Maximum of scaled eigenvalues: dependent input
Arup Bose, Koushik Saha · 2018
In this chapter we try to generalize the results of Chapters 5 and 6 on spectral radius to the situation where the input sequence is dependent. We take { x n } to be an infinite order moving average process, x n = ∑ i = − ∞ ∞ a i ε n − i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429435508/b49d0f83-a479-4a7a-964a-69b22e9e8a46/content/eq945.tif"/> , where { a n ; n ∈ ℤ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429435508/b49d0f83-a479-4a7a-964a-69b22e9e8a46/content/eq946.tif"/> are non-random with Σ n | a n | are i.i.d. In this case the eigenvalues have unequal variances. So, we resort to scaling each eigenvalue by an appropriate quantity and then consider the distributional convergence of the maximum of these scaled eigenvalues of different circulant matrices. This scaling has the effect of (approximately) equalizing the variance of the eigenvalues. Similar scaling has been used in the study of the periodogram (see Walker (1965) , Davis and Mikosch (1999) , and Lin and Liu (2009) ).