ALMOST SURE CONVERGENCE OF THE BARTLETT ESTIMATOR

Periodica Mathematica Hungarica Vol. 51 1, 2005, pp. 11 25 ALMOST SURE CONVERGENCE OF THE BARTLETT ESTIMATOR István Berkes Graz, Budapest, LajosHorváth Salt Lake City, Piotr Kokoszka Logan Qi-man Shao Eugene, Singapore Dedicated to Endre Csáki Pál Révész on the occasion of their 70 th birthdays Abstract We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a. s. behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate normalization, the estimator converges a.s. in the long-range dependent case as well. In both cases, our conditions involve fourth order cumulants assumptions on the rate of growth of the truncation parameter appearing in the definition of the Bartlett estimator. 1. Introduction If {Y i } is a weakly stationary sequence, then under weak conditions which quantify short-range weak dependence 1 n Var Y i σ 2 := γ j, 1 i n j= as n,whereγ j =CovY 0,Y j. Inference for time series modeled by weakly dependent processes requires estimation of the asymptotic variance σ 2.Oneofthe Mathematics subject classification number: 60F15, 62F12. Key words phrases: weak dependence, long-range dependence, variance of the mean, cumulants, increments of partial sums. Research supported by the Hungarian National Foundation for Scientific Research, Grants T-043037 T-037886. Research supported by NSF grant INT-0223262 NATO grant PST.EAP.CLG 980599. Partially supported by NSF grant DMS-0103487, grants R-146-000-038-101 R- 1555-000-035-112 at the National University of Singapore. 0031-5303/2005/$20.00 Akadémiai Kiadó, Budapest c Akadémiai Kiadó, Budapest Springer, Dordrecht

12 i. berkes et al. most popular estimators is the Bartlett estimator defined as s 2 n =ˆγ 0 +2 ω j qnˆγ j, 1.1 1 j qn where ˆγ j = 1 n 1 i n j Yi Ȳn Yi+j Ȳn 1.2 are the sample autocovariances ω j q are the Bartlett weights defined by ω j q =1 j q +1. 1.3 The estimator s 2 n has also been used for long-range dependent observations Y i,see Lo 1991. In the weakly dependent case, σ 2 =2πf0, where f is the spectral density of {Y i }, so s 2 n /2π is also an estimator for f0. An extension of the estimator 1.1 to arbitrary frequencies has been studied extensively in the spectral domain, so most results focused on L 2 convergence. Giraitis et al. 2003 proved that under regularity conditions quantifying weak dependence, s 2 n P σ 2. They also considered the in probability behaviour of s 2 n when {Y i } exhibit long range dependence. In the present paper we establish the almost sure consistency of the variance estimator s 2 n under the conditions used by Giraitis et al. 2003 in the case of weak dependence as well as in the long memory case. Theorem 1.1 below plays a crucial role in Berkes et al. 2003 who developed a procedure for distinguishing between a sequence of long-range dependent observations a sequence of weakly dependent observations with a change point. To lighten the notation, we assume in the following that EY i =0. Recall the definition of the fourth order cumulant κh, r, s givenby κh, r, s =E[Y k Y k+h Y k+r Y k+s ] γ h γ r s + γ r γ h s + γ s γ h r. 1.4 We will also work with the quantity νh, r, s =CovY k Y k+h,y k+r Y k+s = E[Y k Y k+h Y k+r Y k+s ] γ h γ r s = κh, r, s+γ r γ h s + γ s γ h r. 1.5 Theorem 1.1. Suppose {Y k } is a fourth order stationary sequence with EY i =0 γ j =CovY 0,Y j. Suppose the sequence qn is nondecreasing q 2 k+1 sup k 0 q 2 k <. 1.6

