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Corrigendum to “A central limit theorem for quadruple-wise independent arrays of random variables” [Statist. Probab. Lett. 110 (2016) 58–61]
Statistics and Probability Letters 121 (2017) 163
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Corrigendum
Corrigendum to ‘‘A central limit theorem for quadruple-wise independent arrays of random variables’’ [Statist. Probab. Lett. 110 (2016) 58–61] Cristina Tone Department of Mathematics, University of Louisville, 328 Natural Sciences Building, Louisville, KY 40292, United States
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Article history: Received 19 September 2016 Accepted 14 October 2016 Available online 8 November 2016
abstract The goal of this note is to fix an error mentioned in the abstract of my paper ‘‘A central limit theorem for quadruple-wise independent arrays of random variables’’ published in Statistics and Probability Letters, Volume 110, March 2016, Pages 58–61, http://dx.doi.org/ 10.1016/j.spl.2015.11.007.
In the abstract of my paper (Tone, 2016), the statement ‘‘It is still an open question whether for a non-degenerate strictly stationary sequence X := (Xk , k ∈ Z) with finite second moments, quadruple-wise independence implies the CLT.’’ is false. The example of Bradley and Pruss (2009) shows that such a result is false. Also, while N-tuplewise independence with N = 6 implies 5-tuplewise independence, Bradley’s 5-tuplewise independent counterexample to the CLT (see Bradley (2010)) had some extra features (such as a trivial double tail sigma-field and a ‘‘non-linear causal’’ structure and therefore Bernoulli) that were not present in the N-tuplewise independent example given in Bradley and Pruss (2009). It was in those ‘‘extra features’’ that Bradley’s example (Bradley, 2010) gave new information relative to the example in Bradley and Pruss (2009). It is still not known whether there exists a 6-tuplewise independent counterexample to the CLT that has those ‘‘extra features’’ that are not present in the N-tuplewise independent example of Bradley and Pruss (2009). Acknowledgments The author thanks R. Bradley for pointing her out the error in the statement of the abstract. The author would like to apologize for any inconvenience caused. References Tone, C., 2016. A central limit theorem for quadruple-wise independent random vectors. Statist. Probab. Lett. 110, 58–61. Bradley, R.C., Pruss, A.R., 2009. A strictly stationary, N-tuplewise independent counterexample to the central limit theorem. Stochastic Process. Appl. 119, 3300–3318. Bradley, R.C., 2010. A strictly stationary, causal, 5-tuplewise independent counterexample to the central limit theorem. ALEA Lat. Am. J. Probab. Math. Stat. 7, 377–450.
DOI of original article: http://dx.doi.org/10.1016/j.spl.2015.11.007. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.spl.2016.10.017
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Corrigendum
Corrigendum to ‘‘A central limit theorem for quadruple-wise independent arrays of random variables’’ [Statist. Probab. Lett. 110 (2016) 58–61] Cristina Tone Department of Mathematics, University of Louisville, 328 Natural Sciences Building, Louisville, KY 40292, United States
article
info
Article history: Received 19 September 2016 Accepted 14 October 2016 Available online 8 November 2016
abstract The goal of this note is to fix an error mentioned in the abstract of my paper ‘‘A central limit theorem for quadruple-wise independent arrays of random variables’’ published in Statistics and Probability Letters, Volume 110, March 2016, Pages 58–61, http://dx.doi.org/ 10.1016/j.spl.2015.11.007.
In the abstract of my paper (Tone, 2016), the statement ‘‘It is still an open question whether for a non-degenerate strictly stationary sequence X := (Xk , k ∈ Z) with finite second moments, quadruple-wise independence implies the CLT.’’ is false. The example of Bradley and Pruss (2009) shows that such a result is false. Also, while N-tuplewise independence with N = 6 implies 5-tuplewise independence, Bradley’s 5-tuplewise independent counterexample to the CLT (see Bradley (2010)) had some extra features (such as a trivial double tail sigma-field and a ‘‘non-linear causal’’ structure and therefore Bernoulli) that were not present in the N-tuplewise independent example given in Bradley and Pruss (2009). It was in those ‘‘extra features’’ that Bradley’s example (Bradley, 2010) gave new information relative to the example in Bradley and Pruss (2009). It is still not known whether there exists a 6-tuplewise independent counterexample to the CLT that has those ‘‘extra features’’ that are not present in the N-tuplewise independent example of Bradley and Pruss (2009). Acknowledgments The author thanks R. Bradley for pointing her out the error in the statement of the abstract. The author would like to apologize for any inconvenience caused. References Tone, C., 2016. A central limit theorem for quadruple-wise independent random vectors. Statist. Probab. Lett. 110, 58–61. Bradley, R.C., Pruss, A.R., 2009. A strictly stationary, N-tuplewise independent counterexample to the central limit theorem. Stochastic Process. Appl. 119, 3300–3318. Bradley, R.C., 2010. A strictly stationary, causal, 5-tuplewise independent counterexample to the central limit theorem. ALEA Lat. Am. J. Probab. Math. Stat. 7, 377–450.
DOI of original article: http://dx.doi.org/10.1016/j.spl.2015.11.007. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.spl.2016.10.017