Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

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.Comment: 15 pages, 5 figures, 1 tabl

Faculty Publications 2018-2019

The production of scholarly research continues to be one of the primary missions of the ILR School. During a typical academic year, ILR faculty members published or had accepted for publication over 25 books, edited volumes, and monographs, 170 articles and chapters in edited volumes, numerous book reviews. In addition, a large number of manuscripts were submitted for publication, presented at professional association meetings, or circulated in working paper form. Our faculty\u27s research continues to find its way into the very best industrial relations, social science and statistics journal

Distance correlation for long-range dependent time series

We apply the concept of distance correlation for testing independence of long-range dependent time series. For this, we establish a non-central limit theorem for stochastic processes with values in an $L_2$-Hilbert space. This limit theorem is of a general theoretical interest that goes beyond the context of this article. For the purpose of this article, it provides the basis for deriving the asymptotic distribution of the distance covariance of subordinated Gaussian processes. Depending on the dependence in the data, the standardization and the limit of distance correlation vary. In any case, the limit is not feasible, such that test decisions are based on a subsampling procedure. We prove the validity of the subsampling procedure and assess the finite sample performance of a hypothesis test based on the distance covariance. In particular, we compare its finite sample performance to that of a test based on Pearson's sample correlation coefficient. For this purpose, we additionally establish convergence results for this dependence measure. Different dependencies between the vectors are considered. It turns out that only linear correlation is better detected by Pearson's sample correlation coefficient, while all other dependencies are better detected by distance correlation. An analysis with regard to cross-dependencies between the mean monthly discharges of three different rivers provides an application of the theoretical results established in this article