Abstract
The well-known problem of the absence of independent events in some discrete probability spaces (finite or countable) is considered. It is proposed to study such spaces using the minimum absolute value of the correlation coefficient of event indicators (in the case of a countable space, the infimum is taken). Examples of probability space with a prime number of equally possible outcomes, a finite space with weights and irrationality, a geometric space with a prime number of outcomes, and a countable space with probabilities given by the sum of three geometric progressions are considered.
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REFERENCES
O. P. Vinogradov, ‘‘Prime numbers and independence,’’ Sovrem. Probl. Mat. Mekh. 7 (1), 16–21 (2011).
O. P. Vinogradov, ‘‘On independent events in families of discrete distributions,’’ Discrete Math. Appl., 24 (1), 53–59 (2014).
R. C. Shiflett and H. S. Shultz, ‘‘An approach to independent sets,’’ Math. Spectrum 12, 11–16 (1979).
B. Eisenberg and B. K. Ghosh, ‘‘Independent events in a discrete uniform probability space,’’ Am. Stat. 41, 52–56 (1987). https://doi.org/10.1080/00031305.1987.10475443
Yu. M. Baryshnikov and B. Eisenberg, ‘‘Independent events and independent experiments,’’ Proc. Am. Math. Soc. 118, 615–617 (1993). https://doi.org/10.1090/S0002-9939-1993-1146858-9
Zh. Chen, H. Rubin, and R. A. Vitale, ‘‘Addendum to ‘‘Independence and determination of probabilities’’ Proc. Am. Math. Soc. 129, 2817 (2001). https://doi.org/10.1090/S0002-9939-01-05540-X
Zh. Chen, H. Rubin, and R. A. Vitale, ‘‘Independence and determination of probabilities,’’ Proc. Am. Math. Soc. 125, 3721–3723 (1997). https://doi.org/10.1090/S0002-9939-97-03994-4
G. J. Székely and T. F. Móri, ‘‘Independence and atoms,’’ Proc. Am. Math. Soc. 130, 213–216 (2002). https://doi.org/10.1090/S0002-9939-01-06045-2
J. Stoyanov, ‘‘Sets of binary random variables with prescribed independence/dependence structure,’’ Math. Sci. 28 (1), 19–27 (2003).
W. F. Edwards, R. C. Shiflett, and H. S. Shultz, ‘‘Dependent probability spaces,’’ Coll. Math. J. 39, 221–226 (2008). https://doi.org/10.1080/07468342.2008.11922296
E. J. Ionascu and A. A. Stancu, ‘‘On independent sets in purely atomic probability spaces with geometric distribution,’’ Acta Math. Univer. Comenianae 79 (1), 31–38 (2010).
I. M. Sonin, ‘‘Independent events in a simple random experiment and the meaning of independence,’’ arXiv:1204.6731.
J. Stoyanov, Counterexamples in Probability (Wiley, New York, 1997).
J. Grahl and S. Nevo, ‘‘Estimates for probabilities of independent events and infinite series,’’ arXiv:1609.08924.
M. Kovacevic and V. Senk, ‘‘On possible dependence structures of a set of random variables,’’ Acta Math. Hung. 135, 286–296 (2012). https://doi.org/10.1007/s10474-012-0200-0
I. E. Shparlinski, ‘‘Modular hyperbolas,’’ Jpn. J. Math. 7, 235–294 (2012). https://doi.org/10.1007/s11537-012-1140-8
ACKNOWLEDGMENTS
The author thanks O.P. Vinogradov for problem formulation, fruitful ideas, remarks, and suggestions.
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Translated by E. Oborin
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Lebedev, A.V. Correlation Approach to Studying Dependent Discrete Probability Spaces. Moscow Univ. Math. Bull. 76, 9–15 (2021). https://doi.org/10.3103/S0027132221010046
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DOI: https://doi.org/10.3103/S0027132221010046