On the range of validity of the autoregressive sieve bootstrap

0.32 MB·English

by  Jens-Peter Kreiss

TheAnnalsofStatistics
2011,Vol.39,No.4,2103–2130
DOI:10.1214/11-AOS900
(cid:13)c InstituteofMathematicalStatistics,2011
ON THE RANGE OF VALIDITY OF THE AUTOREGRESSIVE
SIEVE BOOTSTRAP
2
1
By Jens-Peter Kreiss, Efstathios Paparoditis and
0
2 Dimitris N. Politis

n
Technische Universit¨at Braunschweig, University of Cyprus and University
a
J of California, San Diego

0
3
Weexplorethelimitsoftheautoregressive(AR)sievebootstrap,

] andshowthatitsapplicabilityextendswellbeyondtherealmoflinear
T time series as has been previously thought. In particular, for appro-
S priate statistics, the AR-sieve bootstrap is valid for stationary pro-
. cesses possessing a general Wold-type autoregressive representation
h
t with respect toawhite noise;in essence, thisincludesall stationary,
a purely nondeterministic processes, whose spectral density is every-
m
where positive. Our main theorem provides a simple and effective
[ tool in assessing whether the AR-sieve bootstrap is asymptotically

valid in any given situation. In effect, the large-sample distribution
1
of the statistic in question must only depend on the first and sec-
v
ond order moments of the process; prominent examples include the
1
1 samplemeanandthespectraldensity.Asacounterexample,weshow
2 how theAR-sievebootstrap is not always valid for thesample auto-
6 covariance even when theunderlyingprocess is linear.
.
1
0 1. Introduction. Duetothedifferentpossibledependencestructuresthat
2
may occur in time series analysis, several bootstrap procedures have been
1
: proposed to infer properties of a statistic of interest. Validity of the different
v
bootstrap procedures depends on the probabilistic structure of the under-
i
X lying stochastic process X=(X :t Z) and/or on the particular statistic
t
r considered.Bootstrapschemesfortim∈eseriesrankfromthoseimposingmore
a
parametric type assumptions on the underlying stochastic process class to
those accounting only for some kind of mixing or weak dependence assump-
tions. For an overview see Bu¨hlmann (2002), Lahiri (2003), Politis (2003)
and Paparoditis and Politis (2009).
Received December 2010; revised April 2011.
AMS 2000 subject classifications. Primary 62M10, 62M15; secondary 62G09.
Key words and phrases. Autoregression, bootstrap, time series.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Statistics,
2011,Vol. 39, No. 4, 2103–2130. This reprint differs from the original in
pagination and typographic detail.
1
2 J.-P. KREISS,E. PAPARODITISANDD. N. POLITIS
A common assumption is that X is a linear time series, that is, that
∞
(1.1) X = b e , t Z,
t j t−j
∈
j=−∞
X
with respecttoindependent,identically distributed(i.i.d.) randomvariables
(e )—often assumed to have mean zero and finite fourth order moments—
t
and forabsolutely summablecoefficients (b ); thisis notto beconfusedwith
j
the Wold representation with respect to white noise, that is, uncorrelated,
errorsthatallstationary,purelynondeterministicprocessespossess.Ifb =0
j
for all j<0, then the linear process is called causal.
Stationary autoregressive (AR) processes of order p are members of the
linear class (1.1) provided the autoregression is defined on the basis of i.i.d.
errors. Model-based bootstrapping in the AR(p) case was among the first
bootstrap proposals for time series; see, for example, Freedman (1984). The
extension to the AR( ) case was inevitable; this refers to the situation
∞
where the strictly stationary process X has the following linear infinite
t
order autoregressive representation
∞
(1.2) X = π X +e , t Z,
t j t−j t
∈
j=1
X
withrespecttoi.i.d.errorse havingmeanzero,variance0<E(e2)=σ2 and
t t e
E(e4)< ; here the coefficients π are assumed absolutely summable and
π(z)t=1∞ ∞ π zj =0for z =1j.Thetworepresentations,(1.1)and(1.2)
− j=1 j 6 | |
are related; in fact, the class (1.2) is a subset of the linear class (1.1). Fur-
P
thermore, it can be shown that the linear AR( ) process (1.2) is causal if
and only if π(z)=1 ∞ π zj =0 for z 1∞.
