I
OWA
S
TATE
U
NIVERSITY
Department of Economics
Ames, Iowa, 50011-‐1070
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Block Bootstrap Consistency Under Weak
Assumptions
Gray Calhoun
Working Paper No. 11017 September 2011
Block bootstrap consistency
under weak assumptions
Gray Calhoun
∗Iowa State University
September 23, 2011
Abstract
This paper weakens the size and moment conditions needed for typical block bootstrap methods (i.e. the moving blocks, circular blocks, and stationary boot-straps) to be valid for the sample mean of Near-Epoch-Dependent functions of mixing processes; they are consistent under the weakest conditions that ensure the original process obeys a Central Limit Theorem (those of de Jong, 1997, Econometric Theory ). In doing so, this paper extends de Jong’s method of proof, a blocking argument, to hold with random and unequal block lengths. This paper also proves that bootstrapped partial sums satisfy a Functional CLT under the same conditions.
jel Classification: C12, C15
Keywords: Resampling, Time Series, Near Epoch Dependence, Functional Cen-tral Limit Theorem
∗Economics Department, Iowa State University, Ames, IA 50011. Telephone: (515) 294-6271.
Email: [email protected]. Web: http://www.econ.iastate.edu/∼gcalhoun. I’d like to thank Helle Bunzel and Dimitris Politis for their comments and feedback on this project.
Block bootstraps, e.g. the moving blocks (Kunsch, 1989; Liu and Singh, 1992), circular block (Politis and Romano, 1992), and stationary bootstraps (Politis and Romano, 1994), have become popular in Economics, partly because they do not require the researcher to make parametric assumptions about the data generating process. They are valid under general weak dependence and moment conditions. Some recent papers (Gon¸calves and White, 2002; Gon¸calves and de Jong, 2003) relax the dependence and moment conditions of the original papers to fit with those commonly used in Econometrics based on Near-Epoch-Dependence (ned).12 But these conditions are still stronger than required for a clt to hold; de Jong (1997) has established the clt under L2-ned with smaller size and moment restrictions.3In this paper, I’ll show that
these block bootstrap methods consistently estimate the distribution of the sample mean under de Jong’s (1997) assumptions, and show that an fclt holds as well. I also relax Gon¸calves and White’s (2002) and Gon¸calves and de Jong’s (2003) requirement that the expected block length be o(n1/2) to the original papers’ requirement that it
be o(n).
The proof will exploit the conditional independence of the blocks in each boot-strap. Each bootstrap proceeds by drawing blocks of M consecutive observations from the original time series, and then pasting these blocks together to create the new boot-strap time series. The moving blocks bootboot-strap does exactly that; the circular block bootstrap “wraps” the observations, so that (Xn−1, Xn, X1, X2), for example, is a
pos-sible block of length four (letting Xt denote the original time series). The stationary
bootstrap wraps the observations and also draws M at random for each block; Politis and Romano (1994) suggest drawing M from the geometric distribution. As the name suggests, the series produced by the stationary bootstrap are strictly stationary, while 1Gon¸calves and White (2002) show that these bootstrap methods can be applied to heterogeneous
L2+δ-ned processes of size −2(r − 1)/(r − 2) on a strong mixing sequence of size −r(2 + δ)/(r − 2),
where r > 2 and δ > 0, when the original series has uniformly bounded 3r-moments. Gon¸calves and de Jong (2003) relax these conditions to L2+δ-ned of size −1 and r + δ moments for the original
series, and size −(2 + δ)(r + δ)/(r − 2) for the underlying mixing series. Both papers require that the expected block length grow with n and be o(n1/2).
2An array {X
nt} is an Lρ-ned process on a mixing array {Vnt} if
kXnt− E(Xnt| Vn,t−m, . . . , Vn,t+m)kρ≤ dntvm (1)
with vm→ 0 as m → ∞ and {dnt} an array of positive constants. It is of size −γ if vm= O(m−γ−δ)
for all δ > 0. Dropping the index “n” gives the series definition.
3
de Jong (1997) proves that the clt holds for averages of L2-ned processes of size −1/2 on a
those produced by the other methods are not. Although the stationary bootstrap was believed to be much less efficient than other block bootstrap methods due to results of Lahiri (1999), Nordman (2009) has shown that it is only slightly less efficient than the other block bootstraps discussed in this paper, and has efficiency identical to that of the non-overlapping block bootstrap. Consequently, there has been renewed interest in the stationary bootstrap, since stationarity of the bootstrap samples is a useful property for theoretical research.
