Supplement to Covariance Matrix Estimation for Stationary Time Series

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1 Supplement to Covariance Matrix Estimation for Stationary ime Series Han Xiao and Wei Biao Wu Department of Statistics 5734 S. University Ave Chicago, IL In this document we give the proofs of Remark 5 and Lemma 9 of the main article, as well as a few remarks on Lemma 9. All the equation, theorem, lemma and remark numbers refer to the main article. he equations and remarks introduced in this document are numbered with an S -prefix. Proof of Remark 5. Set b = 2 1 k=b +1 γ k. Define g,b (θ) = π 1 1 k=b +1 γ k cos(kθ). Since 1 B k=1 kγ k = O(B / ), if we can show that lim P { [ 2π min ˆf,B (θ) E ˆf ],B (θ) g,b (θ) θ } B log B 2 b /5 = 1, (S.1) then (31) will follow by using (30) and similar arguments as those which have led to the lower bound in heorem 2. Let k N be such that A k 1 B < A k. Define D = {θ [0, π] : cos(a k θ) 1/2}, then 2π g,b (θ) b /5 for θ D. Define λ,j = 2πj log(b ) 3 /A k, and set j = max{j : λ,j π}. Using the arguments of Liu and Wu (2010), if p is sufficiently large, we have P [ min 0 j j B where w = 2 log j and f,b (λ,j) Ef,B (λ,j) 2f(λ,j) x w z ] 1 e ex/2, z = (2 log j ) 1/2 (8 log j ) 1/2 (log log j + log(4π)). 1

2 H. Xiao, W.B. Wu/Supplementary File 2 Since the spectral density is bounded away from zero, i.e. ( ) f := min f(θ) A αk 1 θ 2π 4π, and j B /[2 log(b ) 3 ], it follows that { P min [f,b (λ 0 j j,j) Ef,B (λ,j)] f 2B log B hen (S.1) follows by noting that λ,j D for 0 j j. } 1. Proof of Lemma 9. Let w = m /2, and split Q into two parts as Q,1 = 1 X t s=t B a s,t X s and Q,2 = X t t s= a s,t X s, where we make the convention that if a term X s in the previous sum has the subscript s / [1, ], then that term should be replaced by zero. Define Q,1 and Q,2 similarly. We consider Q,2 first. Write Q,2 Q,2 = (X t X t s+w t ) a s,t H s w X s + (X s X s ) a s,t Xt s= t=s+1 + (X t X t t ) a s,t (X s H s w X s ) =: I + II + III. s= For the first term, write I = k=m +1 t P t k X t a s,t H s w X s. s= Since 2w m, we know for each fixed k > m + 1, ( ) t P t k X t a s,t H s w X s s= 1 t is a backward martingale difference sequence with respect to the filtration (F t k ) 1 t. It follows that by (38) I p/2 k=m +1 C p/2 δp (k) C p w B Θ p C p C p/2 Θ p B Θ p (m ).

3 H. Xiao, W.B. Wu/Supplementary File 3 Similarly we have II p/2 C B Θ p (m ). For the third term, using the arguments of Proposition 1 of Liu and Wu (2010), we have III EIII p/2 2 2 C p/2 C p B [Θ p (w ) p (m ) + Θ p (m ) p (w )]. Now we consider Q,1. Observe that Q,1 is nonzero only when B > w. Write Q,1 Q,1 = (X t X t ) + (X t X t ) 1 s=t B a s,t Xs + 1 (X s X s ) s+b t=s+w +1 a s,t Xt s=t B a s,t (X s X s ) =: IV + V + V I. Similarly as II and III, we have V p/2 C p/2 C p Θ p B Θ p (m ) and Write the term IV V I E(V I ) p/2 4 C p/2 C p B Θ p (m ) p (m ). IV = as m k=m +1 l=0 1 P t k X t s=t B a s,t P s l X s. For each fixed pair (k, l), if we remove the the pair (s, t) such that t k = s l from the sum 1 P t k X t a s,t P s l X s, s=t B then by (38) P t k X t t B s<,s l t k a s,t P s l X s 2 C p/2 C p B δ p (k)δ p (l). p/2 herefore, it remains to deal with the term P s lx s t Λ s a s,t P s l X t for 0 l m, where Λ s = [s (w (m l)), (s + B ) ]. Since the sequence (P s l X s t Λ s a s,t P s l X t ) indexed by s is (4B )-dependent, and ( 4B ) E 0 P s l X s a s,t P s l X t 2 4B δ p (l) Θ p (m ); t Λ s p/2 we have by (38) ( E s+b 0 P s l X s a s,t P s l X t) 4 2 C p/2 B δ p (l)θ p (m ). t=s+w +1 p/2 Putting these pieces together, the proof is complete.

4 H. Xiao, W.B. Wu/Supplementary File 4 Remark S.1. If Θ p (m) m α for some α > 0, then the bound becomes C p B m α. If m = O(B ), then this order of magnitude is optimal here, and cannot be improved in general. For example, consider the linear process X t = s=0 a sɛ t s and the quadratic form Q = X 1 s<t tx s 1 0<t s B, where a s = s (1+α), and ɛ s s are iid standard normal random variables. Observe that Q Q is also a quadratic form of ɛ s s which can be written as Q Q = b k,l ɛ k ɛ l, which implies that <k l 2 E 0 Q E 0 Q = 2 b 2 k,k + <k<l Elementary but tedious calculations show that for B < t < B and B /3 k 2B /3, we have b t,t (m +1) k B /3 B /3 a k k=1 b 2 k,l a m +k Cm α. It follows that E 0 Q E 0 Q C B m α, namely the order of magnitude is achieved. Remark S.2. A similar bound was obtained by Liu and Wu (2010): E 0 Q E 0 Q p/2 C p B p (m ). In term of the order of magnitude, our result is better, because Θ p (m) p (m). o be more precise, consider the condition Θ p (m) = O(m α ) for some α > 0, which is the assumption we use for heorem 4. Since Ψ p (m) Θ p (m), we have Ψ p (m) = O(m α ). Conversely, if Ψ p (m) = O(m α+1/2 ), then Θ p (m) = O(m α ). A proof was given by Wu and Zhao (2008). herefore, when using both Θ p (m) = O(m α ) and Ψ p (m) = O(m β ) as assumptions, we necessarily assume α > 0 and α β α + 1/2 to avoid redundancy. Under these two conditions we have p (m) m β/(1+α) min{δ p (k), Cm β } + Θ m ( m β/(1+α) + 1) Cm αβ/(1+α), which implies that p (m) = O(m αβ/(1+α) ). We shall give an example to show that the order cannot be improved. Define { k δ p (k) = (1+α) + 2 βn if k = 2 n for some n N k (1+α) otherwise.

5 H. Xiao, W.B. Wu/Supplementary File 5 It is easily seen that Θ p (m) m α and Ψ p (m) m β. herefore, p (2 n ) = min{k (1+α), 2 βn } 2 βn/(1+α) C (2 n ) αβ/(1+α). min{k (1+α), 2 βn } + Θ p ( 2 βn/(1+α) + 1) In particular, if we only put the condition on Θ p (m), then the largest exponent γ such that p (m) = O(m γ ) for any sequence satisfying Θ p (m) = O(m α ) is γ = α 2 /(1 + α). References Liu, W. and Wu, W. B. (2010). Asymptotics of spectral density estimates. Econometric heory Wu, W. B. and Zhao, Z. (2008). Moderate deviations for stationary processes. Statist. Sinica