− j=1 j 6 | |≤
There is already a large body of literature dealing with applications and
P
properties of the AR-sieve bootstrap. Kreiss (1988, 1992) established valid-
ity of thisbootstrap schemefor differentstatistics includingautocovariances
and autocorrelations. Paparoditis and Streitberg (1992) established asymp-
totic validity of the AR-sieve bootstrap to infer properties of high order
autocorrelations, and Paparoditis (1996) established its validity in a multi-
variatetimeseriescontext.Theaforementionedresultsrequiredanexponen-
tial decay of the AR coefficients π as j ; Bu¨hlmann (1997) extended
j
→∞
the class of AR( ) processes for which the AR-sieve bootstrap works by al-
∞
lowing a polynomially decay of the π coefficients. Furthermore, Bickel and
j
Bu¨hlmann (1999) introduced a mixing concept appropriate for investigating
properties of the AR-sieve bootstrap which is related to the weak depen-
dence concept of Doukhan and Louhichi (1999), while Choi and Hall (2000)
focused on properties of the AR-sieve bootstrap-based confidence intervals.
A basic assumption in the current literature of the AR-sieve bootstrap
is that X is a linear AR( ) process, that is, X is generated by (1.2) with
t
∞
ON THE VALIDITYOFTHE AUTOREGRESSIVESIEVEBOOTSTRAP 3
(e ) being an i.i.d. process. One exception is the case of the sample mean
t
X =n−1 n X , where Bu¨hlmann (1997) proved validity of the AR-sieve
n t=1 t
bootstrap also for the case where the assumption of i.i.d. errors in (1.2)
P
can be relaxed to that of martingale differences, that is, E(e )=0 and
t t−1
|E
E(e2 )=σ2 with =σ( e :s t 1 ) the σ-algebra generated by
t|Et−1 e Et−1 { s ≤ − }
the random variables e ,e ,... . Notice that the process (1.2) with in-
t−1 t−2
{ }
novationsformingamartingaledifferencesequenceisinsomesensenot“very
far” from the linear process (1.2) with i.i.d. errors.In fact, some authors call
the set-up of model (1.1) with martingale difference errors “weak linearity,”
and the same would hold regarding (1.2); see, for example, Kokoszka and
Politis (2011).
To elaborate, for a causal linear process the general L -optimal predic-
2
tor of X based on its past X ,X ,..., namely the conditional expec-
t+k t t−1
tation E(X X ,s t) of X , is identical to the best linear predic-
t+k s t+k
| ≤
tor (X ); here k is assumed positive, and denotes orthogonal
PMt t+k PC
projection onto the set C and =span X :j s , that is, the closed
s j
M { ≤ }
linear span generated by the random variables X :j s . The property of
j
{ ≤ }
linearity of the optimal predictor is shared by causal processes that are only
weakly linear. Recently, under the assumption of weak linearity with (1.2),
Poskitt (2008) claimed validity of the AR-sieve bootstrap for a much wider
class of statistics that are defined as smooth functions of means. However,
this claim does not seem to be correct in general. In particular, our Exam-
ple 3.2 of Section 3 contradicts Theorem 2 of Poskitt (2008); see Remark 3.2
in what follows.
Theaim ofthepresentpaperistoexplorethelimits oftheAR-sieve boot-
strap, and to give a definitive answer to the question concerning for which
classes of statistics, and for which dependence structures, is the AR-sieve
bootstrapasymptotically valid.Moreover,wealsoaddressthequestionwhat
theAR-sievebootstrapreally doeswhenitisappliedtodatastemmingfrom
a stationary process not fulfilling strict regularity assumptions such as lin-
earity orweak linearity. Inordertodothis,weexamineindetailinSection 2
processes possessing a so-called general autoregressive representation with
respect to white noise errors; these form a much wider class of processes
than the linear AR( ) class described by (1.2).
∞
Ourtheoretical resultsinSection3provideaneffective andsimpletoolfor
gauging consistency of the AR-sieve bootstrap. They imply that for certain
classes of statistics the range of the validity of the AR-sieve bootstrap goes
far beyond that of the linear class (1.1). On the other hand,for other classes
of statistics, like for instanceautocorrelations, validity of theAR-sieve boot-
strap is restricted to the linear process class (1.1), while for statistics like
autocovariances, the AR-sieve bootstrap is only valid for the linear AR( )
∞
class(1.2).Buteveninthecaseofthelinearautoregression(1.2)withinfinite
order, the theory developed in this paper provides a further generalization
4 J.-P. KREISS,E. PAPARODITISANDD. N. POLITIS
of existing results since it establishes validity of this bootstrap procedure
under weaker assumptions on the summability of the coefficients π , thus
j
relaxing previous assumptions referring to exponential or polynomial decay
of these coefficients.