Theorem 1 presents our main result, asymptotic normality of the distribution of bootstrapped partial sums. This paper adopt the standard notation that E∗, var∗, etc. are the usual operators with respect to the probability measure induced by the bootstrap and will use explicit stochastic array notation in this paper for precision. Also note that all results are presented for the scalar case but generalize immediately to vector-valued random variables. All of the proofs are presented in the appendix; I only present proofs for the stationary bootstrap, since proofs for the other methods are similar and easier. All limits are taken as n → ∞ unless otherwise noted.
Theorem 1. Suppose the following conditions hold.
1. Xnt is L2-ned of size −1/2 on an array Vnt that is either strong mixing of
size −r/(r − 2) or uniform mixing of size −r/2(r − 1), with r > 2. The ned magnitude indices are denoted {dnt}.
2. E Xnt = µnt, µnt− ¯µn is uniformly bounded, and
√
nk ¯Xn− ¯µnk2 → 1.
3. There exists an array of positive real numbers {cnt} such that maxtcnt is
uni-formly finite, (Xnt− µnt)/cnt is uniformly Lr-bounded, and dnt/cnt is uniformly
bounded.
4. Xnt∗ is generated by the stationary bootstrap with geometric block lengths with success probability pn, pn= cn−a and a, c ∈ (0, 1), or by the moving or circular
block bootstrap with block length Mn such that Mn→ ∞ and Mn/n → 0.
Define B as Brownian Motion and B∗n(γ) = n
−1/2Pbnγc t=1 (X ∗ nt − µ ∗ n). Then √ n( ¯Xn∗− µ∗n) →dN (0, 1) and B∗ n →dB.
Theorem 1 assumes that the series have been normalized by dividing by the square root of the (population) second moment of √n( ¯Xn − ¯µn). Sometimes, though, it’s
dependence and moment conditions listed in Theorem 1 apply to the original array and not the normalized array η−1n Xnt as is clear from inspecting the proofs in this
paper and de Jong (1997).4 Deriving bounds for η−1
n Xnt from primitive conditions on
Xnt would be difficult.
Note that de Jong (1997) allows a little bit more flexibility in his conditions on the array {cnt} (see also Davidson, 1993); essentially, he allows there to be a
sin-gle set of blocks with the maximal {cnt} over each block well-behaved, while this
paper requires that his condition hold for every possible partition of blocks. This ad-ditional restriction is required because the stationary bootstrap will select the blocks randomly.
Consistency for the sample mean follows as an immediate corollary of Theorem 1 and de Jong (1997, Theorem 2).
Corollary 2. Define ¯µn = n−1
Pn
t=1µnt. If the conditions of Theorem 1 hold then
√
n( ¯Xn∗− µ∗ n) and
√
n( ¯Xn− ¯µn) converge in distribution to the same limit.
Corollary 2 justifies using the stationary bootstrap to conduct inference about ¯µn
even though there is considerable heterogeneity. This conclusion is also present in Gon¸calves and White’s (2002) and Gon¸calves and de Jong’s (2003) papers, but those authors introduce and emphasize additional assumptions that ensure the heterogene-ity is asymptotically irrelevant, either because the deviations of µnt from ¯µnare small
or because there are a finite number of breaks that occur in a neighborhood of the first observation. Such assumptions are unnecessary.
But heterogeneity rules out a version of Corollary 2 for the partial sum. It is not hard to see that under the assumptions of Theorem 1,
n−1/2 bγnc X t=1 Xnt− (γn)−1 bγnc X s=1 µnt →dBξ, (2)
where Bξ is a Gaussian process and ξ captures the asymptotic variance of the partial
sums. It would be uncommon for a researcher to be interested in these particular random variables; one would typically be more interested in ¯µnthan (γn)−1
Pbγnc
s=1 µnt,
but these bootstrap methods do not allow us to approximate the distribution of (γn)−1/2Pbγnc
t=1 (Xnt− ¯µn). The same discussion applies if µnt = ¯µnfor all t but Bξ 6= B.
Other methods, such as the local block bootstrap (Dowla et al., 2003; Paparoditis and 4See also Hall and Heyde’s (1980) Theorem 3.2, Corollary 3.1, and following remarks.
Politis, 2002), may be able to capture this heterogeneity, but we do not pursue that possibility further.
