Report title: Heavy-tailed distribution outweigh correlations
Reporter: Academician Joel E. Cohen, Rockefeller University, USA
Reporting time: 8:45-9:45, January 11, 2021
Report location: Zoom meeting (Zoom meeting id: 770 311 8512, password: 378548)
School contact: Wang Peijie firstname.lastname@example.org
Pillai and Meng (2016) speculated that "the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails ....'' We give examples that support this speculation. Under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit. We showed, surprisingly, that this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Also surprisingly , the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative half-line. We show that the asymptotic scaling of Taylor's law (a power-law variance function) for RV rvs is, up to a constant , the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. Th e correlations and heterogeneity do not affect the asymptotic scaling. We analyze the sample kurtosis of heavy-tailed data similarly. We show that the least-squares estimator of the slope in a linear model with heavy-tailed predictor and noise nexpectedly converges much faster than when they have finite variances.
Cohen JE, Davis RA, Samorodnitsky G. 2020 Heavy-tailed distributions, correlations, kurtosis and Taylor’s Law of fluctuation scaling. Proc. R. Soc. A 476:20200610. doi:10.1098/rspa.2020.0610
Pillai NS, Meng X-L. 2016 An unexpected encounter with Cauchy and Lévy. Ann. Stat. 44, 2089–2097. doi:10.1214/15-AOS1407)
Joel E. Cohen is the Abby Rockefeller Mauzé Professor of Populations at The Rockefeller University and Columbia University, New York, USA. He and his colleagues study populations, ecosystems, and environments using mathematical, statistical, and computational tools. His work focuses on phenomena that affect human health, other species humans interact with, and the human environment. Cohen was educated at Harvard University, where he received a BA with highest honors in applied mathematics and two doctorates, one in applied mathematics and another in population sciences and tropical public health. He taught at Harvard until 1975, when he joined Rockefeller as a professor and head of the Laboratory of Populations. Since 1995, he has been a professor in the School of International and Public Affairs at Columbia University. He is affiliated with Columbia's Department of Earth and Environmental Sciences and its Department of Statistics and the University of Chicago's Department of Statistics. Cohen is a member of the US National Academy of Sciences, the American Academy of Arts and Sciences and the American Philosophical Society. He shared the Tyler Prize for Environmental Achievement and the Fred L. Soper Prize of the Pan American Health Organization, Washington, DC, for work on Chagas' disease. He has held honorary and visiting academic appointments in Argentina, China, England, France, and Japan. His 1995 book How Many People Can the Earth Support? received the first Olivia Schieffelin Nordberg Prize, awarded by the Population Council. He has written or edited 13 other books, including two on universal education, and more than 447 scientific papers and chapters.