On cumulative sums of random numbers

Abstract

view Abstract Citations (6) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On cumulative sums of random numbers van Woerkom, A. J. J. Abstract Sections 2, 3 and 4 deal with some general statistical properties of cumulative series of random numbers. it is shown that the first and second summations, if multiplied by the 21 power and the 2 power of the number of steps, fl, respec- tively, and plotted against log n form Markoff chains. The statistical properties of a linear and a parabolic expression, which are fitted to the first and second summation, respectively, are discussed. in the case of the second summation the mean values of the square of the constant and the coefficient of the linear term depend mostly on the order of the initial term while the coefficient of the quadratic term depends principally on the order of the last term in the summation to which the parabolic expression is fitted. A similar result is found for the linear expression fitted to the first summation. in the case of the second summation the mean value of the square of the quadratic term is inversely proportional to the order of the final term. The agreement with experimental values found from a sample of 15 independent summations is satisfactory. Sub- traction of the parabolic or linear expression from the respective summation will result in a relative reduction of the standard deviation of the random numbers. The values found for this reduction factor are 5.1 2.9 and 1.49 .47, respectively. in Section 5 a statistical analysis is made of the annual values of the observed fluctuations in the moon's mean longi- tude for the period from 1820 to 1950. Serial correlations were formed for these annual values, as well as for its first and second differences. The results confirm Brouwer's hypothesis that the change in the rate of rotation of the earth is pri- marily due to cumulative random changes. in Section 6 an analysis is made of the same material as used in Section 5 in order to determine the standard deviation of the random changes from year to year. The method of variate differences is used in which sums of squares of successive differences are analyzed. Differences taken at intervals of I year, 2 years and 4 years yield values for the standard devia- tion 1.14, 1.17 and 1.22, respectively. Publication: The Astronomical Journal Pub Date: February 1953 DOI: 10.1086/106801 Bibcode: 1953AJ.....58...10V full text sources ADS |