Document Type

Article

Department

​​Physics

Publication Date

2-2014

Abstract

Run count statistics serve a central role in tests of non-randomness of stochastic processes of interest to a wide range of disciplines within the physical sciences, social sciences, business and finance, and other endeavors involving intrinsic uncertainty. To carry out such tests, it is often necessary to calculate two kinds of run count probabilities: 1) the probability that a certain number of trials results in a specified multiple occurrence of an event, or 2) the probability that a specified number of occurrences of an event take place within a fixed number of trials. The use of appropriate generating functions provides a systematic procedure for obtaining the distribution functions of these probabilities. This paper examines relationships among the generating functions applicable to recurrent runs and discusses methods, employing symbolic mathematical software, for implementing numerical extraction of probabilities. In addition, the asymptotic form of the cumulative distribution function is derived, which allows accurate runs statistics to be obtained for sequences of trials so large that computation times for extraction of this information from the generating functions could be impractically long.

Comments

Originally published as an Open Access article as:

Open Journal of Statistics, 2014, 4, 122-131 Published Online February 2014 (http://www.scirp.org/journal/ojs) http://dx.doi.org/10.4236/ojs.2014.42014

Copyright © 2014 M. P. Silverman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2014 are reserved for SCIRP and the owner of the intellectual property M. P. Silverman. All Copyright © 2014 are guarded by law and by SCIRP as a guardian.

Creative Commons License


This work is licensed under a Creative Commons Attribution 3.0 License.