TY - JOUR
T1 - Correction
T2 - Asymptotic spectral theory for nonlinear time series (The Annals of Statistics (2007) 35 (1773-1801) DOI: 10.1214/009053606000001479)
AU - Zhang, Yi
AU - Shao, Xiaofeng
AU - Wu, Weibiao
N1 - Publisher Copyright:
© Institute of Mathematical Statistics, 2022.
PY - 2022/10
Y1 - 2022/10
N2 - In this note, we make a correction to an error in Lemma A.4 of [2], as pointed out by Professor Stathis Paparoditis. In Lemma A.4(i), the original statement is max |cov(Ij2,Ik2) − 4fj4δj,k| = O(1/n). j,k≤m There is an error in the constant 4, and the correct statement should be max |cov(Ij2,Ik2) − 20fj4δj,k| = O(1/n). j,k≤m As in the original proof, we only need to consider indecomposable partitions ([1], p. 34) consisting of 4 sets, each containing 2 Xs (one with positive sign and the other with negative sign). In total, there are 20 such partitions. In the original proof, 4 of them are considered, which are {(t1,s1),(t2,s2),(t3,s3),(t4,s4)}, {(t1,s3),(t2,s2),(t3,s1),(t4,s4)}, {(t1,s1),(t2,s4),(t3,s3),(t4,s2)}, {(t1,s3),(t2,s4),(t3,s1),(t4,s2)}. Each partition leads to the sum [A(λj,λk)]4 = fj4δj,k + O(1/n), where A(λj,λk)=2πn1 Σnt1,s1=1 r(t1 − s1)eit1λj−is1λk, and r(k)= cov(X1,Xk+1). As suggested by Professor Paparoditis, there are 16 more indecomposable partitions, 4 of them are {(t1,t2),(s1,s2),(t3,s3),(t4,s4)}, {(t1,t2),(s1,s4),(t3,s3),(t4,s2)}, {(t1,t2),(s2,s3),(t3,s1),(t4,s4)}, {(t1,t2),(s3,s4),(t3,s1),(t4,s2)}. Each partition leads to the sum A(λj,λj)A(λk,λk)[A(λj,λk)]2 = fj4δj,k + O(1/n). The other 12 partitions correspond to {(t1,t4),(s1,s2),(t3,s3),(t2,s4)}, {(t2,t3),(s1,s2),(t1,s3),(t4,s4)}, {(t3,t4),(s1,s2),(t1,s3),(t2,s4)}, {(t1,t4),(s1,s4),(t3,s3),(t2,s2)}, {(t2,t3),(s1,s4),(t1,s3),(t4,s2)}, {(t3,t4),(s1,s4),(t1,s3),(t2,s2)}, {(t1,t4),(s2,s3),(t3,s1),(t2,s4)}, {(t2,t3),(s2,s3),(t1,s1),(t4,s4)}, {(t3,t4),(s2,s3),(t1,s1),(t2,s4)}, {(t1,t4),(s3,s4),(t3,s1),(t2,s2)}, {(t2,t3),(s3,s4),(t1,s1),(t4,s2)}, {(t3,t4),(s3,s4),(t1,s1),(t2,s2)}, and they all lead to fj4δj,k + O(1/n). Another way to heuristically check that the constant factor in max |cov(Ij2,Ik2) − cfj4δj,k| = O(1/n) j,k≤m is indeed c = 20 is to observe that when j = k, var(Ij2)/fj4 = var[(Ij/fj)2] and Ij/fj follows EXP(1) distribution asymptotically. Then we have var(Ij/fj)2 = E(Ij/fj)4 − [E(Ij/fj)2]2 ≈ 4! − (2!)2 = 20, so the factor c is expected to be 20. Many thanks to Professor Paparoditis for pointing out this 15-year-old error.
AB - In this note, we make a correction to an error in Lemma A.4 of [2], as pointed out by Professor Stathis Paparoditis. In Lemma A.4(i), the original statement is max |cov(Ij2,Ik2) − 4fj4δj,k| = O(1/n). j,k≤m There is an error in the constant 4, and the correct statement should be max |cov(Ij2,Ik2) − 20fj4δj,k| = O(1/n). j,k≤m As in the original proof, we only need to consider indecomposable partitions ([1], p. 34) consisting of 4 sets, each containing 2 Xs (one with positive sign and the other with negative sign). In total, there are 20 such partitions. In the original proof, 4 of them are considered, which are {(t1,s1),(t2,s2),(t3,s3),(t4,s4)}, {(t1,s3),(t2,s2),(t3,s1),(t4,s4)}, {(t1,s1),(t2,s4),(t3,s3),(t4,s2)}, {(t1,s3),(t2,s4),(t3,s1),(t4,s2)}. Each partition leads to the sum [A(λj,λk)]4 = fj4δj,k + O(1/n), where A(λj,λk)=2πn1 Σnt1,s1=1 r(t1 − s1)eit1λj−is1λk, and r(k)= cov(X1,Xk+1). As suggested by Professor Paparoditis, there are 16 more indecomposable partitions, 4 of them are {(t1,t2),(s1,s2),(t3,s3),(t4,s4)}, {(t1,t2),(s1,s4),(t3,s3),(t4,s2)}, {(t1,t2),(s2,s3),(t3,s1),(t4,s4)}, {(t1,t2),(s3,s4),(t3,s1),(t4,s2)}. Each partition leads to the sum A(λj,λj)A(λk,λk)[A(λj,λk)]2 = fj4δj,k + O(1/n). The other 12 partitions correspond to {(t1,t4),(s1,s2),(t3,s3),(t2,s4)}, {(t2,t3),(s1,s2),(t1,s3),(t4,s4)}, {(t3,t4),(s1,s2),(t1,s3),(t2,s4)}, {(t1,t4),(s1,s4),(t3,s3),(t2,s2)}, {(t2,t3),(s1,s4),(t1,s3),(t4,s2)}, {(t3,t4),(s1,s4),(t1,s3),(t2,s2)}, {(t1,t4),(s2,s3),(t3,s1),(t2,s4)}, {(t2,t3),(s2,s3),(t1,s1),(t4,s4)}, {(t3,t4),(s2,s3),(t1,s1),(t2,s4)}, {(t1,t4),(s3,s4),(t3,s1),(t2,s2)}, {(t2,t3),(s3,s4),(t1,s1),(t4,s2)}, {(t3,t4),(s3,s4),(t1,s1),(t2,s2)}, and they all lead to fj4δj,k + O(1/n). Another way to heuristically check that the constant factor in max |cov(Ij2,Ik2) − cfj4δj,k| = O(1/n) j,k≤m is indeed c = 20 is to observe that when j = k, var(Ij2)/fj4 = var[(Ij/fj)2] and Ij/fj follows EXP(1) distribution asymptotically. Then we have var(Ij/fj)2 = E(Ij/fj)4 − [E(Ij/fj)2]2 ≈ 4! − (2!)2 = 20, so the factor c is expected to be 20. Many thanks to Professor Paparoditis for pointing out this 15-year-old error.
KW - Periodogram
KW - spectral analysis
UR - https://www.scopus.com/pages/publications/85146131910
U2 - 10.1214/22-AOS2206
DO - 10.1214/22-AOS2206
M3 - Comment/debate
AN - SCOPUS:85146131910
SN - 0090-5364
VL - 50
SP - 3088
EP - 3089
JO - Annals of Statistics
JF - Annals of Statistics
IS - 5
ER -