almost sure convergence of the bartlett estimator 13 Then, i Suppose, in addition, that γ j <, 1.7 sup h j κh, r, s <, 1.8 r,s qn qnlog n 4 = On. 1.9 s 2 n σ2 := j= γ j a.s. 1.10 ii Assume 1 <H<1 1.11 2 γ k c 0 k 2H 2 1.12 for some c 0 > 0. Assume also that qn qn =O nlog n 7/4 4H 1.13 Then sup h qn n r,s n qn 1 2H s 2 n c 2 H = κh, r, s = O n 2H 1. 1.14 c 0 H2H 1 a.s. 1.15 Remark 1.1. The proof of Theorem 1.1 remains valid if the Bartlett weights 1.3 are replaced by arbitrary weights which in addition to 1.6 satisfy ω j q =0 for >q,0 ω j q 1 another condition which is different for parts i ii. For part i, it must be required cf. 2.5 that lim ω jq =1 foreachj. 1.16 q For part ii, it must be required that lim c 0 q q1 2H ω j q γ j = 2H 1H. 1.17 q

14 i. berkes et al. Define also Proof of part i. Let γ j = 1 n n 2. Proofs S k,l = Y i Y i+, l Y i. i=k <n. Then ˆγ j γ j = Ȳn 2 1 n nȳns 1,n + S +1,n =:δ j. As in the proof of Theorem 3.1 of Giraitis et al. 2003, decompose s 2 n as where It suffices to show that v n,1 = v n,2 = s 2 n = v n,1 + v n,2, qn qn γ j, qn+1 δ j. qn+1 v n,1 σ 2 a.s. 2.1 v n,2 0 a.s. 2.2 We first verify the easier relation 2.2. By 1.7 Theorem 3.7.2 of Stout 1974 S 1,n = o n 1/2 log 2 n a.s. 2.3 consequently Hence, by 1.9, qn max S i,j = o n 1/2 log 2 n 1 i j n a.s. Ȳn 2 qn+1 n 2qn+1Ȳ n 2 = o qn/nlog 4 n = o1 a.s.

almost sure convergence of the bartlett estimator 15 1 Ȳn S1,n + S +1,n qn+1 n qn 2qn+1 o log 2 n n 1/2 log 2 n n n 1/2 qnlog 4 n = o = o1 a.s. n This proves 2.2. We now turn to the verification of 2.1. Set Since z n = nv n, Ev n,1 = Ev n,1 = qn qn qn+1 n γ j qn+1 n Yi Y i+ γ. 2.4 j= γ j = σ 2, 2.5 it suffices to verify that z n = on a.s. In the study of z n, we replace qn with q n, where q n is constant on large intervals. This replacement will be done in two steps. First we replace qn in the limit of summation then in the sum itself. We thus define q n =q2 k, if 2 k <n 2 k+1 introduce z n = ẑ n = We will show that q n q n qn+1 q n+1 n n Yi Y i+ γ, 2.6 Yi Y i+ γ. 2.7 z n z n = on a.s., 2.8 We will use the relation z n ẑ n = on a.s. 2.9 sup h ẑ n = on a.s. 2.10 νh, r, s < 2.11 r,s

16 i. berkes et al. which follows immediately from 1.8 1.7. Clearly, where n,2 = z n z n = n,1 = We first show that q n< qn qn+1 n Yi Y i+ γ = n,1 + n,2, 2.12 q n< qn q n< qn For any 2 k <m<n 2 k+1,wewrite n, m,1 = Using 2.11, we have E[a 1n, m] 2 = = q n< qm q n< qm q n< qm n m, j qm q n< qm + qm< qn q n+1 n 1 q n+ 1 qn+1 Yi Y i+ γ n Yi Y i+ γ. n,1 = on a.s. 2.13 n Yi q Y i+ γ n+1 i=m +1 q n+1 n Yi Y i+ γ =: a 1 n, m+a 2 n, m. 2.14 q n< j qm q n< j qm q n< j qm l 4n m <i n m <i n m<i<n m<i <n ν, l, l+ j m j <i n j m j <i n j CovY iy i+, Y i Y i + j ν, i i, i i + j ν, i i j +, i i + Cn mqm. 2.15

