北京大学数量经济与数理金融教育部重点实验室学术报告——The largest eigenvalues of the sample covariance matrix in the heavy-tail case
主 题: 北京大学数量经济与数理金融教育部重点实验室学术报告——The largest eigenvalues of the sample covariance matrix in the heavy-tail case
报告人: Thomas Mikosch (University of Copenhagen)
时 间: 2018-06-11 14:00-15:00
地 点: Room 1560, Sciences Building No. 1
Abstract: This is joint work with Richard A. Davis (Columbia) and Johannes Heiny (Aarhus). Heavy tails of a time series are typically modeled by power law tails with a positive tail index α. We refer to such time series as regularly varying with index α. Regular variation of a time series translates into power law tail behavior of the partial sums of the time series above high threshold. This was observed early on in work by A.V. Nagaev (1969) and S.V. Nagaev (1979) who considered sums of iid regularly varying random variables. These results are referred to as heavy-tail or Nagaev-type large deviations. The goal of this lecture is to argue that heavy-tail large deviations are useful tools when dealing with the eigenvalues of the sample covariance matrix of dimension p×n when p →∞as n →∞in those cases when one can identify the dominating entries in this matrix. These are the diagonal entries in the iid and some other cases. A similar argument allows one to identify the dominating entries if the time series has a linear dependence structure with regularly varying noise. These techniques are an alternative approach to earlier results by Soshnikov (2004,2006), Auffinger, Ben Arous, Peche (2009), Belinschi, Dembo, Guionnet (2009). They also allow one to deal with certain classes of matrices with dependent heavy-tailed entries.
Bio: Thomas Mikosch got his PhD from St. Petersburg University in 1984. Since then he worked at universities in Switzerland, New Zealand, the Netherlands, and the US. Since 2001 he has been professor of actuarial mathematics at the University of Copenhagen. His research interests are in applied probability and statistics. He is Publications Secretary of the Bernoulli Society. He was the EiC of Stochastic Processes and their Applications 2009-2012 and since 2014 he has been EiC of the Springer Extremes journal. In 2018 he received an Alexander von Humboldt Research Award. He is member of the Royal Danish Academy of Sciences and Letters.