We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=Δ^(-d)u(t), where d є (-1/2,1/2) is the fractional integration parameter and u(t) is weakly dependent. The classical condition is existence of q>max(2,(d+1/2)-¹) moments of the innovation sequence. When d is close to -1/2 this moment condition is very strong. Our main result is to show that under some relatively weak conditions on u(t), the existence of q≥max(2,(d+1/2)-¹) is in fact necessary for the FCLT for fractionally integrated processes and that q>max(2,(d+1/2)-¹) moments are necessary and sufficient for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed, which is remarkable because it is the only FCLT where the moment condition has been weakened relative to the earlier condition. As a corollary to our main theorem we show that their moment condition is not sufficient.fractional integration; functional central limit theorem; long memory; moment condition; necessary condition

Morten Ørregaard Nielsen

Søren Johansen

English

Research Papers in Economics

Discussion Papers 
Department of Economics 
University of Copenhagen 
   
 
 
 
 
 
 
 
Øster Farimagsgade 5, Building 26, DK-1353 Copenhagen K., Denmark 
Tel.: +45 35 32 30 01 – Fax: +45 35 32 30 00 
http://www.econ.ku.dk 
 
 
ISSN: 1601-2461 (E) 
 
 
No. 10-29 
 
 
 
 
A Necessary Moment Condition for the Fractional  
Functional Central Limit Theorem 
 
 
 
Søren Johansen, Morten Ørregaard Nielsen 
 
 
 
 
 
 
 
 
 
 
  
  
 
