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.
M. Silverman, "Numerical Procedures for Calculating the Probabilities of Recurrent Runs," Open Journal of Statistics, Vol. 4 No. 2, 2014, pp. 144-153. doi: 10.4236/ojs.2014.42014.
[1] M. P. Silverman and W. Strange, “Search for Correlated Fluctuations in the β+ Decay of Na-22,” Europhysics Letters, Vol. 87, No. 3, 2009, pp. 32001-32006. http://dx.doi.org/10.1209/0295-5075/87/32001 [2] D. Sornette, “Critical Market Crashes,” Physics Reports, Vol. 378, No. 1, 2003, pp. 1-98.
http://dx.doi.org/10.1016/S0370-1573(02)00634-8 [3] J. Wolfowitz, “On the Theory of Runs with Some Applications to Quality Control,” The Annals of Mathematical Statistics, Vol. 14, No. 3, 1943, pp. 380-288.
http://dx.doi.org/10.1214/aoms/1177731421 [4] A. M. Mood, “The Distribution Theory of Runs,” The Annals of Mathematical Statistics, Vol. 11, No. 4, 1940, pp. 367-392.
http://dx.doi.org/10.1214/aoms/1177731825 [5] H. Levene and J. Wolfowitz, “The Covariance Matrix of Runs Up and Down,” The Annals of Mathematical Statistics, Vol. 15, No. 1, 1944, pp. 58-69. http://dx.doi.org/10.1214/aoms/1177731314 [6] M. P. Silverman, W. Strange, C. Silverman and T. C. Lipscombe, “Tests for Randomness of Spontaneous Quantum Decay,” Physical Review A, Vol. 61, No. 4, 2000, Article ID: 042106.
http://dx.doi.org/10.1103/PhysRevA.61.042106 [7] M. P. Silverman and W. Strange, “Experimental Tests for Randomness of Quantum Decay Examined as a Markov Process,” Physics Letters A, Vol. 272, No. 1-2, 2000, pp. 1-9.
http://dx.doi.org/10.1016/S0375-9601(00)00374-1 [8] W. Feller, “Fluctuation Theory of Recurrent Events,” Transactions of the American Mathematical Society, Vol. 67, 1949, pp. 98-119. http://dx.doi.org/10.1090/S0002-9947-1949-0032114-7 [9] D. Branning, A. Katcher, W. Strange and M. P. Silverman, “Search for Patterns in Sequences of Single-Photon Polarization Measurements,” Journal of the Optical Society of America B, Vol. 28, No. 6, 2011, pp. 1423-1430.
http://dx.doi.org/10.1364/JOSAB.28.001423 [10] M. P. Silverman, W. Strange, J. Bower and L. Ikejimba, “Fragmentation of Explosively Metastable Glass,” Physica Scripta, Vol. 85, 2012, Article ID: 065403. http://dx.doi.org/10.1088/0031-8949/85/06/065403 [11] A. M. Mood, F. A. Graybill and D. C. Boes, “Introduction to the Theory of Statistics,” 3rd Edition, McGraw-Hill, New York, 1974, pp. 540-541.