Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin. (English) Zbl 1476.60047

Summary: We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution \(F\) (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of \(F)\). This allows us to illustrate the extent of the ‘failure’ of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer \(K\), new \(K\)-tuplewise independent sequences that are not mutually independent. For \(K \geq 4\), it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.


60F05 Central limit and other weak theorems
60E10 Characteristic functions; other transforms
62E20 Asymptotic distribution theory in statistics
Full Text: DOI arXiv


[1] Avanzi, B., G. Boglioni Beaulieu, P. Lafaye de Micheaux, F. Ouimet, and B. Wong (2021). A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables. J. Math. Anal. Appl. 499(1), 124982. · Zbl 1477.60045
[2] Balbuena, C. (2008). Incidence matrices of projective planes and of some regular bipartite graphs of girth 6 with few vertices. SIAM J. Discrete Math. 22(4), 1351-1363. · Zbl 1229.05130
[3] Biggs, N. (1993). Algebraic Graph Theory. Second edition. Cambridge University Press. · Zbl 0797.05032
[4] Billingsley, P. (1995). Probability and measure. Third edition. Wiley, New York. · Zbl 0822.60002
[5] Böttcher, B., M. Keller-Ressel, and R. L. Schilling (2019). Distance multivariance: new dependence measures for random vectors. Ann. Statist. 47(5), 2757-2789. · Zbl 1467.62104
[6] Bradley, R. C. and A. R. Pruss (2009). A strictly stationary, N-tuplewise independent counterexample to the central limit theorem. Stochastic Process. Appl. 119(10), 3300-3318. · Zbl 1175.60016
[7] Chakraborty, S. and X. Zhang (2019). Distance metrics for measuring joint dependence with application to causal inference. J. Amer. Statist. Assoc. 114(528), 1638-1650. · Zbl 1428.62337
[8] Diestel, R. (2005). Graph Theory. Third edition. Springer, Berlin. · Zbl 1074.05001
[9] Drton, M., F. Han, and H. Shi (2020). High-dimensional consistent independence testing with maxima of rank correlations. Ann. Statist. 48(6), 3206-3227. · Zbl 1461.62078
[10] Etemadi, N. (1981). An elementary proof of the strong law of large numbers. Z. Wahrsch. Verw. Gebiete 55(1), 119-122. · Zbl 0438.60027
[11] Fan, Y., P. Lafaye de Micheaux, S. Penev, and D. Salopek (2017). Multivariate nonparametric test of independence. J. Multivariate Anal. 153, 189-210. · Zbl 1351.62104
[12] Gaunt, R. E. (2019). A note on the distribution of the product of zero-mean correlated normal random variables. Stat. Neerl. 73(2), 176-179.
[13] Genest, C., J. G. Nešlehová, B. Rémillard, and O. A. Murphy (2019). Testing for independence in arbitrary distributions. Biometrika 106(1), 47-68. · Zbl 07051940
[14] Hall, P. and C. C. Heyde (1980). Martingale Limit Theory and its Application. Academic Press, New York. · Zbl 0462.60045
[15] Jin, Z. and D. S. Matteson (2018). Generalizing distance covariance to measure and test multivariate mutual dependence via complete and incomplete V-statistics. J. Multivariate Anal. 168, 304-322. · Zbl 1401.62073
[16] Kantorovitz, M. R. (2007). An example of a stationary, triplewise independent triangular array for which the CLT fails. Statist. Probab. Lett. 77(5), 539-542. · Zbl 1117.60021
[17] Pollard, D. (2002). A User’s Guide to Measure Theoretic Probability. Cambridge University Press. · Zbl 0992.60001
[18] Pruss, A. R. (1998). A bounded N-tuplewise independent and identically distributed counterexample to the CLT. Probab. Theory Relat. Fields 111, 323-332. · Zbl 0910.60008
[19] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. U.S.A. 42(1), 43-47. · Zbl 0070.13804
[20] Takeuchi, K. (2019). A family of counterexamples to the central limit theorem based on binary linear codes. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E102.A(5), 738-740.
[21] Tanabe, K. and M. Sagae (1992). An exact Cholesky decomposition and the generalized inverse of the variance-covariance matrix of the multinomial distribution, with applications. J. R. Stat. Soc. Ser. B Stat. Methodol. 54(1), 211-219. · Zbl 0777.62054
[22] Weakley, L. M. (2013). Some Strictly Stationary, N-tuplewise Independent Counterexamples to the Central Limit Theorem. Ph.D. Thesis, Indiana University.
[23] Yao, S., X. Zhang, and X. Shao (2018). Testing mutual independence in high dimension via distance covariance. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80(3), 455-480. · Zbl 1398.62151
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.