LARGE SAMPLE BEHAVIOUR OF HIGH DIMENSIONAL AUTOCOVARIANCE MATRICES

• 成果类型: Article
• 署名作者: Bhattacharjee, Monika; Bose, Arup
• 署名单位: Indian Statistical Institute; Indian Statistical Institute Kolkata
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1378
• 发表日期: 2016
• 页码: 598-628
• 关键词: dynamic-factor model; time-series; Empirical distribution; Covariance matrices; eigenvalues

摘要

The existence of limiting spectral distribution (LSD) of (Gamma) over cap (u) + (Gamma) over cap (u)*, the symmetric sum of the sample autocovariance matrix (Gamma) over cap (u) 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 (Gamma) over cap (u) + (Gamma) over cap (u)*, (Gamma) over cap (u)(Gamma) over cap (u)*, (Gamma) over cap (u)(Gamma) over cap (u)* + (Gamma) over cap (k)(Gamma) over cap (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.