By D'Agostini G.
Those notes are in response to seminars and minicourses given in quite a few areas during the last 4 years.
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43) 4. 41) may not exist. For any ﬁnite θ, however, the following similar fact is true: If F (x) is a continuous function of [1/Mf , θ], then lim n−1 n→∞ n−1 F [min(ρn,k , θ)] = (2π)−1 k=0 2π F [min(1/f (λ), θ)] dλ. 44) Proof. 1. 9) 2 1 π n−1 ∗ ikλ x Tn x = xk e f (λ)dλ > 0 2π −π k=0 so that for all n min τn,k > 0 k and hence n−1 det Tn = τn,k = 0 k=0 so that Tn (f ) is nonsingular. 2. 1 since f (λ) ≥ mf > 0 ensures that Tn−1 , Cn−1 ≤ 1/mf < ∞. 40 CHAPTER 4. 45) must hold for large enough n. 40), however, if n is large enough, then probably the ﬁrst term of the series is suﬃcient.
The proof of part 4 is motivated by one of Widom . Further results along the lines of part 4 regarding unbounded Toeplitz matrices may be found in . , ﬁnding conditions on f (λ) to ensure that Tn (f ) is invertible. 42 CHAPTER 4. TOEPLITZ MATRICES Parts (a)-(d) can be straightforwardly extended if f (λ) is continuous. For a more general discussion of inverses the interested reader is referred to Widom  and the references listed in that paper. It should be pointed out that when discussing inverses Widom is concerned with the asymptotic behavior of ﬁnite matrices.
2 Let Xn be an autoregressive process with covariance matrix (n) RX with eigenvalues ρn,k . Then (RX )−1 ∼ σ −2 Tn (|a|2 ). 24) where 1/ρn,k are the eigenvalues of (RX )−1 . 25) so that the process is asymptotically stationary. Proof. ) Note that if |a(λ)|2 > 0, then 1/|a(λ)|2 is the spectral density of Xn . If (n) |a(λ)|2 has a zero, then RX may not be even asymptotically Toeplitz and hence Xn may not be asymptotically stationary (since 1/|a(λ)|2 may not be integrable) so that strictly speaking xk will not have a spectral density.