The remaining of the paper is organized as follows. Section 2 develops
the background concerning the Wold-type infinite order AR representation
that is required to study the AR-sieve bootstrap, and states the necessary
assumptions to be imposed on the underlying process class and on the pa-
rameters of the bootstrap procedure. Section 3 presents our main result and
discussesitsimplications bymeansofseveralexamples.Proofsandtechnical
details are deferred to the Appendix.
2. The AR-sieve bootstrap and general autoregressive representation.
Here,andthroughoutthepaper,weassumethatwehaveobservationsX ,...,
1
X stemming from a strictly stationary process X. Let T =T (X ,...,X )
n n n 1 n
be an estimator of some unknown parameter θ of the underlying stochastic
process X. Suppose that for some appropriately increasing sequence of real
numbers c :n N the distribution = (c (T θ)) has a nondegener-
n n n n
{ ∈ } L L −
ated limit. TheAR-sieve bootstrap proposalto estimate the distribution
n
L
goes as follows:
Step 1: Selectanorderp=p(n) N,p n,andfitapthorderautoregres-
∈ ≪
sive model to X ,X ,...,X . Denote by a(p)=(a (p),j =1,2,...,p), the
1 2 n j
Yule–Walker autoregressive parameter estimators, that is, a(p)=Γ(p)−1γ
p
where for 0 h p, b b
≤ ≤
n−|h| b b b
γ (h)=n−1 (X X )(X X ),
X t n t+|h| n
− −
t=1
X
X =n−1 n Xb, Γ(p)=(γ (r s)) and γ =(γ (1),γ (2),...,
n t=1 t X − r,s=1,2,...,p p X X
γ (p))′.
X
P p
Step 2: Let εt(p)b=Xt −b j=1aj(p)Xt−j, t=p+b1,p+b2,...,bn, be the
rbesiduals of the autoregressive fit and denote by F the empirical distri-
P n
bution functioneof the centered rebsiduals ε (p)=ε (p) ε, where ε=(n
t t
p)−1 nt=p+1εt(p). Let (X1∗,X2∗,...,Xn∗) be a set ofbob−servations from th−e
timePseries X∗={Xt∗:t∈Z} where Xt∗= b pt=1aj(ep)Xt∗−j +e∗t and the e∗t’s
are independeent random variables having identical distribution F .
P n
Step 3: Let T∗=T (X∗,X∗,...,X∗) be the sbame estimator as T based
n n 1 2 n n
onthepseudo-timeseriesX∗,X∗,...,X∗ andθ∗ theanalogueofθbassociated
1 2 n
withthebootstrapprocessX∗.TheAR-sievebootstrapapproximationof
n
L
is then given by ∗ = ∗(c (T∗ θ∗)).
Ln L n n −
In theabove (and in whatfollows), ∗,E∗,... will denote probability law,
L
expectation,etc.inthebootstrapworld(conditionalonthedataX ,...,X ).
1 n
ON THE VALIDITYOFTHE AUTOREGRESSIVESIEVEBOOTSTRAP 5
Note that the use of Yule–Walker estimators in Step 1 is essential and
guarantees—among other things—thatthe complex polynomial A (z)=1
p
p a (p)zj has no roots on or within the unit disc z C: z 1 , se−e
j=1 j { ∈ | |≤ }
the discussion before (2.22), that is, the bootstrap process X∗balways is
P
a statibonary and causal autoregressive process.
Thequestion considered in this paperis when can the bootstrap distribu-
tion ∗ correctly approximate the distribution of interest, and moreover
Ln Ln
what the AR-sieve bootstrap does if the latter is not the case. To this end,
let us first discuss a general autoregressive representation of stationary pro-
cesses.
