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Contents of PMS, Vol. 13, Fasc. 2,
pages 253 - 267
 

INÉGALITÉS DE TRACE POUR DES MATRICES DE TŒPLITZ ET APPLICATIONS À DES VRAISEMBLANCES GAUSSIENNES

Malek Bouaziz

Abstract: Let u be an integrable function on the 1-dimensional torus and T (u) n be the Tœplitz matrix with entries ^u(s- t), 0 < s, t < n - 1, where ^u is the Fourier transform of u. In this paper, it is shown that if u ,...,u 1 r are in the Banach algebra of those u that satisfy ||u||= ||u|| + ||u|| < oo , oo 1/2 where ||u|| oo is the L oo -norm of u and sum +o o 21/2 ||u||1/2 = ( - oo |t||^u(t)|) , then

||T (u ...u )- T (u )...T (u )|| < sum ||u|| ||u || prod ||u || , n 1 r n 1 n r 1 i<j i1/2 j 1/2k/=i,j k oo
where the norm on the left is the trace class norm. Using the inequality |tr(A)|< ||A ||1 (tr for trace), it is shown that if boundedness is replaced by continuity, then tr(Tn(u1...ur)- Tn(u1)...Tn(ur)) is convergent (n --> oo ). These results are used to study Whittle’s approximation error for log-likelihoods of stationary Gaussian sequences. It is shown that its moments are bounded or convergent under suitable conditions for spectral densities.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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