almost sure convergence of the bartlett estimator 17 Applying 2.11 again, we obtain E[a 2 n, m] 2 qm< qn qm< j qn n n j i =1 CovY i Y i+,y i Y i + j = n qm< qn qm< qn qm< j qn qm< j qn n l 4n n j i =1 ν, i i, i i + j ν, l, l+ j Cqn qmn. 2.16 Combining 2.14 2.16, we conclude that E[ n, m,1 ] 2 C[n mqn+qn qmn] = C C n i=m+1 n i=m+1 [qn+qi qi 1n] [ q 2 k+1 +qi qi 12 k+1]. 2.17 Using Problem 5 of Billingsley 1968, p. 102, cf. Móricz et al. 1982, we get [ ] 2 E max 2 k+1, m,1 2 k <m 2 k+1 C [ 2 k+ 2 k q 2 k+1 + q 2 k+ q 2 k 2 k+1] log 2 k+1 2, therefore the Chebishev inequality 1.6 yield { } P max 2 k+1, m,1 k 7/4 q 1/2 2 k 2 k/2 2 k <m 2 k+1 C [ 2 k+ 2 k q 2 k+1 + q 2 k+ q 2 k 2 k+1] k 2 k 7/2 q 2 k 2 k C k 3/2. The Borel Cantelli lemma 1.9 give max 2 k+1, m,1 a.s. = o 2 k. 2.18 2 k <m 2 k+1

18 i. berkes et al. Similarly to 2.15, we have, using 2.11 E 2 n,1 = = n q n<, j qn q n<, j qn q n<, j qn q n<, j qn 1 i n 1 i n 1 i,i n l 4n 1 i n j 1 i n j Cov Y i Y i+, Y i Y i + j ν, i i, i i + j ν, i i j +, i i + ν, l, l+ j Cnqn. 2.19 Hence the Chebishev inequality 1.6 give P { 2 k+1,1 k 7/4 q 1/2 2 k 2 k/2} C 2k+1 q2 k+1 k 7/2 q2 k 2 k C k 7/2, so the Borel Cantelli lemma 1.9 yield 2 k+1,1 a.s. = o2 k+1. 2.20 Now 2.13 follows from 2.18 2.20. Next we show n,2 = on a.s. 2.21 Observe that n,2 = qn q n qn+1 q n< qn q n< qn q n+1 For any 2 k <m<n 2 k+1,wewrite n,2 m,2 = q n< qn q n< qm n q n+1 q n+1 n Yi Y i+ γ. n q n+1 Yi Y i+ γ Yi Y i+ γ m Yi Y i+ γ

almost sure convergence of the bartlett estimator 19 = q n< qm + qm< qn q n+1 q n+1 = a 3 n, m+a 4 n, m. Following the proofs of 2.15 2.16, we obtain n i=m +1 n Yi Y i+ γ Yi Y i+ γ E[a 3 n, m] 2 Cn mqm, 2.22 E[a 4 n, m] 2 Cqn qmn 2.23 E 2 n,2 Cnqn. 2.24 Just as relations 2.15, 2.16 2.19 implied 2.13, relations 2.22, 2.23 2.24 imply 2.21. Relation 2.8 follows from 2.13 2.21. Observing that z n ẑ n = qn q n qn+1 q n q n+1 n Yi Y i+ γ, following the proof of 2.21, one can easily verify that 2.9 holds. Also, similarly to 2.15 2.24, for any 2 k <m,n 2 k+1,wehave E[ẑ n ẑ m ] 2 Cn mq n Eẑ 2 n Cnq n. Hence, relation 2.10 follows from Problem 5 of Billingsley 1968, the Chebyshev inequality, 1.6 the Borel Cantelli lemma. Proof of part ii. The idea of the proof is similar to that used in part i but different bounds are needed. We use the same notation as in the proof of part i. As verified on p. 291 of Giraitis et al. 2003 qn 1 2H Ev n,1 = qn 1 2H qn ω j qn γ j c 0 H2H 1 = c2 H. 2.25 We first show that qn 1 2H v n,2 0 a.s. Observe that by 1.12 it follows easily ESk,l 2 Cl k +1 2H

