Large sample behaviour of high dimensional autocovariance matrices

Article Type

Research Article

Publication Title

Annals of Statistics

Abstract

The existence of limiting spectral distribution (LSD) of Λu + Λ ∗ u, the symmetric sum of the sample autocovariance matrix Λu of order u, is known when the observations are from an infinite dimensional vector linear process with appropriate (strong) assumptions on the coefficient matrices. Under significantly weaker conditions, we prove, in a unified way, that the LSD of any symmetric polynomial in these matrices such as Λ u + Λ ∗ u, Λu Λ ∗ u, Λ u Λ ∗ u + Λk Λ ∗ k exist. Our approach is through the more intuitive algebraic method of free probability in conjunction with the method of moments. Thus, we are able to provide a general description for the limits in terms of some freely independent variables. All the previous results follow as special cases. We suggest statistical uses of these LSD and related results in order determination and white noise testing.

First Page

598

Last Page

628

DOI

10.1214/15-AOS1378

Publication Date

4-1-2016

Comments

Open Access; Hybrid Gold Open Access

Link to Full Text

Share

COinS