For completeness, I’ll present a consistency result under the additional condi-tion that the heterogeneity vanishes asymptotically (similar to Assumpcondi-tion 2.2 of Gon¸calves and White, 2002). This result is a corollary of Theorem 1 and de Jong and Davidson (2000, Theorem 3.1).
Corollary 3. Let Bn(γ) = n−1/2P bnγc
t=1 (Xnt − ¯µn) and suppose the conditions of
Theorem 1 hold, supt|µnt− ¯µn| → 0, and
n−1
bγnc
X
s,t=1
E(Xns− µns)(Xnt− µnt) → γ (3)
for all γ ∈ [0, 1]. Then Bn and B∗n converge in distribution to the same limit.
We can develop some more intuition by looking at the argument behind Theo-rem 1. For concreteness, let Xn1, . . . , Xnndenote the original array and let Xn1∗ , . . . , Xnn∗
a hypothetical series generated by the stationary bootstrap. Conditional on the num-ber of blocks, Nn, and the block lengths, Mn1, . . . , MnNn, we can see that
√ n ¯Xn∗ = √1 Nn Nn X i=1 (r Nn n Kn,i X t=Kn,i−1+1 Xnt∗ ) , with Kn0 = 0, Kni = i X j=1 Mnj, (4) is the sum of Nn independent random variables. So
√
n( ¯Xn∗− ¯Xn) should obey a clt
under very weak moment conditions and satisfy a relationship like ¯ Xn∗− µ∗ n var∗( ¯X∗ n | Mn1, . . . , Mn,Nn) →d∗ N (0, 1). (5)
This argument applies directly to the moving blocks and circular bootstraps, since Nnand Mniare deterministic. But they are stochastic for the stationary bootstrap and
the randomness of Nn, in particular, complicates a direct argument; conditioning on
Nnwould require us to work with the distributions of Mnjgiven Mn1+· · ·+Mn,Nn = n.
This information should not matter in the limit, though, and so, as a first step, I’ll show that we can replace Nnwith its expected value without affecting the asymptotic
distribution induced by the bootstrap.
1. Draw Mn1, . . . , Mn,bnpnc independently from the geometric distribution with
suc-cess probability pn, let mn =P bnpnc
i=1 Mni, and let Mn= (Mn1, . . . , Mn,bnpnc).
2. Draw bnpnc blocks from Xn1, . . . , Xnn (with the observations “wrapped” as in
the stationary bootstrap), with Mni the length of the ith block.
Let {Xnt∗∗} be a hypothetical array generated by this procedure, let {X∗
nt} be an array
generated by the stationary bootstrap, define µ∗n = µ∗∗n = ¯Xn, and define B∗∗n (γ) =
m−1/2n P bγmnc
t=1 (X
∗∗
nt − µ∗∗n ). Under the conditions of Theorem 1, B∗n and B∗∗n converge
to the same limit.
It is easy to see that if Xn1∗ , . . . , Xnn∗ is a hypothetical sequence generated by the moving or circular blocks bootstrap with block length Mn and Xn1∗∗, . . . , X
∗∗
n,mn is a
bootstrap generated as in Lemma 4 but with bn/Mnc blocks of length Mn, then the
conclusion of Lemma 4 still holds.
Given Lemma 4 and the preceding discussion, consistency for the distribution of the sample mean can be expected to hold if
¯ Xn∗∗− µ∗∗ n var∗∗( ¯X∗∗ n | Mn) →d∗∗ N (0, 1) (6)
and var∗∗(m1/2n X¯n∗∗ | Mn) →p 1. Lemma 5 establishes a stronger result necessary for
partial sums.
Lemma 5. If the conditions of Theorem 1 hold then var∗∗(m−1/2n P bγmnc
t=1 X
∗∗
nt | Mn) →p
γ for all γ ∈ [0, 1].
The crux of Lemma 5 is the recognition that var∗∗(m1/2n X¯n∗∗| Mn) can be written
as an average over squared blocks of consecutive observations of the original series;5
i.e. var∗∗(m1/2n X¯n∗∗| Mn) = n−1 n X τ =1 bnpnc X j=1 Yn,τ,j2 (7) where Yn,τ,j = n−1/2P In,τ,j t=In,τ,j−1+1(Xnt− ¯Xn) and 0 = In,τ,0 < In,τ,1 < · · · < In,τ,bnpnc=
mn (the specifics are given in the proof and use different notation). Standard
argu-ments from de Jong (1997) establish thatP
jY 2
n,τ,j →p 1 for each τ .