Recall that by the well-known Wold representation, every purely non-
deterministic, stationary and zero-mean stochastic process X= X :t Z
t
{ ∈ }
can be expressed as
∞
(2.1) X = b u +u ,
t j t−j t
j=1
X
where ∞ b2 < and u =X (X ) is a zero mean, white noise
j=1 j ∞ t t −PMt−1 t
“innovation” process with finite variance 0<σ2 =E(u2)< ; recall that
P u t ∞
=span X :j s .
s j
M { ≤ }
Less known is that for all purely nondeterministic, stationary and zero-
mean time series unique autoregressive coefficients (a :k N) exist that
k
∈
only depend on the autocovariance function of the time series (X ), such
t
that for any n N,
∈
n
(2.2) (X )= a X +e , t Z,
PMt−1 t k t−k t,n ∈
k=1
X
where (e :t Z) is stationary and e span X :s t n 1 .
t,n t,n s
Underthea∈dditionalassumptionthat∈thecoe{fficient≤s (a−,k−N})areabso-
k
lute summable, that is, ∞ a < , one then obtains an a∈utoregressive,
k=1| k| ∞
Wold-type representation of the underlying process given by
P
∞
(2.3) X = a X +ε , t Z.
t k t−k t
∈
k=1
X
Here again (ε :t Z) denotes a white noise, that is, uncorrelated, process
t
with finite varian∈ce σ2=Eε2 which fulfills
ε t
∞
(2.4) σ2=γ (0) a γ (k),
ε X − k X
k=1
X
where γ () denotes the autocovariance function of X.
X
·
Under the absolute summability assumption on the autoregressive coeffi-
cients (a )—conditions for which will begiven in Lemma2.1 in the sequel—
k
we have that X (X )=X ∞ a X ; this implies that the
t −PMt−1 t t − k=1 k t−k
P
6 J.-P. KREISS,E. PAPARODITISANDD. N. POLITIS
white noise process (u ) appearing in (2.1) coincides with the white noise
t
process (ε ) in (2.3). Notice that this does not mean that if we have an
t
arbitrary one sided moving average representation of a time series (X ),
t
even with summable coefficients, that this moving average representation
is the Wold representation of the process; see Remark 2.1 for an example.
Furthermore, let f be the spectral density of X, that is,
X
f (λ)=(2π)−1 γ(h)exp iλh , λ [ π,π].
X
{− } ∈ −
h∈Z
X
Then, from (2.3) one immediately obtains that
∞ 2 σ2
(2.5) 1 a e−ikλ f (λ)= ε, λ [ π,π],
k X
(cid:12) − (cid:12) · 2π ∈ −
(cid:12) Xk=1 (cid:12)
(cid:12) (cid:12)
which implies(cid:12)that for strictly(cid:12)positive spectral densities f thepower series
X
(cid:12) (cid:12)
A(z):=1 ∞ a zk hasnozeroes with z =1.For moredetails of theau-
− k=1 k | |
toregressiveWoldrepresentation(2.3)seePourahmadi(2001),Lemma6.4(b),
P
(6.10) and (6.12). It is worth mentioning that in the historical evolution of
Wold decompositions the autoregressive variant preceded the moving aver-
age one.
Remark 2.1. If we consider a purely nondeterministic and stationary
time series possessing a standard one-sided moving average representation,
andifweadditionallyassumethatthespectraldensityisboundedawayfrom
zero and that the moving average coefficients b are absolutely summable,
j
then this would imply that the polynomial B(z)=1+ ∞ b zj has no ze-
j=1 j
roes with magnitude equal to one. There may of course exist zeroes within
P
the unit disk. But since the closed unit disk is compact and B(z) repre-
sents a holomorphic function there could exist only finitely many zeroes
with magnitude less than one. Following the technique described in Kreiss
and Neuhaus [(2006), Section 7.13] one may switch to another moving av-
erage model for which the polynomial has no zeroes within the unit disk.
This procedure definitely changes the white noise process; for example, if
the white noise process in the assumed moving average representation con-
sists of independent random variables, this desirable feature typically is lost
when switching to the moving average model with all zeroes within the unit
disk removed. In fact, only the property of uncorrelatedness is preserved.
The modified moving average process allows then for an autoregressive rep-
resentation of infinite order and this process, because of the uniqueness of
the autoregressive representation, coincides with the one in (2.3).