20 i. berkes et al. hence E max 1 l n S 0,l 2 Cn 2H 2.26 by a maximal inequality of Billingsley 1968, p. 94. We will now verify that 2.26 implies max S 0,l = o n H log n a.s. 2.27 1 l n Fix 1/2 <p<1 note that for any ɛ>0 P max S 0,l >ɛ2 kh k p 1 1 l 2 k ɛ 2 1 k< 1 ɛ 2 1 k< 1 k< = C 1 ɛ 2 1 k< [ 2 kh k p] 2 E max 1 l 2 k S 0,l 2 [ 2 kh k p] 2 C2 2kH k 2p <. Hence, by the Borel Cantelli lemma max S 0,l = o 2 kh k p 1 l 2 k a.s. as k. Now, for any n, choosingk such that 2 k 1 <n 2 k,weobtain max S 0,l max S 0,l = o 2 kh k p = o n H log n a.s., 1 l n 1 l 2 k establishing 2.27. Using the definition of v n,2 2.27, it is easy to see that qn 1 2H v n,2 = o1qn 2 2H log 2 nn 2H 2 [ ] 2 2H qn = o1 log n1/1 H a.s. n Therefore, assumption 1.13 implies that qn 1 2H v n,2 0 a.s. We now show that qn 1 2H v n, Ev n,1 0 a.s. This will be accomplished by showing that z n z n = o nqn 2H 1 a.s., 2.28 z n ẑ n = o nqn 2H 1 a.s. 2.29 ẑ n = o nqn 2H 1 a.s., 2.30 with z n, z n ẑ n defined, respectively, by 2.4, 2.6 2.7.

almost sure convergence of the bartlett estimator 21 We obtain some inequalities for the second moments of z n z n, z n ẑ n ẑ n their increments on the intervals 2 k, 2 k+1]. The inequality in Problem 5 of Billingsley 1968, p. 102 cf. also Corollary 3.1 in Móricz et al. 1982 with the Borel Cantelli lemma then yields 2.28 2.30. First we use the decompositions in 2.12 2.14. Similarly to 2.15, for 2 k <m<n 2 k+1, we have, using the first relation of Lemma 2.1, E[a 1 n, m] 2 n m ν, l, l+ j qm j qm l 4n Cn mqn 2H n 2H 1. 2.31 As in 2.16, we have, using the second relation of Lemma 2.1, E[a 2 n, m] 2 n ν, l, l+ j qm< qn qm< j qn l 4n C qn qmn 2H qn 2H 1. 2.32 Hence, using also 1.13 we get { P max 2 k+1, m,1 ɛ2 k+1 [ q 2 k+1] } 2H 1 2 k <m 2 k+1 Ck2 ɛ 2 Ck2 ɛ 2 2 k+ 2 k[ q 2 k+1] 2H 2 2H 1k + q 2 k+ q 2 k[ q 2 k+1] 2H 1 2 2kH q 2 k+1 2 k+1 2 2k [q2 k+1 ] 4H 2 2 2H Ck 3/2 ɛ 2, 2.33 so by the Borel Cantelli lemma we have max 2 k+1, m,1 = o 2 k+1 [ q 2 k+1] 2H 1 2 k <m 2 k+1 a.s. 2.34 Similarly, E 2 n,1 Cnqn 2H, so by 1.6, 1.13 the Borel Cantelli lemma we have a.s. = o 2 k+1 q 2 k+1 2H 1. Hence we conclude that 2 k+1,1 n,1 = o nqn 2H 1 a.s. 2.35

