Statistica Sinica 28 (2018), 2657-2670

APPLICATIONS OF PETER HALL'S MARTINGALE

LIMIT THEORY TO ESTIMATING AND TESTING

HIGH DIMENSIONAL COVARIANCE MATRICES

Danning Li, Lingzhou Xue and Hui Zou

Jilin University, Pennsylvania State University and University of Minnesota

Abstract: Martingale limit theory is increasingly important in modern probability theory and mathematical statistics. In this article, we give a selected overview of Peter Hall's contributions to both the theoretical foundations and the wide applicability of martingales. We highlight his celebrated coauthored book, Hall and Heyde (1980) and his ground-breaking paper, Hall (1984). To illustrate the power of his martingale limit theory, we present two contemporary applications to estimating and testing high dimensional covariance matrices. In the first, we use the martingale central limit theorem in Hall and Heyde (1980) to obtain the simultaneous risk optimality and consistency of Stein's unbiased risk estimation (SURE) information criterion for large covariance matrix estimation. In the second application, we use the central limit theorem for degenerate U-statistics in Hall (1984) to establish the consistent asymptotic size and power against more general alternatives when testing high-dimensional covariance matrices.

Key words and phrases: Degenerate U-statistics, hypothesis testing, large covariance matrix, martingale limit theory, Stein's unbiased risk estimation.