The following simple example, taken from Brockwell and Davis [(1991),
Example 3.5.2] illustrates these points. Based on i.i.d. random variables (e )
t
withmeanzeroandfiniteandnonvanishingvarianceσ2,constructthesimple
e
ON THE VALIDITYOFTHE AUTOREGRESSIVESIEVEBOOTSTRAP 7
MA(1)-process
(2.6) X =e 2e , t Z.
t t t−1
− ∈
This MA(1)-model is not invertible to an autoregressive process. However,
ageneralautoregressive representation as describedaboveexists.Inorderto
obtain this representation denote by L the usual lag-operator and consider
B(L):=1 2L as well as B(L):=1 0.5L. Of course
− −
B(L)
(2.7) eX =B(L) e .
t t
B(L)
e
Since B(e−iλ)2/B(e−iλ)2=4, we obtain that
e
| | | |
∞ j−1
B(L) 3 1
eε := e =e e .
t t t t−j
B(L) − 2 2
j=1(cid:18) (cid:19)
X
Again (εt) is a (uncorreleated) white noise process with variance σε2 =4σe2.
Moreover, we have
∞
(2.8) X =ε 0.5ε = 0.5jX +ε .
t t t−1 t−j t
− −
j=1
X
Obviously ε =X (X ) which means that (2.8) and not (2.6) is the
t t−PMt−1 t
Wold representation of the time series (X ). This also means that the mod-
t
ified moving average (or Wold) representation of the process (X ) possesses
t
only uncorrelated innovations (ε ) instead of independent innovations (e ).
t t
Buttherepresentation(2.8)withuncorrelatedinnovationshastheadvantage
that it indeed possesses an autoregressive representation of infinite order.
Of course, via the described modification, we do not change any property of
the process (X ). But, and this is essential, the modification leading to the
t
general AR( )-representation typically destroys a existing independence
∞
property of the white noise in a former moving average representation.
To elaborate, the problem of understanding the stochastic properties of
the innovation process in linear time series has been thoroughly investigated
in the literature. Breidt and Davis (1992) showed that time reversibility
of a linear process is equivalent to the fact that the i.i.d. innovations e are
t
Gaussian andusedthis resultto derive for aclass of linear processes unique-
ness of moving average representations with i.i.d. non-Gaussian innovations
and to discuss the stochastic properties of the innovation process appearing
in alternative moving average representations for the same process class.
Breidt, Davis and Dunsmuir (1995) used such results to initialize autore-
gressive processes in Monte Carlo generation of conditional sample paths
running autoregressive processes backward in time and Andrews, Davis and
8 J.-P. KREISS,E. PAPARODITISANDD. N. POLITIS
Breidt (2007) for estimation problems for all-pass time series models. Prop-
erties of the innovation process in non-Gaussian, noninvertible time series
have been also discussed in Lii and Rosenblatt (1982, 1996).
As we have seen the variances of e and ε do not coincide and the same
t t
is true for the fourth order cumulant E(e4)/σ4 3 which will be of some
t e −
importancelater.Usingthefactthatε isdefinedviaalineartransformation
t
on the i.i.d. sequence (e ) we obtain by straightforward computation
t
E(ε4) 2E(e4) 6
(2.9) t 3= 1 ,
(E(ε2))2 − 5 σ4 − 5
t e
which only equals E(e4)/σ4 3 in case the latter quantity is equal to 0, for
1 e −
example, when the e are normally distributed. The normally distributed
t
case always leads to the fact that uncorrelatedness and independence are
equivalent, thus implying that the white noise process in the general au-
toregressive representation always consists of independentand normally dis-
tributed randomvariables which leads for theautoregressive sieve bootstrap
in some cases to a considerable simplification as we will see later.
In order to get conditions which ensure the absolute summability of the
autoregressivecoefficients(a ,k N),onecangobacktoanimportantpaper
k
∈
by Baxter (1962). Informally speaking it is the smoothness of the spectral
density f which ensures summability of these coefficients. To be more pre-
X
cise, we have the following result.
Lemma 2.1. (i) If f is strictly positive and continuous and if
X
∞
hr γ (h) <
X
| | ∞
h=0
X
for some r 0, then
≥
∞
(2.10) hr a < .
h
| | ∞
h=0
X
(ii) If f is strictly positive and possesses k 2 derivatives, then
X
≥
∞
(2.11) hr γ (h) < r<k 1.