5Obviously, this argument needs to hold for the partial sums as well, but I’ll just discuss the
As a final point, the consistency of the bootstrap variance follows as a corollary to Lemmas 4 and 5.
Corollary 6. If the conditions of Theorem 1 hold then var∗(n1/2X¯∗
n) = var(n1/2X¯n) +
op.
The key insight of this paper is that proving that bootstrapped partial sums obey an fclt reduces to proving consistency of the bootstrap variance (conditional on the block lengths, if they are random), and that the conditional bootstrap variance can be expressed in terms of blocks of consecutive observations of the original series. Since convergence of these blocks is essential to the clt in general,6 I conjecture that the same approach will hold under most forms of dependence that allow for a clt or fclt. Finally, this result allows for considerable heterogeneity in the original process when approximating the distribution of a sample mean but not a partial sum.
Appendix: Supporting Results and Proofs
Lemma A.1. Let αn(x) = 1 + ((x − 1) mod n), let Nn = (Mn1, . . . , Mn,Nn), and
suppose that the conditions of Theorem 1 hold. For any t0 and k such that the index
of the following summations are well defined,
E∗ t0+k X t=t0+1 Xnt∗ Nn ! = E∗∗ t0+k X t=t0+1 Xnt∗∗ Mn ! = k ¯Xn, (8)
6See, for example, McLeish (1974), Hall and Heyde (1980, Chapter 3), and Davidson (1994,
and var∗ t0+k X t=t0+1 Xnt∗ Nn ! = var∗∗ t0+k X t=t0+1 Xnt∗∗ Mn ! = n−1 n X τ =1 "(Kn,κn0 X t=t0+1 (Xn,αn(τ +t)− ¯Xn) )2 + κn1−1 X j=κn0+1 ( Knj X t=Kn,j−1+1 (Xn,αn(τ +t)− ¯Xn) )2 + ( t0+k X t=Kn,κn1−1+1 (Xn,αn(τ +t)− ¯Xn) )2# (9)
where κn0 and κn1denote the blocks containing t0 and t0+k, so Kn,κn0−1 < t0 ≤ Kn,κn0
and Kn,κn1−1 < t0+ k ≤ Kn,κn1
Proof of Lemma A.1. Equation (8) is immediate. For (9), notice that
var∗ t0+k X t=t0+1 Xnt∗ Nn ! = var∗ Kn,κn0 X t=t0+1 Xnt∗ Nn ! + κn1−1 X j=κn0+1 var∗ Knj X t=Kn,j−1+1 Xnt∗ Nn ! + var∗ t0+k X t=Kn,κn1−1+1 Xnt∗ Nn ! , (10)
since the blocks are independent given Nn. Now, for each block,
var∗ Knj X t=Kn,j−1+1 Xnt∗ Nn ! = n−1 n X τ =1 ( Knj X t=Kn,j−1+1 (Xn,αn(τ +t)− ¯Xn) )2 , (11)
with similar formulae for the first and last blocks. The same argument holds for var∗∗, completing the proof.
Lemma A.2. Suppose that Mn1, . . . , Mn,bnpnc are i.i.d. geometric random variables
with success parameter pn = cn−a with a, c ∈ (0, 1), and that `n = (pnlog p−1n ) −1.
Then (i) maxi=1,...,bnpncMni/n →p 0 and (ii) maxi=1,...,bnpncMni/`1+n →p 0 as n → ∞
for any positive .
that xnpn → ∞,
Pr[max
i Mni ≤ xn] = (1 − (1 − pn) xn
)bnpnc→ lim exp(−npn(1 − pn)xn). (12)
Moreover, npn(1 − pn)xn → lim npne−xnpn. To prove (i), let xn = nx for any positive
number x. Then Pr[max
i Mni/n ≤ x] → lim exp(−npne −npnx
) = exp(0) = 1. (13) Since x is arbitrary, maxiMni/n →p 0.
For (ii), let xn = `1+n x and note that
pn`1+n ≥ p
−(−δ−δ)
n = c
−(−δ−δ)
na(−δ−δ)≡ bna(−δ−δ) (14)
for any δ > 0 and large enough n. Choose δ small enough that > δ(1 + ). Then npnexp(−`1+n pn) ≤ npnexp(−bna(−δ−δ)) = cv
1−a a(−δ−δ)
n exp(−bvn) → 0, (15)
with vn = na(−δ−δ). Consequently,
Pr[max
i Mni/` 1+
n ≤ x] → exp(0) = 1 (16)
as well.
