Our company succeeded to derive exact formulae for the distributions of words. The power of tests that depend on the statistics of words shows that these tests are superior compared to conventional tests. We can provide high performance statistical tests for random numbers, DNA analysis, etc.
The distributions of words are the number of the occurrences of words in finite strings. For example, the words 01 and 11 appear three and two times in the string 010011101. The distributions of words play important roles in information theory, statistics, DNA analysis, computer science, etc. In 17th century, French mathematician Pascal found distributions of letters, which are called binomial distributions, however the explicit formulae for the distributions of words such as 11 are obtained only after 1980 for limited cases.
Computers cannot generate true random numbers． Random numbers are often substituted by pseudo random numbers that are generated by computers. Pseudo random numbers are used in cryptography, computer simulation, Monte Carlo method,etc., and it is important to secure their randomness. We can provide high performance statistical tests that are not supported by standard test tools (NIST, die hard). For more details see Slides (ICIAM2023), Poster (ICIAM2023).