X
| | ∞ ∀ −
h=0
X
Proof. Cf. Baxter (1962), pages 140 and 142. (cid:3)
The uniquely determined autoregressive coefficients (a ) are closely re-
k
lated to the coefficients of an optimal (in the mean square sense) autore-
gressive fit of order p, or equivalently, to prediction coefficients based on the
ON THE VALIDITYOFTHE AUTOREGRESSIVESIEVEBOOTSTRAP 9
finite past. To be precise, denote the minimizers of
p 2
(2.12) E X c X
t r t−r
−
!
r=1
X
by a (p),...,a (p), which of course are solutions of the following Yule–
1 p
Walker linear equations:
γ (0) γ (p 1) c γ (1)
X X 1 X
(2.13) ... ·..·.· ...− ... = ... .
    
γ (p 1) γ (0) c γ (p)
X X p X
− ···
    
Recall from Brockwell and Davis [(1991), Proposition 5.1.1] that the co-
variance matrix Γ(p) on the left-hand side is for all p invertible provided
γ (0)>0 and γ (h) 0 as h .
X X
→ →∞
NowbyslightmodificationsofBaxter(1962),Theorem2.2[cf.alsoPourah-
madi (2001), Theorem 7.22], we obtain the following helpful result relating
the coefficients a (p) of the pth order autoregressive fit to the (a ) of the
k k
general autoregressive representation.
Lemma 2.2. Assume that f is strictly positive and continuous and
X
that ∞ (1+h)r γ (h) < for some r 0. Then there exists p N and
h=0 | X | ∞ ≥ o∈
C >0 (both depending on f only) such that for all p p ,
X o
P ≥
p ∞
(2.14) (1+k)r a (p) a C (1+k)r a
k k k
| − |≤ · | |
k=0 k=p+1
X X
as well as
∞
(2.15) (1+k)r a < .
k
| | ∞
k=1
X
This means that we typically can achieve a polynomial rate of convergence
of a (p) toward a .
k k
As already mentioned, γ (0)>0 and γ (h) 0 as h ensure non-
X X
→ →∞
singularity of all autocovariance matrices appearing in the left-hand side
of(2.13).Sincethesematricesarepositivesemidefinitethismeansthatunder
these conditions Γ(p) actually is positive definite. This in turn with Kreiss
and Neuhaus [(2006), Section 8.7] implies that the polynomial A (z) =
p
1 p a (p)zk has no zeroes in the closed unit disk. We can even prove
− k=1 k
a slightly stronger result.
P
Lemma 2.3. Assume that f is strictly positive and continuous, that
X
∞ γ (h) < and γ (0)>0. Then there exists δ>0 and p N such
h=0| X | ∞ X o∈
P
10 J.-P. KREISS,E. PAPARODITISANDD. N. POLITIS
that for all p p ,
o
≥
p
(2.16) inf 1 a (p)zk δ>0.
k
|z|≤1+1/p(cid:12) − (cid:12)≥
(cid:12) Xk=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The uniform convergence o(cid:12)f A (z) toward(cid:12)A(z) on the closed unit disk
p
immediately implies the following corollary to Lemma 2.3.
Corollary 2.1. Under the assumption of Lemma 2.3, we have
∞
(2.17) A(z)=1 a zj =0 z 1.
j
− 6 ∀| |≤
j=1
X
Lemma2.3andCorollary2.1nowenableustoinvertthepowerseriesA(z)
as well as the polynomial A (z). Let us denote
p
∞ −1 ∞
(2.18) 1 a zj =1+ α zj z 1
j j
− ∀| |≤
!
j=1 j=1
X X
and for all p large enough (because of Lemma 2.3)
p −1 ∞
1
(2.19) 1 a (p)zj =1+ α (p)zj z 1+ .
j j
− ∀| |≤ p
!
j=1 j=1
X X
From (2.19), one immediately obtains that
−j
1
(2.20) α (p) C 1+ j N.
j
| |≤ · p ∀ ∈
(cid:18) (cid:19)
A further auxiliary result contains the transfer of the approximation prop-
erty of a (p)for a totherespectivecoefficients α (p)and α of theinverted
j k k k
series. For such a result, we make use of a weighted version of Wiener’s
lemma; cf. Gro¨chenig (2007).
Lemma 2.4. Under the assumptions of Lemma 2.3 and additionally
∞ (1+h)r γ (h) < for some r 0 there exists a constant C >0 such
h=0 | X | ∞ ≥
that for all p large enough
P
∞ ∞
(2.21) (1+j)r α (p) α C (1+j)r a 0.
j j j p→∞
| − |≤ · | |→
j=1 j=p+1
X X
In a final step of this section, we now move on to estimators of the co-
efficients a (p). The easiest one might think of is to replace in (2.13) the
k