22 i. berkes et al. Similarly, along the lines of the proof of 2.21 2.35, we get n,2 = o nqn 2H 1 a.s., 2.36 completing the verification of 2.28. The proofs of 2.29 2.30 are similar to that of 2.28 are therefore omitted. We conclude this section with Lemma 2.1 which was used in the proof of Theorem 1.1. The proof of Lemma 2.1 uses Lemma 2.2 which follows the proof of Lemma 2.1. Lemma 2.1. Suppose {X k } is a fourth order stationary sequence. If 1.12 is satisfied with 1/2 <H<1 assumptions 1.13 1.14 hold, then νh, r, r + s = O qn 2H n 2H 1 2.37 h qn s qn ν h,r,r+ s qm< h qn qm< s qn = O qn qmn 2H 1 qn 2H 1. 2.38 Proof. We note that by 1.6, relation 1.14 remains valid if the summation domain n r, s n is changed to 4n r, s 4n. Thus κh, r, r + s = O qnn 2H 1, h qn s qn by Lemma 2.2 [ γr γ h r+ s + γr+ s γ h r ] h qn s qn qn sup h s qn γ h r+ s γ r +sup h γ h r s qn γr+ s = O qn 2H n 2H 1. Since H>1/2, we have qn =o qn 2H, so 2.37 follows from the identity 1.5.

almost sure convergence of the bartlett estimator 23 Using 1.14 again, we have qm< h qn qm< s qn κ h,r,r+ s = O qn qmn 2H 1 by Lemma 2.2, qm< h qn qm< s qn γr γ h r+ s =2 qm< h qn 2 qm< h qn qm<s qn i 2qn γ r γ i r γr γ h s r Cqn qmn 2H 1 qn 2H 1. Similarly, by 1.12 Lemma 2.2, qm< h qn = qm< h qn qm< s qn qm< s qn γr+ s γ h r Cqn qm 1+ + 1 r 4n 4n r 1 Cqn qmn 2H 1 qn 2H 1. 0 r 4n r 2H 2 r 2H 2 + 4n r 1 qm< h qn qm< s qn γ r+ s γ h r γ h r γr+ s Lemma 2.2. Suppose {X k } is a fourth order stationary sequence. If 1.12 is satisfied with 1/2 <H<1, then uniformly in v =0, ±1, ±2,... k γ i+v Ck 2H 1

24 i. berkes et al. Proof. Let v 1, then k k γ i+v C i + v 2H 2 k C i 2H 2 Ck 2H 1. Let v 0. If v k 1thenwehave k k k γ i+v C i + v 2H 2 C i k 1 2H 2 C If k 1 v 0, we have k i 2H 2 Ck 2H 1. k γ i+v 2C 2k i 2H 2 Ck 2H 1. References I. Berkes, L. Horváth, P. Kokoszka Q-M. Shao, On discriminating between longrange dependence changes in mean, Technical Report, Utah State University, 2003. P. Billingsley, Convergence of probability measures, Wiley, New York, 1968. L. Giraitis, P. S. Kokoszka, R. Leipus G. Teyssière, Rescaled variance related tests for long memory in volatility levels, Journal of Econometrics 112 2003, 265 294. A. W. Lo, Long-term memory in stock market prices, Econometrica 59 1991, 1279 1313. F. A. Móricz, R. J. Serfling W. F. Stout, Moment probability bounds with quasi-superadditive structure for the maximum partial sums, Ann. Probab. 4 1982, 1032 1040. W. F. Stout, Almost sure convergence, Academic Press, New York, 1974. Received: April 25, 2005 István Berkes Department of Statistics Graz University of Technology Steyrergasse 17, A-8010 Graz Austria A. Rényi Institute of Mathematics Hungarian Academy of Sciences P.O. Box 127, H-1364 Budapest Hungary

almost sure convergence of the bartlett estimator 25 Lajos Horváth Department of Mathematics University of Utah 155 South 1440 East Salt Lake City, UT 84112-0090 USA Piotr Kokoszka Department of Mathematics Statistics Utah State University 3900 Old Main Hill Logan, UT 84322-3900 USA Qi-man Shao Department of Mathematics University of Oregon Eugene, OR 97403-1222 USA Department of Statistics Applied Probability National University of Singapore Singapore 117543 Singapore