Lemma A.3. Suppose the conditions of Theorem 1 hold, let `n = (p log p−1)−1, and
define ZC
nj(τ, Mn), Mnj(τ ), and Knj(τ ) as in the proof of Lemma 5. Now let Jn be
the number of block lengths Mnj(τ ) that are greater that `n; let jn1, . . . , jn,Jn be their
indices, so that jn1 is the index of the first such block, jn2 is index of the second, etc.;
and define Wni = (Vn,Ii−1+1, . . . , Vn,Ii) with I0 = 0 and, for i > 0,
Ii =
Kn,ji jn,i+1− jni = 1,
b(Kn,jn,i+1 + Kn,jn,i)/2c otherwise.
(17)
Then {ZnjiC (τ, Mn), Fni}, with Fni= σ(Mn, Wni, Wn,i−1, . . . ), is an L2-mixingale,
conditional on Mn, of size −1/2 with magnitude indices
for some > 0, with D ≥ 2 maxt(dnt/cnt) maxt(cnt) and B a finite constant that
depends only on the mixingale coefficients of Xnt.7
Proof of Lemma A.3. Similar to de Jong (1997, Lemma 5), we’ll show that Zc
nji(τ, Mn)2−
Zc
nji(τ, Mn)2 is an L2-mixingale (conditional on Mn) by first showing that it is L2
-ned (also conditional on Mn); the conclusion then follows immediately. By the same
argument as in de Jong’s Equation (A.32), for k > 1
EhZnjiC (τ, Mn) − E(ZnjiC (τ, Mn) | Mn; Wn,i−k, . . . , Wn,i+k)
i2 Mn
12
≤ k−1/2−· CDMn,ji(τ )3/2n−1`−1/2−n , (19)
with > 0 and (for k = 0)
EhZnjiC (τ, Mn) − E(ZnjiC (τ, Mn) | Mn; Wn,i)
i2 Mn
12
≤ CDBMn,jin−1 (20)
Davidson’s (1994) Theorem 17.5 completes the proof.
Lemma A.4. Define Mnj(τ ) and Knj(τ ) as in the proof of Lemma 5. Under the
conditions of Theorem 1, n−1 bγnpnc X j=2 E Kn,j−1(τ ) X t=1 Knj(τ ) X s=Kn,j−1(τ )+1 (Xnt− ¯µn)(Xns− ¯µn) Mn ! →p 0 (21) uniformly in γ and τ .
Proof of Lemma A.4. Define `n = (pnlog p−1n )
−1. Then n−1 bγnpnc X j=2 E Kn,j−1(τ ) X t=1 Knj(τ ) X s=Kn,j−1(τ )+1 (Xnt− ¯µn)(Xns− ¯µn) Mn ! = Op(`npn) + n−1 bγnpnc X j=2 E Kn,j−1(τ ) X t=1 Knj(τ ) X s=Kn,j−1(τ )+`n+1 (Xnt− ¯µn)(Xns− ¯µn) Mn ! (22)
uniformly in τ and γ by McLeish (1975, Theorem 1.6). Then `npn → 0 by
construc-tion and the remainder of the proof follows the same argument as de Jong’s (1997) Lemma 4.
