We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x_{t} = Delta^{-d} u_{t}, where d in (-1/2,1/2) is the fractional integration parameter and u_{t} is weakly dependent. The classical condition is existence of q≥2 and q>1/(d+1/2) 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 when d in (-1/2,0) and under some relatively weak conditions on u_{t}, the existence of q≥1/(d+1/2) moments is in fact necessary for the FCLT for fractionally integrated processes, and that q>1/(d+1/2) moments are necessary for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed. As a corollary to our main theorem we show that their moment condition is not sufficient, and hence that their result is incorrect.Fractional integration, functional central limit theorem, long memory, moment condition, necessary condition

Morten Ørregaard Nielsen

Søren Johansen

English

Research Papers in Economics

QED
Queen’s Economics Department Working Paper No. 1244
A necessary moment condition for the fractional functional
central limit theorem
Søren Johansen
University of Copenhagen and CREATES
Morten Ørregaard Nielsen
Queen‘s University and CREATES
Department of Economics
Queen’s University
94 University Avenue
Kingston, Ontario, Canada
K7L 3N6
10-2010A necessary moment condition for the fractional
functional central limit theorem￿
Słren Johansen
University of Copenhagen and CREATES
Email: Soren.Johansen@econ.ku.dk
Morten ￿rregaard Nielsen
Queen￿ s University and CREATES
Email: mon@econ.queensu.ca
April 22, 2011
Abstract
We discuss the moment condition for the fractional functional central limit theo-
rem (FCLT) for partial sums of xt = ￿￿dut, where d 2 (￿1=2;1=2) is the fractional
integration parameter and ut is weakly dependent. The classical condition is exis-
tence of q ￿ 2 and q > (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
when d 2 (￿1=2;0) and under some relatively weak conditions on ut, the existence of
q ￿ (d + 1=2)￿1 moments is in fact necessary for the FCLT for fractionally integrated
processes, and that q > (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. As a corollary to our main theorem we show that their
moment condition is not su¢ cient, and hence that their result is incorrect.
Keywords: Fractional integration, functional central limit theorem, long memory,
moment condition, necessary condition.
JEL Classi￿cation: C22.
Proposed running head: Necessary moment condition for fractional FCLT
Corresponding author: Morten ￿rregaard Nielsen
Address: Department of Economics, Dunning Hall Room 307, Queen￿ s University, 94
University Avenue, Kingston, Ontario K7L 3N6, Canada.
Email: mon@econ.queensu.ca
￿We are grateful to Benedikt P￿tscher, three anonymous referees, and James Davidson for comments,
and 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.
1Necessary moment condition for 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 ￿￿d = (1 ￿ L)￿d is the fractional
di⁄erence operator de￿ned in (3) below and "t is i.i.d. with mean zero. Davydov (1970)
proved the fractional FCLT under a moment condition of the form Ej"tjq < 1 for q ￿ 4
and q > ￿4d=(d + 1=2), which was subsequently improved by Taqqu (1975) to q ￿ 2 and
q > q0 = (d+1=2)￿1. The standard moment condition from Donsker￿ s Theorem is q ￿ 2, see
Billingsley (1968, chapter 2, section 10), and because the condition q ￿ q0 is only stronger
than q ￿ 2 when d < 0 we consider only d 2 (￿1=2;0).
The fractional FCLT has been extended and generalized in numerous directions. For ex-
ample, Marinucci and Robinson (2000) replace "t by a class of linear processes, assuming the
moment condition q > q0. The latter authors proved FCLTs for so-called type II fractional
processes, whereas Davydov (1970) and Taqqu (1975) discussed type I fractional processes,
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) claim in their Theorem 3.1 that for some
