The explicit formula of the distributions of the nonoverlapping words and its applications to statistical tests for random numbers

Bassino et al. 2010 and Regnier et al. 1998 showed the generating functions of the distributions of the number of the occurrences of words (distributions of words for short) in finite string in the form of rational functions. However the coefficients of the expansion of the rational functions are complicated and we do not have a simple formula of the exact distributions of words from rational functions. In this paper we study the finite dimensional generating functions of the distribution of nonoverlapping words for each fixed sample size and show the explicit formula of the distributions of words for Bernoulli model. We demonstrate that 1) the tests based on the distributions of words reject the random number generator in BSD Library with p-value almost zero and 2) computation of the distributions of words in the human DNA size strings.