  
 A necessary moment condition for the fractional
functional central limit theorem￿
Słren Johanseny
University of Copenhagen
and CREATES
Morten ￿rregaard Nielsenz
Queen￿ s University
and CREATES
October 21, 2010
Abstract
We discuss the moment condition for the fractional functional central limit theorem
(FCLT) for partial sums of xt = ￿￿dut, where d 2 (￿1=2;1=2) is the fractional inte-
gration parameter and ut is weakly dependent. The classical condition is existence of
q > max(2;(d+1=2)￿1) moments of the innovation sequence. When d is close to ￿1=2
this moment condition is very strong. Our main result is to show that under some rela-
tively weak conditions on ut, the existence of q ￿ max(2;(d+1=2)￿1) is in fact necessary
for the FCLT for fractionally integrated processes and that q > max(2;(d + 1=2)￿1)
moments are necessary for more general fractional processes. Davidson and de Jong
(2000) presented a fractional FCLT where only q > 2 ￿nite moments are assumed,
which is remarkable because it is the only FCLT where the moment condition has been
weakened relative to the earlier condition. As a corollary to our main theorem we show
that their moment condition is not su¢ cient.
Keywords: Fractional integration, functional central limit theorem, long memory,
moment condition, necessary condition.
JEL Classi￿cation: C22.
￿We are grateful to the Social Sciences and Humanities Research Council of Canada (SSHRC grant 410-
2009-0183) and the Center for Research in Econometric Analysis of Time Series (CREATES, funded by the
Danish National Research Foundation) for ￿nancial support.
yAddress: Department of Economics, University of Copenhagen, ￿ster Farimagsgade 5, DK-1353 Copen-
hagen K. Denmark, Email: Soren.Johansen@econ.ku.dk
zCorresponding author. Address: Department of Economics, Dunning Hall Room 307, Queen￿ s Univer-
sity, 94 University Avenue, Kingston, Ontario K7L 3N6, Canada. Email: mon@econ.queensu.ca
1A necessary moment condition for the fractional FCLT 2
1 Introduction
The fractional functional central limit theorem (FCLT) is given in Davydov (1970) for partial
sums of the fractionally integrated process ￿￿d"t, where "t is i.i.d. with mean zero under a
moment condition of the form Ej"tjq < 1 for q > q0 = max(2;(d + 1=2)￿1).
This result has been extended and generalized in numerous directions. For example, Mar-
inucci and Robinson (2000) replace "t by a class of linear processes, assuming the same mo-
ment condition. The latter authors proved FCLTs for so-called type II fractional processes,
whereas Davydov (1970) discussed a type I fractional process, but the distinction between
type I and type II processes is not relevant for our discussion of the moment condition.
Davidson and de Jong (2000, henceforth DDJ) state in their Theorem 3.1 that for some
near-epoch dependent (NED) processes with uniformly bounded q￿ th moment the fractional
FCLT holds, but with a much weaker moment condition than previous results, namely q > 2.
To the best of our knowledge, Theorem 3.1 of DDJ is the only fractional FCLT for which
the moment condition has been weakened relative to the earlier condition.
In the next section we give some de￿nitions and construct an i.i.d. sequence and a frac-
tional linear process which are central to our results. In Section 3 we present our main
results which state that if the fractional FCLT holds for any class of processes U(q) contain-
ing these processes, then it follows that q ￿ q0 if the fractional FCLT is based on fractional
integration coe¢ cients and q > q0 if the coe¢ cients are more general. The proofs of both
results are based on counter examples which are constructed in a similar way as a counter
example in Wu and Shao (2006, Remark 4.1). In Section 4 we discuss the results and give
two applications. In particular, it follows from our main result that if the FCLT holds for
NED processes with uniformly bounded q moments, then q ￿ q0. Hence DDJ￿ s Theorem 3.1
and all their subsequent results do not hold under the assumptions stated in their theorem.
Throughout, c denotes a generic ￿nite constant, which may take di⁄erent values in dif-
ferent places.
2 De￿nitions