Proof of Theorem 1
By Lemma 4, it suffices to prove that B∗∗n obeys the fclt. We have
m−1/2n bγmnc X t=1 (Xnt∗∗− µ∗∗n ) = (np)−1/2 bγnpnc X j=1 n (npn/mn)1/2 Knj X t=Kn,j−1+1 (Xnt∗∗− µ∗∗n )o + m−1/2n sgn(γmn− Kn,bγnpnc) max(bγmnc,Kn,bγnpnc) X t=min(bγmnc,Kn,bγnpnc)+1 (Xnt∗∗− µ∗∗n). (23)
We can show that the second sum in (23) vanishes asymptotically. To simplify the presentation, assume that γmn> Kn,bγnpnc; the same argument works for the reverse
inequality as well. By Lemma A.1,
E∗∗m−1/2n
bγmnc
X
t=Kn,bγnpnc+1
(Xnt∗∗−µ∗∗n ) = ( ¯Xn− ¯Xn) E∗∗(bγmnc−Kn,bγnpnc)/m1/2n = 0 (24)
(remember that mn is random) and
var∗∗m−1/2n bγmnc X t=Kn,bγnpnc+1 (Xnt∗∗− µ∗∗n ) (25) = E∗∗(mnn)−1 n X τ =1 ( κn−1 X j=bγnpnc+1 Knj X t=Kn,j−1+1 (Xnt− ¯Xn) !2 (26) + bγmnc X t=Kn,κn−1+1 (Xnt− ¯Xn) !2) →p 0 (27)
uniformly in γ by McLeish (1975, Theorem 1.6), with κn indicating the block
con-taining bγmnc, so Kn,κn−1 < bγmnc ≤ Kn,κn. It now suffices to prove that the first
term in (23) satisfies an fclt. Let Znj∗∗ = pnpn/mn PKnj t=Kn,j−1+1(X ∗∗ nt − µ ∗∗ n); clearly Z ∗∗ nj is an mds conditional
on Mn that has finite variance and is globally covariance stationarity condition by
Lemma 5. Moreover, (npn)−1 P j(Z ∗∗ nj)2 →p ∗∗
1 by the lln (e.g. Davidson, 1994, The-orem 19.7) and maxj(Znj∗∗)2 = op∗∗(n) given Mn by uniform integrability. Then the
random functions γ 7→ (npn)−1/2
Pbγnpnc
j=1 Z
∗∗
nj converge in distribution to B conditional
depend on Mn, it holds unconditionally as well.
Proof of Lemma 4
We’ll use a coupling argument. If mn> n, then ¯Xn∗ d = ¯Yn∗∗≡ n−1Pn t=1X ∗∗ nt. Similarly, if mn < n, ¯Xn∗∗ d = ¯Zn∗ ≡ n−1Pmn t=1X ∗
nt. So the result holds if m 1/2 n ( ¯Xn∗∗ − ¯Yn∗∗) →p 0 and n1/2( ¯X∗ n− ¯Z ∗ n) →p 0; i.e. m−1/2n bγnc X t=bγmnc+1 (Xnt∗ − µ∗n) →p 0, (n−1/2− m−1/2n ) bγnc X t=1 (Xnt∗ − µ∗n) →p 0 (28) and n−1/2 bγmnc X t=bγnc+1 (Xnt∗∗− µ∗∗n ) →p 0, (m−1/2n − n−1/2) bγmnc X t=1 (Xnt∗∗− µ∗∗n ) →p 0 (29)
uniformly in γ, where summations over empty index sets are defined to be zero. I’ll prove (28) as the argument for (29) is identical
Now let Nn = (Mn1, . . . , Mn,Nn). By Lemma A.1,
E∗m−1/2n bγnc X t=bγmnc+1 (Xnt∗ − µ∗n) Nn = (bγnc − bγmnc) + n1/2 ( ¯Xn− ¯Xn) = 0, (30) E∗(n−1/2− m−1/2n ) bγnc X t=1 (Xnt∗ − µ∗n) Nn = 0, (31)
and var∗m−1/2n bγnc X t=bγmnc+1 (Xnt∗ − µ∗n) Nn (32) = (mnn)−1 n X τ =1 "( Kn,κn0 X t=bγmnc+1 (Xn,αn(τ +t)− ¯Xn) )2 (33) + κn1−1 X j=κn0+1 ( Knj X t=Kn,j−1+1 (Xn,αn(τ +t)− ¯Xn) )2 (34) + ( bγnc X t=Kn,κn1−1+1 (Xn,αn(τ +t)− ¯Xn) )2# . (35) By McLeish (1975, Theorem 1.6), E var∗(γmn)−1/2 bγnc X t=bγmnc+1 (Xnt∗ − µ∗n) Nn ≤ γ−1E(mnn)−1 n X τ =1 EhAn0(Kn,κn0 − bγmnc) + κn1−1 X j=κn0+1 AnjMnj+ An,κn1(bγnc − Kn,κn1 − 1) i → 0 (36)
uniformly in γ, since the Anj are uniformly bounded constants that only depend on
the mixingale magnitude indices of {Xnt}. Also,
var∗(n−1/2− m−1/2n ) bγnc X t=1 (Xnt∗ − µ∗n) | Nn →p 0 (37)
uniformly by a similar argument. McLeish (1975, Theorem 1.6) also ensures that the conditional variances are uniformly integrable, giving (28).
