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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 31, Fasc. 2,
pages 313 - 329
 

EIGENVALUE DISTRIBUTION OF LARGE SAMPLE COVARIANCE MATRICES OF LINEAR PROCESSES

Oliver Pfaffel
Eckhard Schlemm

Abstract: We derive the distribution of the eigenvalues of a large sample covariance matrix when the data is dependent in time. More precisely, the dependence for each variable i = 1,...,p is modelled as a linear process (X  )      = (∑ ∞  cZ    )
  i,t t=1,...,n      j=0  j i,t-j t=1,...,n  , where (Z  )
  i,t are assumed to be independent random variables with finite fourth moments. If the sample size n and the number of variables p = p
     n  both converge to infinity such that y = lim     n∕p > 0
      n→ ∞    n , then the empirical spectral distribution of p-1XXT  converges to a non-random distribution which only depends on y and the spectral density of (X   )
   1,tt∈ℤ  . In particular, our results apply to (fractionally integrated) ARMA processes, which will be illustrated by some examples.

2000 AMS Mathematics Subject Classification: Primary: 15A52; Secondary: 62M10.

Keywords and phrases: Eigenvalue distribution, fractionally integrated ARMA process, limiting spectral distribution, linear process, random matrix theory, sample covariance matrix.

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