near-epoch dependent (NED) processes with uniformly bounded q￿ th moment the fractional
FCLT holds, but (incorrectly, as we shall see below) assume 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 which claims a moment condition that is weaker than 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) with uniformly
bounded q￿ th moment containing these two processes, then it follows that q ￿ q0 if the frac-
tional FCLT is based on fractional 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 results 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 de￿ne the class U0(q) as the class of processes ut which satisfy the moment
condition
sup
￿1<t<1
Ejutj
q < 1 for some q ￿ 2 (1)
and have 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 two subclasses of U0(q):Necessary moment condition for fractional FCLT 3
￿ ULIN(q) ￿ U0(q) is the class of linear processes ut 2 U0(q) satisfying 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) ￿ U0(q) is the class of zero mean covariance stationary processes ut 2 U0(q)
which are L2-NED of size ￿1
2 on vt with dt = 1, where vt is either an ￿￿mixing se-
quence of size ￿q=(1￿q) or a ￿￿mixing sequence of size ￿q=(2(1￿q)); see Assumption
1 of DDJ.
Note that the condition
P1
n=0
P1
j=n ￿2
j < 1 in the de￿nition of the class ULIN(q) neither
implies nor is implied by (2). Also note that if ut 2 ULIN(q) then ut is zero mean and
covariance stationary.
Next we de￿ne the fractional and general fractional processes.
De￿nition 2 For any ut 2 U0(q) we we de￿ne the fractional process
xt = ￿
￿dut =
1 X
j=0
bj(d)ut￿j for ￿ 1=2 < d < 0; (3)
where bj(d) = (￿1)j￿￿d
j
￿
= d(d + 1):::(d + j ￿ 1)=j! ￿ cjd￿1 are the fractional coe¢ cients,
i.e., the coe¢ cients in the binomial expansion of (1￿z)￿d, and ￿￿￿means that the ratio of
the left- and right-hand sides converges to one. The general fractional process is de￿ned by
xt =
1 X
j=0
aj(d)ut￿j for ￿ 1=2 < d < 0; (4)
where aj(d) ￿ c‘(j)jd￿1 and ‘(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). The processes (3) and (4) are well de￿ned because, with jjxjj2 denoting the L2-norm,
we have from (1) that
jjxtjj2 ￿ jjutjj2 + c
1 X
j=1
‘(j)j
d￿1jjut￿jjj2 ￿ c for d < 0
using Karamata￿ s Theorem, see Bingham, Goldie, and Teugels (1989, p. 26).
We base our results on the construction of the following two speci￿c processes.
De￿nition 3 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).Necessary moment condition for fractional FCLT 4
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
".
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=1
1 X
j=n
j
￿2d￿4 ￿ c
1 X
n=1
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 this purpose
we de￿ne the scaled partial sum process
XT(￿) = ￿
￿1
T
[T￿] X
t=1
xt; 0 ￿ ￿ ￿ 1; (5)
where ￿2
T = E(
PT
t=1 xt)2 and [z] is the integer part of the real number z.
Fractional FCLT for U(q): Let XT(￿) be given by (5) and suppose xt is linear in ut. We
say that the functional central limit theorem (FCLT) for fractional Brownian motion
holds for a set U(q) ￿ U0(q) of processes if, for all ut 2 U(q), it holds that
XT(￿)
D ! X(￿) in D[0;1]; (6)
where X(￿) is fractional Brownian motion.
Here,
D ! denotes convergence in distribution (weak convergence) in D[0;1] endowed with
the metric d0 in Billingsley (1968, chapter 3, section 14), which induces the Skorohod topol-
ogy and under which D[0;1] is complete.
3 The necessity results
Our ￿rst main result is the following theorem.
Theorem 1 Let XT(￿) be de￿ned by (5) for xt given by the fractional coe¢ cients bj(d) in
(3) with ￿1=2 < d < 0, and let U(q) ￿ U0(q) be such that uit 2 U(q) for i = 1;2. If the
fractional FCLT holds for all ut in U(q) for some q ￿ 2, then q ￿ q0.Necessary moment condition for fractional FCLT 5
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 (5), 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; (7)
