Uniform random numbers are the fundamental building blocks for the generation of random numbers of any distribution. The key criterion for a good sequence of uniform random numbers is that it is as random as possible" which means that there should not be any "clustering" or "patterns" in the generated values.
Early uniform random generators were recursive in nature - they usually start with a seed and obtain the next value in the sequence by applying a simple formula to the previous value.
(1) Pseudo random number generators
- Pros: Numbers can be generated very quickly and efficiently
- Cons: Numbers generally exhibit clustering or patterns when plotted in certain ways.
The most common such pseudo random number generator is the linear congruential generator, which generates an integer in {0; 1; : : : ; n - 1} by the rule:
Note that a random integer between 0 and n - 1 can be converted to a random number between 0 and 1 by dividing by n. It can be shown that random integers generated as above has a period, and the sequence repeats once this period is reached. Clearly, it is desirable to have the period as large as possible to avoid repetition.
Reference: https://en.wikipedia.org/wiki/Linear_congruential_generator
(2) Low discrepancy sequences: designed to generate components of vectors to fill the space of given dimension as uniformly as possible.
A popular low discrepancy sequence is the Sobol sequence.
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