De￿nition 1 We assume that ut is a zero mean covariance stationary stochastic process
which satis￿es the moment condition
sup
￿1<t<1
Ejutj
q < 1 for some q ￿ 2 (1)
and has long-run variance
￿
2
u = lim
T!1
T
￿1E(
T X
t=1
ut)
2; 0 < ￿
2
u < 1: (2)
For such processes we de￿ne the two classes:
￿ Ulin(q) is the class of linear processes ut =
P1
n=0 ￿n"t￿n, where
P1
n=0
P1
j=n ￿2
j < 1
and "t is i.i.d. with mean zero and variance ￿2
" > 0.
￿ UNED(q) is the class of processes ut which are L2-NED of size ￿1
2 on vt with dt = 1,
where vt is either an ￿￿mixing sequence of size ￿q=(1 ￿ q) or a ￿￿mixing sequence
of size ￿q=(2(1 ￿ q)); see Assumption 1 of DDJ.A necessary moment condition for the fractional FCLT 3
We base our results on the construction of the following two speci￿c processes.
De￿nition 2 Let "t be i.i.d. with mean zero, variance ￿2
" > 0, and ￿nite q￿ th moment for
some q ￿ 2 to be chosen later. For such "t we de￿ne the two processes:
￿ u1t = "t.
￿ u2t = "t + ￿1+d"t.
For these two processes we note the following connection with the classes Ulin(q) and
UNED(q). Because q ￿ q0 is only stronger than q ￿ 2 when d < 0 we consider only this case.
Lemma 1 For d 2 (￿1=2;0) and for i = 1;2, uit 2 Ulin(q) \ UNED(q) and the long-run
variance of uit is ￿2
".
Proof. Clearly, u1t = "t is contained in both Ulin(q) and UNED(q) and has long-run variance
￿2
".
Let bj(d) = (￿1)j￿￿d
j
￿
denote the coe¢ cients in the binomial expansion of (1 ￿ z)￿d,
which satisfy jbj(d)j ￿ cjd￿1. The process u2t = "t + ￿1+d"t = "t +
P1
j=0 bj(￿d ￿ 1)"t￿j is a
linear process and
1 X
n=0
1 X
j=n
bj(￿d ￿ 1)
2 ￿ c
1 X
n=0
1 X
j=n
j
￿2d￿4 ￿ c
1 X
n=0
n
￿2d￿3 ￿ c for d 2 (￿1=2;0);
so that u2t is in Ulin(q). To see that u2t is in UNED(q), we calculate
jju2t￿E(u2tj"t￿m;:::;"t+m)jj2 = jj
1 X
n=m+1
bn(￿d￿1)"t￿njj2 ￿ c(
1 X
n=m+1
n
￿2d￿4)
1=2 ￿ cm
￿d￿3=2:
Because 3=2 + d > 1=2 for d 2 (￿1=2;0); this shows that u2t is L2-NED of size ￿1=2 on "t,
and hence u2t is also in UNED(q): The generating function for u2t is f(z) = 1 + (1 ￿ z)1+d
and for z = 1 we ￿nd because 1 + d > 0 that f(1) = 1. Therefore the long-run variance of
u2t is limT!1 T ￿1E(
PT
t=1("t + ￿1+d"t))2 = f(1)2V ar("t) = ￿2
".
We next give a general formulation of the FCLT for fractional processes. For any process
ut which satis￿es (1) and (2), we construct a fractional process by de￿ning
xt = ￿
￿dut =
1 X
j=0
bj(d)ut￿j for ￿ 1=2 < d < 0: (3)
This process is well de￿ned because, with jjxjj2 denoting the L2-norm, we have from (1) that
jjxtjj2 ￿ c
1 X
j=0
j
d￿1jjut￿jjj2 ￿ c for d < 0:
We also de￿ne the scaled partial sum process
XT(￿) = ￿
￿1
T
[T￿] X
t=1
xt; 0 ￿ ￿ ￿ 1; (4)
where ￿2
T = E(
PT
t=1 xt)2 and [z] is the integer part of the real number z.A necessary moment condition for the fractional FCLT 4
Fractional FCLT for U(q): We say that the functional central limit theorem (FCLT) for
fractional processes holds for a set U(q) of processes if, for ut 2 U(q), it holds that
XT(￿)
D ! X(￿) in D[0;1]; (5)
where X(￿) is fractional Brownian motion.
Here,
D ! denotes convergence in distribution (weak convergence) in D[0;1] endowed with
a suitable metric, see Billingsley (1968).
3 The necessity result
Our main result is the following theorem.
Theorem 1 Let XT(￿) be de￿ned by (3) and (4) with ￿1=2 < d < 0 and let U(q) be any
class containing u1t and u2t. If the fractional FCLT holds for U(q) for some q ￿ 2, then
q ￿ q0.
Proof. We prove the theorem by assuming that there is a q1 2 [2;q0) for which the FCLT
holds for U(q1), and show that this leads to a contradiction by a careful construction of "t
and therefore u1t and u2t.
For uit;i = 1;2; we de￿ne xit and XiT by (3) and (4), and because uit is in U(q1) the
fractional FCLT holds by the maintained assumption for uit and hence XiT(￿) converges in
distribution to fractional Brownian motion.
(i) The normalizing variance for X1T. The variance of
PT
t=1 x1t =
PT
t=1 ￿￿du1t = PT
t=1 ￿￿d"t can be found in Davydov (1970), see also Lemma 3.2 of DDJ,
￿
2
1T = E(
T X
t=1
x1t)
2 ￿ ￿
2
"VdT
2d+1; (6)
where Vd = 1
￿(d+1)2( 1
2d+1 +
R 1
0 ((1+￿)d ￿￿d)2d￿) is a constant and ￿￿￿means that the ratio
of the left- and right-hand sides converges to one.
(ii) The normalizing variance for X2T: We write x2t and X2T in terms of x1t and X1T;