where Vd = 1
￿(d+1)2( 1
2d+1 +
R 1
0 ((1 + ￿)d ￿ ￿d)2d￿).
(ii) The normalizing variance for X2T: We write x2t and X2T in terms of x1t and X1T;
using (3) and (5),
x2t = ￿
￿du2t = x1t + "t ￿ "t￿1; (8)
X2T(￿) = ￿
￿1
2T
[T￿] X
t=1
x2t = ￿1T￿
￿1
2TX1T(￿) + ￿
￿1
2T("[T￿] ￿ "0): (9)
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)):
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: (10)
(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: (11)Necessary moment condition for fractional FCLT 6
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 (9) 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: (12)
We have seen in (7) and (10) that ￿2
2T ￿ ￿2
1T+c and ￿2
1T ￿ ￿2
"VdT 1+2d ! 1 for d 2 (￿1=2;0),
so that ￿1T￿
￿1
2T ! 1. Therefore, both ￿
￿1
1T￿2TX2T(￿) and X1T(￿) converge in distribution
by the previous results and it follows from (12) 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-behavior
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 result that shows when the moment condition q > q0 is
necessary instead of q ￿ q0. The former is the moment condition applied by, e.g., Taqqu
(1975) and Marinucci and Robinson (2000).
Theorem 2 Let XT(￿) be de￿ned by (5) for xt given by the general fractional coe¢ cients
in (4) with ￿1=2 < d < 0, and let U(q) ￿ U0(q) be such that uit 2 U(q) for i = 1;2. If the
fractional FCLT holds for all slowly varying functions ‘(￿) and all ut in 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 (4) and (5) 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.Necessary moment condition for fractional FCLT 7
(iii) We choose the tail of "t as in (11) 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.
We ￿rst discuss the implications of Theorem 1 for the results of DDJ who state (in our
notation) the following claim (given as Theorem 3.1 in DDJ):
DDJ claim: If XT(￿) is de￿ned by (3) and (5) where jdj < 1=2, and ut 2 UNED(q) for
q > 2, then XT(￿)
D ! X(￿) in D[0;1] where X(￿) is fractional Brownian motion.
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 weaker than q ￿ q0 if d < 0. The following corollary to Theorem 1 shows
how our result disproves Theorem 3.1 in DDJ.
Corollary 1 Let XT(￿) be de￿ned by (3) and (5) with ￿1=2 < d < 0. If the fractional
FCLT holds for all ut in UNED(q) then q ￿ q0.
Proof. From Lemma 1 we know that u1t and u2t are in UNED(q) which by Theorem 1 proves
the corollary.
It follows from Corollary 1 that Theorem 3.1 of DDJ (and their subsequent results relying
on Theorem 3.1) does not hold under their Assumption 1. We ￿nish with an application of
our results to the processes in Marinucci and Robinson (2000).
Corollary 2 Let XT(￿) be de￿ned by (4) and (5) with ￿1=2 < d < 0. If the fractional
FCLT holds for all slowly varying functions ‘(￿) and all ut in ULIN(q) then q > q0. Thus,
the moment condition (1) with q > q0 is both necessary and su¢ cient 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 ofNecessary moment condition for fractional FCLT 8
Marinucci and Robinson (2000) (translated to type I processes) was in fact used to de￿ne
the class ULIN(q) and the coe¢ cients aj(d).
It follows from Corollary 2 that q > q0 is both necessary and su¢ cient for the fractional
FCLT when using the coe¢ cients aj(d) to de￿ne a general fractional process and ut is a
linear process of the type in ULIN(q). 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.
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parametric Inference and Probability 50, pp. 20￿ 30.

A necessary moment condition for the fractional functional central limit theorem

http://qed.econ.queensu.ca/working_papers/papers/qed_wp_1244.pdf

A necessary moment condition for the fractional functional central limit theorem

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