Proof of Lemma 5
For any τ in {1, . . . , n}, let κn(τ ) indicate the block containing τ , so Kn,κn(τ )−1 < τ ≤
Kn,κn(τ ). Also let
Mnj(τ ) = (min(n, Kn,βn(κn(τ )+j)− τ ) − max(1, Kn,βn(κn(τ )−1+j)+ 1 − τ ))+, (38)
with βn(x) = (x − 1) modbnpnc + 1, and let Knj(τ ) =
Pj i=1Mnj(τ ) and Kn0(τ ) = 0. Now define Znj(τ, Mn) = Knj(τ ) X t=Kn,j−1(τ )+1 n−1/2(Xnt− ¯µn), j = 1, . . . , bγnpnc. (39) By Lemma A.1, var∗∗m−1/2n bγmnc X t=1 Xnt∗∗| Mn = n−1 n X τ =1 bγnpnc X j=1 Znj(τ, Mn)2 + op (40)
uniformly in γ. The advantage of this representation is that each series {Znj(τ, M)}j
is formed from consecutive blocks of the original series.
Define the lower bound `n on the lengths Mni as `n = (pnlog p−1n )−1. Since bγnpnc X j=1 E Znj(τ, Mn)21{Mnj(τ ) > `n} ≤ (1 − pn)1/pn pn`n bγnpnc X j=1 E Znj(τ, Mn)2 = e−pn`nO(1) → 0, (41)
with the last equality a consequence of McLeish (1975, Theorem 1.6), we have
var∗∗(m1/2n X¯n∗∗| Mn) = n−1 n X τ =1 bγnpnc X j=1 Znj(τ, Mn)21{Mnj(τ ) > `n} + op. (42)
Now, truncate Znj(τ, Mn) similar to de Jong (1997, Lemma 5): define ZnjC(τ, Mn) = 0 Mnj(τ ) ≤ `n, Znj(τ, Mn) Mnj(τ ) > `n, |Znj(τ, Mn)| ≤ CpMnj(τ )/n CpMnj(τ )/n Mnj(τ ) > `n, Znj(τ, Mn) > CpMnj(τ )/n −CpMnj(τ )/n Mnj(τ ) > `n, Znj(τ, Mn) < −CpMnj(τ )/n. (43) McLeish (1975, Theorem 1.6) establishes that the family {nZnj(τ, M)2/Mnj(τ ); n, j, τ, M}
is uniformly integrable,8 so Z
nj(τ, Mn)2− Znjc (τ, Mn)2 can be made arbitrarily small
by choosing a large enough C. So it suffices to prove
bγnpnc X j=1 ZnjC(τ, Mn)2− E(ZnjC(τ, Mn)2 | Mn) →p 0 (44) and n−1 n X τ =1 bγnpnc X j=1 E(Znj(τ, Mn)2 | Mn) →p γ. (45)
McLeish (1975, Theorem 1.6) and Lemma A.3 imply that, for some > 0,
E (bγnpnc X j=1 ZnjC(τ, Mn)2 − E(ZnjC(τ, Mn)2 | Mn) )2 Mn ! ≤ A2C2 bγnpnc X j=1 (Mnj(τ )/n)2max(1, Mnj(τ )/`1+n ), (46)
which converges to zero if maxjMnj/n →p 0 and maxjMnj/`1+n →p 0. Both follow
from Lemma A.2, proving (44).
To prove (45), let dn be a sequence of integers such that dn → ∞ and dn/n → ∞
and let ζn(γ) be a sequence of functions such that ζn(x) = ζn(x − 1) for all x and
(2dn)−1
bγnc+dn
X
s,t=bγnc−dn+1
E(Xns− ¯µn)(Xnt − ¯µn) − ζn(γ) → 0 (47)
for all γ ∈ (0, 1]. It is obvious that (45) holds for γ = 0. Lemmas A.3 and A.4 and 8See also Davidson (1992, Lemma 3.2) and de Jong (1997, Lemma 5)
the fact that Pn
τ =1n −1ζ
n(x + τ /n) → 1 uniformly in x by assumption imply that
n−1 n X τ =1 bγnpnc X j=1 E(Znj2 (τ, Mn) | Mn) = n X τ =1 bγnpnc X j=1 ζn(τ /n + j/npn) n2p n + op (48) = γ + op, (49)
completing the proof.
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