using (3) and (4),
x2t = ￿
￿du2t = x1t + "t ￿ "t￿1 (7)
X2T(￿) = ￿
￿1
2T
[T￿] X
t=1
x2t = ￿1T￿
￿1
2TX1T(￿) + ￿
￿1
2T("[T￿] ￿ "0); (8)
We next ￿nd that the variance of
PT
t=1 x2t =
PT
t=1 ￿￿du2t =
PT
t=1(x1t + "t ￿ "t￿1) is
￿
2
2T = E(
T X
t=1
x2t)
2 = E(
T X
t=1
(x1t + "t ￿ "t￿1))
2 = E("T ￿ "0 +
T X
t=1
x1t)
2
= E("T ￿ "0)
2 + E(
T X
t=1
x1t)
2 + 2E(
T X
t=1
￿
￿d"t("T ￿ "0)):A necessary moment condition for the fractional FCLT 5
The ￿rst term is constant, the next is ￿2
1T, and letting 1fAg denote the indicator function of
the event A, the last term consists of
E(
T X
t=1
￿
￿d"t"T) =
T X
t=1
1 X
k=0
bk(d)E("t￿k"T) = ￿
2
"
T X
t=1
1 X
k=0
bk(d)1fk=t￿Tg = ￿
2
"b0(d) = ￿
2
"
and
E(
T X
t=1
￿
￿du1t"0) =
T X
t=1
1 X
k=0
bk(d)E("t￿k"0) = ￿
2
"
T X
t=1
1 X
k=0
bk(d)1fk=tg ￿ c
T X
t=1
t
d￿1 ￿ c for d < 0:
Therefore,
￿
2
2T ￿ ￿
2
1T + c: (9)
(iii) The contradiction. We now construct the i.i.d. process "t so that it has no moment
higher than q1, that is Ej"tjq = 1 for q > q1, by choosing the tail to satisfy
P(j"tj
q1 ￿ c) ￿
1
c(logc)2 as c ! 1: (10)
In this case we still have Ej"tjq1 < 1. We then ￿nd
P(￿
￿1
1T max
1￿t￿T
j"tj < c) = P(￿
￿1
1Tj"1j < c)
T = P(j"1j
q1 < c
q1￿
q1
1T)
T
= (1 ￿ P(j"1j
q1 ￿ c
q1T
q1=q0))
T
￿
￿
1 ￿
1
cq1T q1=q0(q1(logc + q
￿1
0 logT))2
￿T
￿ exp(￿
T 1￿q1=q0
cq1(q1(logc + q
￿1
0 logT))2) ! 0
as T ! 1 because q1 < q0. Thus, ￿
￿1
1T max1￿t￿T j"tj
P ! 1 because the normalizing constant
￿1T = ￿"V
1=2
d T 1=q0 = ￿"V
1=2
d T 1=2+d < ￿"V
1=2
d T 1=q1 is too small to normalize max1￿t￿T j"tj
correctly.
The de￿nition (8) implies the evaluation
max
0￿￿￿1
j"[T￿]j ￿ max
0￿￿￿1
j"[T￿] ￿ "0j + j"0j ￿ max
0￿￿￿1
j￿2TX2T(￿)j + max
0￿￿￿1
j￿1TX1T(￿)j + j"0j
such that
￿
￿1
1T max
0￿￿￿1
j"[T￿]j ￿ max
0￿￿￿1
j￿
￿1
1T￿2TX2T(￿)j + max
0￿￿￿1
jX1T(￿)j + ￿
￿1
1Tj"0j: (11)
We have seen in (6) and (9) that ￿2
2T ￿ ￿2
1T +c and ￿2
1T ￿ ￿2
"VdT 1+2d ! 1 for d > ￿1=2, so
that ￿1T￿
￿1
2T ! 1. Therefore, both ￿
￿1
1T￿2TX2T(￿) and X1T(￿) converge in distribution by the
previous results and it follows from (11) that ￿
￿1
1T max0￿￿￿1 j"[T￿]j is OP(1). This contradicts
that ￿
￿1
1T max1￿t￿T j"tj
P ! 1, and hence completes the proof of Theorem 1.
The proof of Theorem 1 implies that the issue is that the rate of convergence, T ￿(d+1=2),
of
P[T￿]
t=1 ￿￿du1t can be very slow for d close to ￿1=2. Thus, more control on the tail-behaviorA necessary moment condition for the fractional FCLT 6
of the ut sequence is needed when d 2 (￿1=2;0), and this is achieved through the moment
condition (1).
We end this section by giving a complementary result that shows when the moment
condition q > q0 is necessary instead of q ￿ q0. The former is the moment condition applied
by Davydov (1970) and Marinucci and Robinson (2000), and indeed all other fractional
FCLT results of which we are aware (with the exception of DDJ).
De￿ne coe¢ cients aj(d) which satisfy aj(d) ￿ c‘(j)jd￿1, where ‘(j) is a (normalized)
slowly varying function, see Bingham, Goldie, and Teugels (1989, p. 15). Note that the
bj(d) coe¢ cients from the fractional di⁄erence ￿lter are a special case of aj(d). We now
de￿ne the general fractional process,
xt =
1 X
j=0
aj(d)ut￿j for ￿ 1=2 < d < 0; (12)
and let the partial sum process XT(￿) be de￿ned in (4) as before. We then obtain the
following result.
Theorem 2 Let XT(￿) be de￿ned by (12) and (4) with ￿1=2 < d < 0 and let U(q) be any
class containing u1t and u2t. If the fractional FCLT holds for U(q) for some q ￿ 2, then
q > q0.
Proof. We assume that there is a q1 2 [2;q0] for which the FCLT holds for U(q1) and show
that this leads to a contradiction. For uit;i = 1;2; we de￿ne xit and XiT by (12) and (4)
and use the proof of Theorem 1 with the following modi￿cations.
(i) From Karamata￿ s Theorem, see Bingham, Goldie, and Teugels (1989, p. 26), we ￿nd
that the normalizing variance is ￿2
1T ￿ c‘(T)2T 2d+1 = c‘(T)2T 1=q0.
(iii) We choose the tail of "t as in (10) in the proof of Theorem 1 and take ‘(T) = (logT)￿1
and ￿nd
P(￿
￿1
1T max
1￿t￿T
j"tj < c) = P(￿
￿1
1Tj"1j < c)
T = P(j"1j
q1 < c
q1￿
q1
1T)
T
= (1 ￿ P(j"1j
q1 ￿ c
q1T
q1=q0‘(T)
q1))
T
￿
￿
1 ￿
1
cq1T q1=q0‘(T)q1(q1(logc + q
￿1
0 logT + log‘(T)))2
￿T
￿ exp(￿
T 1￿q1=q0‘(T)￿q1
cq1(q1(logc + q
￿1
