77 through 95

  100.6228  100.3901   99.5440  100.2714   99.1937   99.6922   99.7770  100.2451   99.4812   99.2587  100.0448  100.6201   98.9741   99.1781   99.5432   99.6016  100.5860  100.1544   98.4221

  Columns 96 through 100

  100.5930  101.2390   98.7681  101.2734   99.6102

as you can see, this number is near the number of possible integer values

but the output almost never repeats itself if there are enough ouput values possible. E.g. if you have 1000 possible outputs and a sequence of 1e6 values, the probability of an exact repetition is (1/1000)^(1e6)

note again: random numbers are random, if you have uniform distribution, the probability of a specific number is always the the same, regardless if you have drawn a number for the first time or for the 1000th time.

[Sofie Brammer]

Thank you very much for your answer! :)

I was already worrying that the term "period length" might be confusing, I just could not find a better phrase.

What I mean by that is not that a single random number would have a period length, but that if i generate a veeeery long sequence (or vector) of random numbers (e.g. using rand()), then this would (for my understanding) at some point repeat.

Because what we generate are not really random umbers, but pseudo-random numbers according to some algorithm, e.g. Mersenne-Twister MT19937ar (https://en.wikipedia.org/wiki/Mersenne_Twister). And these algorithms are periodic.

In the "Handbook of Monte Carlo Methods" (Kroese) I read that this MT19937ar algorithm was (at 2011) the default for random number generation in MATLAB.

But I am neither sure that it is explicitely (and still) behind rand() nor that randn() uses rand() - so I am not sure if I can conclude that the period length is that of MT19937ar.

Still, I guess I don't need to worry that the period length might be too short using MATLAB build-in functions :D

Would just be nice to know for sure about what is going on.