English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The distributions of the number of occurrences of words (the distributions ofwords for short) play key roles in information theory, statistics, probabilitytheory, ergodic theory, computer science, and DNA analysis.  Bassino et al. 2010 and Regnier et al. 1998 showed generating functions ofthe distributions of words for all sample sizes. Robin et al. 1999 presentedgenerating functions of the distributions for the return time of words anddemonstrated a recurrence formula for these distributions. These generatingfunctions are rational functions; except for simple cases, it is difficult toexpand them into power series. In this paper, we study finite-dimensionalgenerating functions of the distributions of nonoverlapping words for eachfixed sample size and demonstrate the explicit formulae for the distributionsof words for the Bernoulli models. Our results are generalized tononoverlapping partial words. We study statistical tests that depend on thenumber of occurrences of words and the number of block-wise occurrences ofwords, respectively. We demonstrate that the power of the test that depends onthe number of occurrences of words is significantly large compared to the otherone. Finally, we apply our results to statistical tests for pseudo randomnumbers.

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