0 logT + log‘(T)))2) ! 0
as T ! 1 because q1 ￿ q0. Note that even with q1 = q0 (and q0 > 2 because d < 0) we have
the factor exp(￿c(logT)q1￿2) ! 0 which ensures the convergence to zero. The contradiction
follows exactly as in the proof of Theorem 1.
4 Discussion
In this section we present two corollaries which demonstrate how our results apply to the
processes in Marinucci and Robinson (2000) and to those in DDJ, respectively, and we discuss
some implications for the results of DDJ.A necessary moment condition for the fractional FCLT 7
Corollary 1 Let XT(￿) be de￿ned by (12) and (4) with ￿1=2 < d < 0. If the fractional
FCLT holds for Ulin(q) then q > q0. Thus, the moment condtion (1) with q > q0 is necessary
for Theorem 1 of Marinucci and Robinson (2000).
Proof. The ￿rst statement follows from Theorem 2 because u1t and u2t are in Ulin(q) by
Lemma 1. The second statement follows because the univariate version of Assumption A of
Marinucci and Robinson (2000) (translated to type I processes) was in fact used to de￿ne
the class Ulin(q).
It follows from Corollary 1 that the moment condition applied by Marinucci and Robinson
(2000) is in fact necessary for their results. That is, using the coe¢ cients aj(d) to de￿ne
a general fractional process where ut is a linear process, our results show that q > q0 is
necessary for the fractional FCLT. However, it does not follow from our results that q ￿ q0 is
necessary for the FCLT when ut is an i.i.d. or ARMA process because the process u2t needed
in the construction is neither i.i.d. nor ARMA.
We next discuss the implications of Theorem 1 for the results of DDJ who state in their
Theorem 3.1 that the fractional FCLT (5) holds for UNED(q) if q > 2. It is noteworthy that
UNED(q) allows ut to have a very general dependence structure through the NED assumption,
but in particular that DDJ assume only that supt Ejutjq < 1 for q > 2, which is much weaker
than (1) if d < 0.
The following corollary to Theorem 1 shows how our result applies to DDJ.
Corollary 2 Let XT(￿) be de￿ned by (3) and (4) with ￿1=2 < d < 0. If the fractional FCLT
holds for UNED(q) then q ￿ q0. Thus, the moment condtion (1) with q ￿ q0 is necessary for
Theorem 3.1 of DDJ.
Proof. >From Lemma 1 we know that u1t and u2t are in UNED(q) which by Theorem 1
proves the ￿rst statement. The last statement follows because Assumption 1 of DDJ was
used to de￿ne UNED(q).
It follows from Corollary 2 that Theorem 3.1 of DDJ (and their subsequent results relying
on Theorem 3.1) does not hold under their Assumption 1. It is well known, e.g. Billingsley
(1968, chp. 15), that the fractional FCLT holds upon proving convergence of the ￿nite-
dimensional distributions and tightness (stochastic equicontinuity). Since ￿nite-dimensional
convergence holds from standard central limit theorems for UNED(q), and in particular holds
with only q > 2 moments, it is the proof of tightness that fails in DDJ and requires the
stronger moment condition.
References
1. Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
2. Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1989). Regular Variation, Cam-
bridge University Press, Cambridge.
3. Davidson, J. and de Jong, R. M. (2000). The functional central limit theorem and weak
convergence to stochastic integrals II: fractionally integrated processes. Econometric
Theory 16, pp. 643￿ 666.
4. Davydov, Yu. A. (1970). The invariance principle for stationary processes. Theory of
Probability and Its Applications 15, pp. 487￿ 498.A necessary moment condition for the fractional FCLT 8
5. Marinucci, D. and Robinson, P. M. (2000). Weak convergence of multivariate fractional
processes. Stochastic Processes and their Applications 86, pp. 103￿ 120.
6. Wu, W. B. and Shao, X. (2006). Invariance principles for fractionally integrated non-
linear processes. IMS Lecture Notes￿ Monograph Series: Recent Developments in Non-
parametric Inference and Probability 50, pp. 20￿ 30.

A Necessary Moment Condition for the Fractional Functional Central Limit Theorem

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A Necessary Moment Condition for the Fractional Functional Central Limit Theorem

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