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Uniform Probability Distribution and Basic Pseudo- and Quasi-Random Number Generators

- Theoretical evaluation is the first stage in rejecting bad generators. Theoretical research provides the basis for better understanding of generator properties, such as its period length, lattice structure, discrepancy, or equidistribution. The results obtained through theoretical testing refer to a basic generator used over the entire period.
- Empirical tests ensure that the remaining generators are of acceptable quality by testing a generator over a small fraction of the period actually used.

^{31}, which can be exhausted within seconds on modern Intel® processors. Taking into account that good statistical behavior of the generator is observed only over a fraction of its period (B.D. Ripley [Ripley87] recommends to take no more than a square root of the period length) you may not consider such period length acceptable. However, in Monte Carlo applications with a relatively small quantity of random numbers to be used, such generators may be useful because of the speed, small memory requirements for keeping the generator state, and efficient methods available for generation of random subsequences. For example, while estimating a global solution to an integral equation through the Monte Carlo method, the same random numbers should be used for different parameters [Mikh2000]. Modern computational capacities require BRNGs of at least 2

^{60}period length. All other VS BRNGs meet these requirements.

`that result in good quality properties of the output sequence in terms of period length, lattice structure, discrepancy, equidistribution, and so on. In particular, ifk, ai, m`

`is a prime number selected with proper coefficientsm`

`, a period length of orderai`

`may be obtained. Nevertheless,mk`

`is often taken as 2m`

`(p`

`>1) because of efficient modulop`

`reduction. Some authors do not recommend usingm`

`in the form of a power of 2 as the lower bits of the generated random numbers prove to be non-random on the whole. For example, see D. Knuth [Knuth81], P. L’Ecuyer [L'Ecu94]. However, this is irrelevant for most of Monte Carlo applications. Moreover, even ifm`

`is a prime number, you should also be careful when selecting random bits in the output sequence.m`

`as well. Quasi-random sequences filling a space according to a non-uniform distribution can be generated by transforming a sequence produced by a basic quasi-random number generator. In most cases, tests designed for pseudorandom number generators cannot be used for quasi-random number generators. Special batteries of tests should be designed for basic quasi-random number generators.Basic Random Number Generators`